%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % % % This is a plain TeX file. % % % %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \footline={\ifnum\pageno=1{\hfill}\else{\hss\tenrm\folio\hss}\fi} \input mssymb.tex \magnification=\magstep1 \baselineskip=13pt \mathsurround=2pt \tolerance=20000 \def\wphi{{\widetilde\phi}} \def\wK{{\widetilde K}} \def\wk{{\widetilde k}} \def\hydr{{\rm hydr}} \def\Coul{{\rm Coul}} \def\atom{{\rm atom}} \def\cS{{\cal S}} \def\ve{{\varepsilon}} \def\cE{{\cal E}} \def\TF{{\rm TF}} \def\HF{{\rm HF}} \def\C0i{{C_0^\infty}} \def\Kb{{K_\beta}} \def\Ha{{H_\alpha}} \def\oEpm{{\overline E^\pm}} \def\Epm{{E^\pm}} \def\oE{{\overline E}} \def\oP{{\overline P}} \def\oH{{\overline H}} \def\ochi{{\overline\chi}} \def\otheta{{\overline\theta}} \def\dd{\delta\delta} \def\BC{{\Bbb C}} \def\BR{{\Bbb R}} \def\BZ{{\Bbb Z}} \def\hf{{\hat f}} \def\hg{{\hat g}} \def\hx{{\hat x}} \def\vp{{\varphi}} \def\ind{{\rm ind}} \def\infspec{{\rm inf\, spec}\,} \def\const{{\rm const}} \def\supp{{\rm supp}\, } \def\tr{{\rm tr}\,} \def\Ran{{\rm Ran}\, } \def\Prf{\noindent {\bf Proof.} \ \ } \def\sq1-e{\sqrt {1-\varepsilon^2}} \def\xax{{|x_a|\over |x|}} \def\xbx{{|x^b|\over |x|}} \def\xbta{{|x_b|\over t^\alpha}} \def\xta{{|x|\over t^\alpha}} \def\At{{|A|\over t}} \font\bigbf=cmbx10 scaled\magstep2 \font\eightrm=cmr8 at 8pt \def\stackbel #1#2{\mathrel{\mathop{#2}\limits_{#1}}} \def\ii{{\mathop{\int\int}\limits}} \def\squish{{\scriptstyle \partial}\!e} \def\frac#1#2{{#1\over #2}} \def\Ran{{\rm Ran} \, } \def\const{{\rm const}\, } \def\det{{\rm det}} \def\sgn{{\rm sgn}} \def\sign{{\rm sign}} \centerline{\bf Asymptotics of the Ground State} \centerline{\bf Energies of Large Coulomb Systems\footnote*{\rm Partially supported by NSERC under Grant NA7901.}} \vglue .5in \baselineskip=12pt \centerline {V. Ja. Ivrii} \centerline {Centre de Math\'ematiques} \centerline {Ecole Polytechnique} \centerline {F-91128 Palaiseau} \centerline {France} \bigskip \bigskip \centerline {I.M. Sigal\footnote\dag{I.W. Killam Research Fellow}} \centerline {Department of Mathematics} \centerline {University of Toronto} \centerline {Toronto, Ontario M5S 1A1} \centerline {Canada} \vglue .75in \baselineskip=20pt \centerline{\bf Abstract} \bigskip \centerline {We prove the Scott conjecture for molecules.} \vskip 1in \settabs 4\columns \+&&{\it Submitted to}\cr \+&&{\it Annals of Mathematics}\cr \vfill\eject \vglue .5in \centerline{\bf TABLE OF CONTENTS} \vglue .25in \midinsert\narrower \item{1.\ \ } Introduction and Result \dotfill 3 \item{2.\ \ } Mean Field Theory\dotfill 10 \item{3.\ \ } TF Gas and Weyl Asymptotic\dotfill 26 \item{4.\ \ } Energy Bounds\dotfill 31 \item{5.\ \ } Approximate Evolution \dotfill 41 \item{6.\ \ } Estimates of the Evolution \dotfill 51 \item{7.\ \ } Estimates of the Local Traces\dotfill 54 \item{8.\ \ } Multiscale Analysis \dotfill 61 \item{9.\ \ } Decoupling of Singularities \dotfill 77 \item{10.\ } Coulomb Problem\dotfill 85 \item{11.\ } Estimates of Schwartz Kernels \dotfill 87 \item{\ \ \ \ \ } Appendix\dotfill 97 \item{\ \ \ \ \ } Supplement\dotfill 103 \item{\ \ \ \ \ } References\dotfill 108 \endinsert \vfill\eject \baselineskip=20pt \beginsection 1. Introduction and Result Consider a molecule consisting of $N$ electrons, each of the charge $-1$ and $M$ nuclei of charges $Z_1,\ldots, Z_M$. The position and spin space of a single electron are $\BR^3$ and $\BZ_2$, respectively. We assume that the system is neutral, i.e. $$ N=\sum\limits_{i=1}^M Z_i\ , $$ and that the nuclei are infinitely heavy and located at positions $R_1,\ldots,R_M$. The Schr\"odinger operator of such a system in appropriate units is $$ H(Z,R)\ = \ \sum_{i=1}^N \big( - {1\over2}\Delta_i - V_Z (x_i,R)\big) + \sum_{i0$ independent of $Z$ and let the mutual distances between $R_j$ (which are allowed to depend on $Z$) be bounded from below by $|Z|^{-{5\over9}+\ve}$ with $\ve>0$. Then for any $\delta>0$ $$ E(Z,R)\ = \ E^{\rm TF} (Z,R) + {1\over4}\sum Z_j^2 + O\big( a^{-{1\over3}} |Z|^{{16\over9}+\delta}\big) \ , \eqno (1.8) $$ where $a=\min(|Z|^{{1\over3}} |R_i-R_j|,\, i\ne j ;\, 1)$ and the estimate is uniform in the $Z_j |Z|^{-1}$ and in the $|Z|^{1\over3} R_j$. \par It will follow from the analysis below that the leading term on the r.h.s. represents the quasiclassical energy of the bulk of electrons and the second term, the quantum spectrum of Coulomb singularities. The leading term in (1.6) was obtained in [LiebSim 1977] (see also Lieb 1981 and Thirr 1981). The second term of asymptotics was conjectured by J.M.C. Scott in 1952 as a contribution of those electrons which move very close to the nuclei (see [LiebSim 1977 and Lieb 1981] for a discussion). For atoms the Scott conjecture was proven in [Hughes 1990, SiedWeik 1987,1989] (see also [SiedWeik 1990]). Thus the new result of this paper is a proof of Scott conjecture for molecules. A proof of the next, $Z^{5\over3}$ term for atoms is announced in [FeffSeco 1990]. The approach in [Hughes 1990, SiedWeik 1987,1989,1990, FeffSeco 1990] is based on an expansion in angular momentum channels. This is possible since the electron interaction with the nucleus $V(x,R)$, is spherically symmetric in atoms. The problem is then reduced to a one-dimensional one which is treated by the standard WKB method. The proof in this paper is rather general and is discussed below. The philosophy of our approach is the same as that of Hughes-Siedentop-Weikard and can be discerned from [Lieb 1981]. Namely, on the first step the ground state energy of fully interacting electrons is approximated up to the order $O(|Z|^{5\over3})$ by the ground state energy of independent Fermions moving in an effective exterior potential, $-\phi (x)$. The latter is composed of the original attractive potential between a given electron and the nuclei and an electro-static potential produced by an averaged out electron charge density. This mean electron charge density must be found in a self-consistent way and it turns out to be the density which minimizes the Thomas-Fermi functional, i.e. the Thomas-Fermi density. We mention here that such an approach is rather common in Physics and is called the mean field approximation. After that the problem of finding the ground state energy of a system of $N$ independent Fermions is reduced to determining the sum of the first $N$ eigenvalues (counting the multiplicities, including those due to the spins) of the basic one-particle Schr\"odinger operator, $P=-{1\over2}\Delta - \phi(x)$. A simple scaling maps the large $|Z|$ problem into the quasiclassical problem with $|Z|^{-{1\over3}}$ playing the role of a Planck constant. On the second step one studies quasiclassical asymptotics for the sum of the first $N$ eigenvalues of $P$, or, in general, a class of one-particle Schr\"odinger operators whose potentials have Coulomb singularities. The point here is that the Weyl term of the asymptotics will be identified with the Thomas-Fermi contribution to the original ground state energy while the second term, with the Scott correction. The physics of the problem suggests that the second term should come from the singularities of the potential. The difficulty in finding such an asymptotic expansion is twofold. First of all the problem of determining the second term in the quasiclassical and spectral asymptotics is notoriously difficult (see e.g. [H\"orm III, IV, Ivrii 1990-91, Robert 1987, Hux 1988, HelffRob 1990] and references therein). Secondly, all known approaches use pseudodifferential or Fourier integral operator Calculus and consequently require smooth potentials. Our method originates in general ideas of [Ivrii 1986, 1990-91], related to an approach of [Beals Feff 1974, Beals 1975] (see also [Tam 1984]). There are two ingredients in our proof. First of all we estimate global quantities through local ones. For instance, we study $$ {\rm tr}\big(\psi(x) g(P)\big)\ ,\eqno (1.7) $$ where $g(\lambda)=\lambda$ for $\lambda\le 0$ and $=0$ for $\lambda\ge 0$. If $\psi\equiv 1$, then the trace above is just the sum of negative eigenvalues of $P$, the quantity we want to estimate. We take for $\psi$ smooth functions localized outside of the singularities of the potentials. Then it is not difficult to obtain asymptotic expansion in the quasiclassical parameter $\beta$ of the trace (1.7). Pseudodifferential calculus provides convenient tools for such a purpose. Adapting a standard technique, one represents $g(P)$ as $$ g(P)\ = \ \int \hat g (t) e^{-iPt} dt\ , $$ where $\hat g (t)$ is the Fourier transform of $g$. The evolution operator $e^{-iPt}$ is then approximated for sufficiently small times and to any power in $\beta$ by Fourier integral operators in the spirit of the geometrical optics. Such an approximation is possible because of finite speed of propagation of singularities (properly defined) for the Schr\"odinger equation, provided the energy is bounded from above (i.e. $\sup({\rm supp}\, g)<\infty$) and the coordinate is localized in a domain in which the potential is bounded from below (cf. [SigSof 1988]). In the latter case, for sufficiently short times the bicharacteristics do not reach the singularities of $\phi$. In fact, the Fourier integral operators in question are constructed to have $C_0^\infty$ symbols and their analysis is well within the realm of the second year Calculus student. The approximating Fourier integral operators are then expanded by the method of stationary phase. The second ingredient is a multiscale analysis. There are three scales in the problem: momentum scale determined by the quasiclassical parameter $\beta\sim|Z|^{-{1\over3}}$, space scale, $\ell(x)$, determined by how the potential behaves under differentiation and the energy scale, $f(x)$, determined by the size of the potential. The first scale is constant while the other two depend on $x$. In our problem $$ \ell(x)\ = \ \hbox{dist. of $x$ to the singularities} $$ and $f(x)=\ell(x)^{-1}$. At each given point outside of the singularities we rescale the problem using the scales at this point in such a way that the problem is mapped into one on a unit ball with a smooth potential which is bounded together with all its derivatives independently of the scales involved and with a new quasiclassical parameter, $\alpha$, defined in terms of the old one and all the scales. The new problem admits a quasiclassical expansion (in powers of $\alpha$), as discussed above. This implies a quasiclassical expansion for the original problem outside of small balls around the singularities of the effective potential $\phi(x)$. The remainder in this expansion is bounded in terms of an explicit combination of all the scales times an absolute constant. The remarkable fact here is that since the subprincipal symbol of $P$ is $0$, the second term in the asymptotic expansion is zero. In estimating the remainder the fact that the sum of the eigenvalues is given by the trace of a function of $P$ which is once differentiable at the origin is crucial. In a small ball around a singularity we replace our operator by one whose potential represents the leading singularity at this point. We estimate the spectral function of the new operator more carefully using its special form: that of hydrogen-type Hamiltonian. In fact, we use the spherical symmetry of the new potential. Similarly to the way [SiedWeik 1987,1989,1991] proceeded with the original problem we decompose our local trace into the angular momentum channels, use the explicit form of the eigenvalues to sum up the low angular momentum channels and our original quasiclassical expansion, in order to treat the high ones. At this stage, when we sum up explicitly the contribution of the low angular momentum channels of the Hydrogen Hamiltonian the Scott correction is produced.\hfill \bigskip {\bf Notation.} We use the following standard conventions for the derivatives: $\partial_x = {\rm grad}_x$ (the gradient in $x$), $\partial_t={\partial\over\partial t}$, $\partial_{xt}^2=\partial_x\partial_t$, etc., and $\partial_x^\alpha = \prod\limits_{i=1}^{\rm dim} \partial_{x_i}^{\alpha_i}$ for a multi-index $\alpha=(\alpha_1, \alpha_2,\ldots)$. Moreover, $\nabla$ will stand for the gradient in all the variables involved, $\|\cdot\|_1$ will denote the trace norm: $\| A\|_1={\rm tr}(A^*A)^{1\over2}$ and $e_i(A)$ will stand for the $i$-th eigenvalue of a self-adjoint operator $A$, counting the multiplicities. $\langle x\rangle = (1 +|x|^2)^{1\over2}$. In the rest of the paper $C$ stands for various constants independent either of $N$, $Z$ and $R$ or of $\beta$ and $y$, or of $\alpha$. \hfill \bigskip {\bf Acknowledgement.} This work was done while the first author was visiting Universit\'es Paris VII and Paris-Sud and the second author, ETH-Z\"urich. The authors are grateful to A.M. Boutet de Monvel, L. Boutet de Monvel, B. Lascar, R. Lascar, J. Sj\"ostrand and to J. Fr\"ohlich and W. Hunziker, respectively, for the hospitality. The second author takes an opportunity to thank J. Fr\"ohlich, G.-M. Graf, C. G\'erard, W. Hunziker, H. Siedentop and, especially, V. Bach for many fruitful discussions. The authors are also grateful to the referee for pointing out many omissions and misprints, and demanding vigorously details and clarifications. We suffered while the paper and the reader gained.\hfill \vfill\eject \vfill\eject \beginsection 2. Mean Field Theory In this section we reduce the problem of estimating the ground state energy of $H(Z,R)$ to one of finding the quasiclassical asymptotics for the sum of negative eigenvalues of an effective one-particle Hamiltonian. The latter describes the motion of a particle in the external Thomas-Fermi potential, i.e. the nuclear potential $V(x,R)$, screened by a mean electron charge distribution. In the rest of this section we use extensively results from the Thomas-Fermi theory of Coulomb systems. We refer the reader to [Lieb 1981] for an excellent review of the subject. Here we mention the following facts. In order to define the first term in ${\cal E}^{\rm TF}(\rho)$ one has to take $\rho\in L^{5\over3}$. The physical origin of $\rho$ dictates $\rho\in L^1$. This already suffices for ${\cal E}^{\rm TF}(\rho)$ to be well defined and bounded from below. To see this one observes that $|x|^{-1}\in L^{5\over2}+L^\infty$ which implies that $$ \int |x|^{-1} \rho\ \le\ C_1\|\rho\|_{5\over3} + C_2\|\rho\|_1 \ . $$ The latter inequality yields that the second and the third terms in ${\cal E}^{\rm TF}(\rho)$ are finite in absolute value as well as that ${\cal E}^{\rm TF}(\rho)$ is bounded from below as $$ \eqalign{ {\cal E}^{\rm TF}(\rho)\ \ge\ &\gamma \|\rho\|_{5\over3}^{5\over3} - C_1|Z|\|\rho\|_{5\over3}\cr &-C_2|Z|\|\rho\|_1\cr} $$ with $C_1$ and $C_2$ independent of $Z$, $R$ and $M$. Furthermore, it is readily shown that ${\cal E}^{\rm TF} (\rho)$ defined on the set $L^1\cap L^{5\over3}$, $\rho\ge 0$ and $\int\rho=N$, is lower semicontinuous in the weak topology of $L^{5\over3}$. This and Banach-Alaoglu theorem imply that the infimum of ${\cal E}^{\rm TF}(\rho)$ with the side conditions $\int\rho=|Z|$ and $\rho\ge 0$ is attained in $L^1\cap L^{5\over3}$. The inequality above shows that the $L^p$-norms with $1\le p\le {5\over3}$ of the $\rho$ minimizing ${\cal E}^{\rm TF}(\rho)$ are bounded by $C|Z|^{{9p-5\over 4p}}$ with $C$ independent of $R$ and $M$. Next, analyzing directly expression (1.4) for $\cE^\TF(\rho)$, one sees that this functional is strictly convex $$ \cE^\TF \big(t\rho_1 + (1-t)\rho_2\big)\ <\ t\cE^\TF (\rho_1) + (1-t) \cE^\TF (\rho_2)\ , $$ provided $0\ 0 \eqno (2.2) $$ (provided $|Z|=N$) (see [LiebSim 1977, thms 11.20 and IV.3]). Note that the equation (2.1) implies that in the sense of distributions $$ -\Delta \phi_Z (x)\ =\ 4\pi\big( \Sigma Z_j \delta (x-R_j)-\rho_Z (x,R)\big)\ . $$ The Thomas-Fermi theory has a simple scaling property which we will use later. Using that $$ V_Z (x,R)\ =\ \beta^{-4} V_{\beta^3 Z} (\beta^{-1} x ,\beta^{-1} R) \eqno (2.3) $$ one concludes readily that the Thomas-Fermi functional for the potential $V_Z(x,R)$ evaluated at a density $\rho(x)$ equals $\beta^{-7}$ times the Thomas-Fermi functional for the potential $V_{\beta^3 Z}(x,\beta^{-1} R)$ evaluated at the density $\beta^6\rho (\beta x)$. Using the uniqueness of the minimizer we conclude that $$ \rho_Z (x,R)\ =\ \beta^{-6} \rho_{\beta^3 Z} (\beta^{-1} x , \beta^{-1} R) \eqno (2.4) $$ and that (1.5) holds. Using relation (2.4) together with either (2.1) and (2.3) or with (2.2), we conclude that $$ \phi_Z (x,R)\ =\ \beta^{-4} \phi_{\beta^3 Z} (\beta^{-1} x, \beta^{-1} R)\ .\eqno (2.5) $$ The Thomas-Fermi energy $E^{\rm TF}(Z,R)$ is, in fact, the quasiclassical energy of a gas of $|Z|$ non-interacting Fermions in the external potential $-\phi_Z(x,R)$. Indeed, the quasiclassical Fermi energy of such a gas in the ground state is $0$: $$ \eqalign{ 2\int_{p\le 0} dx d\xi\ &=\ 2^{3\over2} (3\pi^2)^{-1} \int \phi_Z (x,R)^{3\over2} dx\cr &=\ \int \rho_Z (x,R) dx\ =\ |Z|\ ,\cr} $$ where $p(x,\xi)={1\over2} |\xi|^2-\phi_Z(x,R)$, the classical Hamiltonian of a single Fermion. Here we have used Thomas-Fermi equation (2.2) and the normalization of $\rho_Z$. Hence the quasiclassical energy of this gas is $$ 2\int_{p\le 0} p dx d\xi\ =\ -{2\over 15\pi^2} \int \phi_Z^{5\over3}\ . $$ Now the Thomas-Fermi equation (2.3) implies $$ \int\phi_Z^{5\over3} \ =\ (3\pi^2)^{5\over3} \int p_Z^{5\over3}\quad {\rm and}\quad \int\phi_Z \rho_Z \ =\ (3\pi^2)^{2\over3} \int \rho_Z^{5\over3}\ . $$ The last three equations together with the definition of $E^{\rm TF} (Z,R)$ (see e.g. eqn (1.4)) yield $$ 2\int_{p\le 0} p dx d\xi\ =\ E(Z,R) + D_{\rm TF}\ .\eqno (2.6) $$ This relation is, in fact, the Virial theorem of the Thomas-Fermi theory (see [LiebSim 1977, thm II.23, Lieb 1981, thm 2.14]) and it is a consequence of a scaling property of the Thomas-Fermi potential under the map $\rho(x)\to t^3\rho (tx)$. Our next task is to derive estimates on $\phi_Z (x,R)$ needed later. These estimates extend somewhat earlier results of [LiebSim 1977, thms IV.5, IV.10] and their proof proceeds along similar lines. To make future references more convenient we change the symbols for parameters and in the next theorem deal with $\phi_\lambda (x,y)$ instead of $\phi_Z (x,R)$. We introduce the scale functions $$ \ell(x)\ \equiv\ \ell(x,y)\ \equiv\ \min_j |x - y_j | \ ,\eqno (2.7) $$ $$ f(x)\ =\ \ell(x)^{-{1\over2}}\langle \ell (x)\rangle^{-{3\over2}} \ .\eqno (2.8) $$ \proclaim Theorem 2.1. The Thomas-Fermi potential $\phi$ is smooth (in $x$) outside the points $y_1,\ldots, y_M$ and obeys the estimates $$ |\partial^\nu\phi_\lambda(x,y)|\ \le\ C_\nu (\min\,\lambda_j)^{-|\nu|} f(x)^2\ell (x)^{-|\nu|} \eqno (2.9) $$ for any multi-index $\nu$ and uniformly in $y$, $M$ and in $\lambda$. Here $\ell(x)$ and $f(x)$ are defined in (2.7) and (2.8), respectively.\par {\bf Proof.} We begin with \proclaim Lemma 2.2. Denote by $\phi_z^{at}(x)$ the Thomas-Fermi potential of the $z$-atom. Then $$ \eqalign{ &\sum\lambda_j \phi_{z=1}^{at} (x-y_j)\ \le\ \phi_\lambda(x,y)\cr &\le \Big(\sum\lambda_j\phi_{z=1}^{at}(x-y_j)^{3\over2}\Big)^{2\over3}\ .\cr} \eqno (2.10) $$ \par {\bf Proof.} We follow closely the proof of [LiebSim 1977, thm V.12]. We omit the subindex $\lambda$ at $\rho$ and $\phi$. Note first that by the definition of $\phi_\lambda(x,y)$ $$ \Delta \phi\ =\ 4\pi\Big( - \sum_{j=1}^M \lambda_j \delta (x-y_j)+\rho(x)\Big)\ . $$ Let $B=\{x\in\BR^d\bigm|\rho(x)\ge\sum\lambda_j\rho_j(x)\}$, where $\rho_j(x)=\rho^{at}(x-y_j)$ with $\rho^{at}(x)$, the Thomas-Fermi density of the $(z=1)$-atom. Let $\psi=\phi-\sum\lambda_j\phi_j$, where $\phi_j=\phi_{z=1}^{at} (x-y_j)$. Then $-\Delta \psi = 4\pi (-\rho + \sum\lambda_j\rho_j)$ and therefore $-\Delta\psi<0$ on $B$, i.e. $\psi$ is subharmonic on $B$. Hence it attains its maximum either on $\partial B$ or at infinity. Since $\psi=0$ on $\partial B$ and at $\infty$, we conclude that $\psi\le 0$ on $B$. On the other hand, $B\subseteq\{ x\in\BR^d \bigm| \psi(x)>0\}$ and therefore $B=\phi$. The latter fact is equivalent to the inequality $\rho\le\sum \lambda_j \rho_j$, which, due to the Thomas-Fermi equation (2.2), can be written as the upper bound in (2.10). To prove the lower bound in (2.10) we consider the same function $\psi$ as above and the set $D=\{x\in\BR^d\bigm|\psi(x)<0\}$. Since $\sum\lambda_j\phi_j\le(\sum\lambda_j\phi_j^{3\over2})^{2\over3}$, which is true due to $\sum\lambda_j=1$, we have that $$ \eqalign{ &\sum\lambda_j\rho_j-\rho\ =\ \sum\lambda_j\phi_j^{3\over2}- \phi^{3\over2}\cr \ge\ &(\sum\lambda_j\phi_j)^{3\over2}-\phi^{3\over2}\ .\cr} $$ Hence $\sum\lambda_j\rho_j-\rho>0$ and therefore $-\Delta\psi>0$ on $D$. Thus $\psi$ is superharmonic on $D$. Consequently it takes its minimum on $\partial D$ or at infinity. Since $\psi=0$ on $\partial D$ and at $\infty$, we have that $\psi\ge 0$ on $D$, i.e. $D=\phi$. Thus $\psi=\phi-\sum\lambda_j\phi_j\ge 0$, which is our lower bound. $\square$ Now the comparison argument, given in the proof of [LiebSim 1977, thm IV.10], applied to $\phi_{z=1}^{at}(x)$, yields $$ C_1\min (|x|^{-1},|x|^{-4})\ \le\ \phi_{z=1}^{at} (x)\ \le\ C_2\min (|x|^{-1},|x|^{-4}) \eqno (2.11) $$ for some (absolute) constants $0$ the total number of negative eigenvalues of $P$. Denote $$ \nu(x)\ = \ \sum_{i=1}^N |\psi_i(x)|^2\eqno (2.23) $$ and $$ \gamma(x,y)\ = \ \sum_{i=1}^N \psi_i(x) \overline{\psi_i(y)}\ .\eqno (2.24) $$ Note that $\nu$ and $\gamma$ are the one-particle density and one-particle density matrix, respectively for the ground state of $H^{\rm ind}(Z,R)$. In this section we use the shorthand $$ \int\int {\rho(x)\sigma(y)\over |x- y|} dx dy\ \equiv\ \int\int {\rho\sigma\over |x-y|}\ . $$ We begin with a result known to experts (see [Lieb 1979, NarnThirr 1981, SiedWeik 1987,1989]). \proclaim Theorem 2.4. Let $\mu=\nu-\rho_Z$. Then there is a constant $C$ independent of $Z$ and $M$ s.t. $$ \eqalign{ &{1\over2} \int\int {\mu\mu\over |x-y|} - {1\over2}\int\int {|\gamma|^2\over |x-y|}\cr \ge\ &E (Z,R)-E^{\rm ind} (Z,R) \ge - CM^{1\over2} (\Sigma Z_j)^{5\over3}\cr}\eqno (2.25) $$ \par {\bf Proof.} We begin with a lower bound. Namely, we show that uniformly in $Z$, in $M$ and in $R$ $$ E(Z,R)\ \ge\ E^\ind (Z,R) - C M^{1\over2}|Z|^{5\over3}\ .\eqno (2.26) $$ Let $\psi$ be any function in $\bigwedge\limits_{i=1}^N L^2(\BR^3\times\BZ_2)$, normalized as $\sum\limits_{\rm spins}\int |\psi|^2=1$. (Remember $N=|Z|$.) We associate with it the one-electron density $$ \rho_\psi(x_1)\ = \ N\sum\limits_{\sigma_1,\ldots,\sigma_N} \int |\psi(x_1,\sigma_1\ldots,x_N,\sigma_N)|^2 dx_2,\ldots,dx_N\ .\eqno (2.27) $$ Note that $$ \int\rho_\psi\ = \ N \ . $$ It is shown in [Lieb 1979] that $$ \eqalign{ &\langle \psi, \sum_{i0$ independent of $\beta$. Then the number of eigenvalues of $K_\beta$ not greater than $\mu\le -\beta^A$ for some $A\ge 0$ verifies the following asymptotic expansion $$ \# \{ e_i (K_\beta)\le\mu\}\ =\ 2\beta^{-3} \int_{k\le\mu} dx d\xi + O(\beta^{-2})\ .\eqno (2.53) $$ Here the remainder estimate is uniform in $ \mu,y$ and in $\lambda$. \par {\bf Proof.} The statement follows from theorem 8.9 with $s=0$ and $d=3$ from the relation $$ \#\{e_i(K_\beta)\le\mu\}\ = \ \tr E(\mu,K_\beta) \eqno (2.54) $$ where $E(\mu,A)$ is the spectral projection on $(-\infty,\mu]$ for an operator $A$. The factor 2 comes from the fact that $K_\beta$ in theorems 2.4 and 8.6 acts on $L^2(\BR^3\times\BZ_2)$ and on $L^2(\BR^3)$, respectively. $\square$ This result must be compared with analogous results in [Chaz 1980, Ivrii 1986, Rob 1987, Tam 1984] for assumptions on the potentials which are somewhat stronger than ours. \bigskip \noindent {\bf Remark 2.6.} The leading terms in (2.53) can be computed more explicitly: $$ 2\int_{k\le\mu} dx d\xi \ = \ {2^{3\over2}\over6\pi^2}\int (\phi_\lambda+\mu)_+^{3\over2} dx\ . \eqno (2.55) $$ Next, let $e_N=e_{N}(K_\beta)$ if $N\le$ the total number of eigenvalues of $K_\beta$ and $e_N=0$ otherwise. In other words, $e_N$ is the Fermi energy. Here $N=|Z|=\beta^{-3}$, the number of electrons. \proclaim Theorem 2.7. (Estimate of the Fermi Energy) We have uniformly in the $y$ and $\lambda$ $$ e_N \ = \ O\big( \beta^ {4\over3})\ .\eqno (2.56) $$ \par {\bf Proof.} First, recall that $\phi_\lambda$ is the potential of the neutral Thomas-Fermi theory with $\sum\lambda_i$ electrons. Hence the Thomas-Fermi density $\rho_\lambda$ satisfies (2.2) and $$ \int\rho_\lambda\ = \ \Sigma\lambda_i\ .\eqno (2.57) $$ Now rewrite the expression (2.56) for $\mu=0$: $$ \eqalign{ 2\int_{k\le0} dx d\xi \ &=\ {2^{3\over2}\over3\pi^2} \int\phi_\lambda^{3\over2} dx\cr &=\ \Sigma\lambda_i\ .\cr} \eqno (2.58) $$ According to our convention $e_N$ is either $0$ or $e_N<0$ and obeys, due to (2.58) and theorem 2.5 with $\mu=e_N$, $$ \beta^3 N\ = \ {2^{3\over2}\over3\pi^2} \int (\phi_\lambda+e_N)_+^{3\over2} + O(\beta)\ . \eqno(2.59) $$ Substracting (2.57) from this equation and remembering that $\beta^{-3}\Sigma\lambda_i=N$, we obtain $$ \int [(\phi_\lambda+e_N)_+^{3\over2}-\phi_\lambda^{3\over2}]\ = \ O(\beta)\ .\eqno (2.60) $$ We estimate the l.h.s. from above. Remembering that $e_N\le 0$, we obtain $$ \eqalign{ &\int [\phi_\lambda^{3\over2} - (\phi_\lambda + e_N)_+^{3\over2}]\cr &\quad \ge \ \int_{\phi_\lambda\le -e_N} \phi_\lambda^{3\over2}\ .\cr} \eqno(2.61) $$ This inequality together with eqns (2.12) and (2.60) shows already that $-e_N\le 1$. Next, using (2.12), we derive $$ \int_{\phi_\lambda\le -e_N}\phi_\lambda^{3\over2}\ \ge\ \delta_1(\min\lambda_j)^{3\over2}\int_{\ell(x)^{-4}\le -e_N}\ell(x)^{-6} \eqno (2.62) $$ for some $\delta_1>0$ independent of $\beta$, of $M$, of the $\lambda_j$'s and of the $y_j$'s. Using an elementary estimate $$ \eqalign{ &\int_{\ell(x)\ge\rho}\ell(x)^{-6} dx\ \ge\ \int_{2\rho\ge\ell(x) \ge \rho} \ell(x)^{-6} dx\cr =\ & \sum_{j} \int_{{2\rho\ge |x-y_j|\ge \rho\atop \ell(x)=|x-y_j|}} |x-y_j|^{-6} dx\cr \ge\ & (2\rho)^{-6} \sum_j {\rm meas} \{ x\bigm| \rho\le\ell(x)\le 2\rho,\ \ell(x)=|x-y_j|\}\cr \ge\ &(2\rho)^{-6} {\rm meas}\{ x\bigm| \rho\le\ell(x)\le2\rho\}\cr =\ &{7\pi\over 16}\rho^{-3} \ ,\cr} $$ we obtain furthermore $$ \int_{\phi_\lambda\le-e_N} \phi_\lambda^{3\over2} \ \ge\ \delta (\min\,\lambda_j)^{3\over2} (-e_N)^{3\over4} \eqno (2.63) $$ with $\delta>0$ independent of $\beta$, of $M$, of the $\lambda_j$'s and of the $y_j$'s. The last two inequalities imply $$ \int [ \phi_\lambda^{3\over2} - (\phi_\lambda + e_N)_+^{3\over2}]\ \ge\ \delta (\min\lambda_j)^{3\over2}(-e_N)^{3\over4} $$ with $\delta$ independent of $\beta$ and the $y_j$'s, which together with (2.60) implies (2.56). $\square$ \medskip Denote by $e(x,y,\mu,K_\beta)$ the Schwarz kernel of the spectral projection $E(\mu,K_\beta)$. Let $$ \eqalignno{ e_0(x,\mu,K_\beta)\ = \ & \beta^{-3} \int_{k\le\mu} d\xi\ . & (2.64) \cr = \ & \beta^{-3} {2^{3\over2}\over3\pi^2} (\phi_\lambda+\mu)_+^{3\over2}\ . \cr} $$ We restate here corollary 11.7 from section 11 for $d=3$. \proclaim Theorem 2.8. Let $K_\beta$ be the Schr\"odinger operator defined in (2.51) (and $d=3$). Assume $\min\lambda_j\ge\delta$ with $\delta$ independent of $\beta$. Then for any $\mu\le 0$ $$ \eqalign{ \Big( \int |e(x,x,\mu,K_\beta) &- e_0 (x,\mu,K_\beta)|^p dx\Big)^ {1\over p}\cr &\le\ C\beta^{-2}\cr} \eqno (2.65) $$ with $C$ independent of $\beta$, $\lambda$, $y$ and $\mu$. \par We use this theorem in the proof of the following \proclaim Lemma 2.9. Let, as above, $\mu=\nu-\rho_Z$, where $\nu$ is given in (2.5). Then $$ \int\int {\mu\mu\over |x-y|}\ \le\ C |Z|^{5\over3}\ .\eqno (2.66) $$ \par {\bf Proof.} Due to (2.51) we have for $Z=\beta^{-3}\lambda$ and $R=\beta y$ $$ \psi_i(x)\ = \ \beta^{-{3\over2}} \varphi_i(\beta^{-1} x)\ , \eqno (2.67) $$ where $\varphi_i$ are the eigenfunctions of $K_\beta$ considered on $L^2(\BR^3\times\BZ_2)$ corresponding to the eigenvalues $e_i=e_i(K_\beta)=\beta^4 E_i$, if $i\le$ the total number of eigenvalues of $K_\beta$, and $=0$ otherwise. Hence $$ \sum_{i=1}^N |\psi_i(x)|^2\ =\ \beta^{-3} \sum_{i=1}^N |\varphi_i (\beta^{-1}x)|^2\ .\eqno (2.68) $$ By the definition of $e_N$ $$ \sum_{i=1}^N |\varphi_i (x)|^2 \ = \ e(x,x,e_N,K_\beta)\ , \eqno (2.69) $$ where, recall, $e(x,y,\lambda,K_\beta)$ is the Schwartz kernel of $E(\lambda,K_\beta)$. Hence $$ \nu(x)\ \equiv \ \sum_{i=1}^N |\psi_i(x)|^2 \ = \ \beta^{-3}e(\beta^{-1}x, \beta^{-1}x, e_N,K_\beta)\ .\eqno (2.70) $$ Next using (2.56) and (2.64) and the Thomas-Fermi equation (2.2), we find $$ e_0(x,e_N,K_\beta)\ = \ \beta^{-3}\rho_\lambda (x,y) + O\big(M^{4\over3}\beta^{-{5\over3}} |\phi|^{1\over2}\big)\ .\eqno (2.71) $$ Thus remembering the scaling property (2.48) of the Thomas-Fermi density and scaling relation (2.70), we derive $$ \mu(x)\ = \ \beta^{-3}\mu_1 (\beta^{-1}x)\ ,\eqno (2.72) $$ where $$ \eqalign{ \mu_1(x)\ = \ &e(x,x,e_N,K_\beta)\cr &- e_0(x,e_N,K_\beta) + O\big( \beta^{-{5\over3}}\phi^{1\over2}\big)\ . \cr}\eqno (2.73) $$ Using this, we rescale the integral $$ \int\int {\mu\mu\over |x-y|}\ = \ \beta^{-1} \int\int {\mu_1\mu_1\over |x-y|}\ .\eqno (2.74) $$ Next, we apply the weak Young inequality $$ \int\int {\mu_1\mu_1\over |x-y|} \ \le\ C\| \mu_1\|_{6/5}^2\ . \eqno (2.75) $$ Next using (2.73), theorem 2.8 and the fact that $\int\phi^{3\over5}< \infty$, we find $$ \|\mu_1\|_{6\over5} \ \le\ C\beta^{-2}\ . \eqno (2.76) $$ Equations (2.74)--(2.76) imply (2.66). $\square$\hfill \bigskip Theorem 2.4 and lemma 2.9 yield equation (2.21). Theorem 2.3 is proven. $\square$\hfill \medskip \noindent {\bf Remark 2.10.} There is a trade-off between analysis of this section and that of section 8 required for theorem 2.5. The latter can be simplified (at the expense of the former) if one replaces the potential $\phi_\lambda (x,y)$ in definition (2.51) of $K_\beta$ by a deformed potential $\phi_{\lambda,\beta}(x,y)$ which differs from $\phi_\lambda (x,y)$, more precisely, which changes the sign at large $x$. Such a potential can be defined as follows $$ \eqalign{ \phi_{\lambda,\beta} (x,y)\ =\ &\phi_\lambda (x,y)\chi \big( \beta^{2\over3}\ell_\beta (x,y)\big)\cr &-\beta^2\ell_\beta (x,y)^{-1} \overline\chi \big(\beta^{2\over3} \ell_\beta(x,y)\big)\ ,\cr}\eqno (2.77) $$ where $\chi\in C_0^\infty (\BR)$ and is supported in $(-2,2)$ and $=1$ in $[0,1]$, $\overline\chi=1-\chi$ and $\ell_\beta(x,y)= \omega_\beta(x)*\ell(x,R)$. Here $\omega_\beta(x)=\beta^{2\over3}\ve^{-1}\omega\Big( {x\over\ve\beta^{-{2\over3}}}\Big)$ with $\ve>0$ sufficiently small and $\omega\in C_0^\infty$ and supported in $B(0,2)$ and $=1$ on $B(0,1)$. Equation (2.9) implies that the deformed potential $\phi_{\lambda,\beta}(x,y)$ obeys the estimates $$ |\partial^\nu\phi_{\lambda,\beta} (x,y)|\ \le\ C_\nu (\min\,\lambda_j)^ {-|\nu|} f_1(x)^2 \ell(x)^{-|\nu|}\ ,\eqno (2.78) $$ where $$ f_1(x)\ =\ \max\big( f(x), \beta\ell (x)^{-{1\over2}}\big)\ . \eqno (2.79) $$ \vfill\eject \beginsection 3. TF Gas and Weyl Asymptotics In this section we establish a quasiclassical asymptotics for the sum of negative eigenvalues of $K_\beta$ as $\beta\to 0$. The proof is obtained by patching together results of sections 8--10. After that we relate the mentioned asymptotics for $K_\beta$ to that for the ground state energy $E^{\rm ind}(Z,R)$ as $Z\to\infty$. Combining this with theorem 2.1 we conclude that theorem 1.1 is valid. Since $H^{\rm ind}(Z,R)$ acts on $\bigwedge\limits_{i=1}^N L^2 (\BR^3\times\BZ_2)$ and since the variables $x_1,\ldots,x_N$ in it separate, we have $$ E^{\rm ind}(Z,R)\ = \ \sum_{i=1}^N E_i - D_{\rm TF}\ ,\eqno (3.1) $$ where, recall, $E_i$ are the eigenvalues of $P$ labelled in order of their increase and counting their multiplicity (see the paragraph after equation (2.22)). This is a well known relation in Quantum Physics and is a consequence of the Pauli principle: at most two electrons (the double degeneracy corresponding to $\BZ_2$) for an energy level. We begin with \proclaim Theorem 3.1. Consider the Schr\"odinger operator $K_\beta$ on $L^2(\BR^3)$ with a potential $\phi(x)$ obeying (2.7)--(2.9) with $a=\min\{|y_i-y_j|\,|i\ne j\}\ge\beta^{{2\over3}-\varepsilon}$ for some $\varepsilon>0$. Assume $\min\lambda_j\ge\delta_1$ with $\delta_1>0$ independent of $\beta$. Then for any $\delta>0$ $$ \sum_{e_i\le 0} e_i (K_\beta)\ = \ {\rm Weyl} \ + \ {\rm Scott}\ + \ O\big(a^{-{1\over3}}\beta^{-{4\over3}-\delta}\big) \ ,\eqno (3.2) $$ where the remainder estimate is uniform in the $y_j$'s and in $\lambda_j$'s, restricted as above $$ {\rm Weyl}\ =\ \beta^{-3} \int\int_{k\le0} k\, dxd\xi \eqno (3.3) $$ and $$ {\rm Scott}\ = \ {\Sigma \lambda_j\over 8} \beta^{-2}\ . \eqno (3.4) $$ \par {\bf Proof.} Let $$ g(\sigma)\ = \ \cases{ \sigma&if $\sigma\le0$\cr 0&if $\sigma>0.$\cr}\eqno (3.5) $$ Then $$ \sum_{e_i\le 0} e_i(K_\beta)\ = \ \tr g(K_\beta)\ .\eqno (3.6) $$ Introduce a smooth partition of unity, $\psi_0,\ldots,\psi_M$, $$ \sum_{i=0}^M \psi_i \ = \ 1 \eqno (3.7) $$ with the properties $$ \eqalign{ {\rm for}\ i\ge 1,\ \ &\psi_i\ {\hbox{is supported in}} \ B(y_i,2r)\cr &\psi_0\ \ {\hbox{is supported in}}\ \BR^3\backslash \bigcup_{i=1}^M B(y_i,r)\ ,\cr} \eqno (3.8) $$ where $r>0$ obeys $$ {1\over4}\beta^{1-\delta}\ \le\ r\ \le\ {1\over3}a\eqno (3.9) $$ for some $\delta>0$. We will choose $r$ later. Then $$ \tr g(K_\beta)\ = \ \sum_{i=0}^M \tr\big( \psi_i g(K_\beta)\big) \ .\eqno (3.10) $$ Since $g$ obeys the conditions of theorem 8.1 with $s=1$, theorem 8.9 with $d=3$ and $s=1$ is applicable and yields $$ \eqalign{ \tr\big( \psi_0 g(K_\beta)\big)\ =\ &\beta^{-3} \int\int g(k)\psi_0 dx d\xi\cr &+ O\big(\beta^{-1}r^{-{1\over2}}\big) \ .\cr} \eqno (3.11) $$ By theorem 9.1 with $d=3$ $$ \eqalign{ &|\tr\big( \psi_i g(K_\beta)-\psi_i g(K_{i,\beta})\big)\cr &-\beta^{-3}\int\int\psi \big( g(k)-g(k_i)\big) dxd\xi|\cr &\le\ C (\beta^{-1} r^{-{1\over2}} + a^{-1}\beta^{-2} r ) \ ,\cr} \eqno (3.12) $$ provided $\beta^{{2\over3}-\delta}\le r\le {1\over3}a$ for some $\delta>0$, $i\ge 1$, $$ K_{i,\beta}\ = \ -{1\over2}\beta^2 \Delta - {\lambda_i\over |x-y_j|} \eqno (3.13) $$ and $$ k_i(x,\xi)\ = \ {1\over2} |\xi|^2 - {\lambda_i\over |x-y_i|}\ .\eqno (3.14) $$ Finally, by theorem 10.1 $$ \eqalign{ &\tr\psi_i g(K_{i,\beta})\cr =\ &\beta^{-3} \int\int \psi_i g(k_i) dx d\xi\cr - &{\lambda_j^2\over 8} \beta^{-2} + O(\beta^{-1}r^{-{1\over2}})\cr}\eqno (3.15) $$ with $i\ge 1$. Equations (3.12) and (3.15) yield for $i\ge 1$ $$ \eqalign{ |\tr\big(&\psi_i g(K_\beta)\big)-\beta^{-3} \int\int \psi_i g(k) dx d\xi \cr &+ {\lambda_i^2\over 8} \beta^{-2}| \ \le\ C( \beta^{-2} r a^{-1} + \beta^{-1}r^{-{1\over2}})\ .\cr} \eqno (3.18) $$ This relation together with equations (3.10)--(3.11) implies $$ \eqalign{ |\tr g(K_\beta)- &\beta^{-3} \int\int g(k) dx d\xi\cr &+ {1\over8} \Sigma \lambda_i^2 \beta^{-2}| \ \le\ C(\beta^{-2} r a^{-1}+ \beta^{-1}r^{-{1\over2}}) \ .\cr} \eqno (3.19) $$ Comparing this with (3.6) and choosing $r= \beta^{{2\over3}-\delta} a^{2\over3}$, we arrive at (3.2). $\square$\hfill \bigskip {\bf Proof of theorem 1.1.} First we show that $$ \#\{ 0>e_i (K_\beta)>e_N\}\ =\ O\big(\beta^{-2}\big)\ .\eqno (3.20) $$ Indeed, due to (2.53), $$ \eqalign{ &\#\{ 0>e_i(K_\beta)>e_N\}\cr =\ &\beta^{-3}\int_{0\ge k\ge e_N} dx d\xi + O\big( \beta^{-2}\big)\ .\cr} \eqno (3.21) $$ Equations (2.55) and (2.60) yield $$ \int_{0\ge k\ge e_N} dx d\xi\ =\ O\big(\beta\big)\ ,\eqno (3.22) $$ which together with (3.21) implies (3.20). Equations (3.20) and (2.53) yield $$ \eqalign{ \sum_{i>N} |e_i (K_\beta)| \ &\le\ |e_N| \#\{ 0>e_i(K_\beta) > e_N\}\cr &\le\ C\beta^{-{2\over3}}\ .\cr} \eqno (3.23) $$ Recall that $k(x,\xi)={1\over2}|\xi|^2-\phi_\lambda (x,y)$, where $\phi_\lambda$ is the potential of the neutral Thomas-Fermi theory with nuclei of charges $\lambda_1,\ldots,\lambda_M$ located at $y_1,\ldots,y_M$. Equations (2.5) and (2.6) yield $$ 2\beta^{-3} \ii_{k\le 0} k dxd\xi\ =\ \beta^{-3} E^\TF (\lambda,y) + D_\TF\ .\eqno (3.24) $$ Equations (3.23) and (3.24), together with (3.2), imply $$ \sum_{i=1}^{N} e_i (K_\beta)\ = \ \beta^{-3} E^\TF (\lambda,y) + {\Sigma\lambda_i^2\over 4} \beta^{-2} + D_\TF + O\big(a^{-{1\over3}}\beta^{-{4\over3}-\delta}\big) \ , \eqno (3.25) $$ provided $|y_i-y_j|\ge\beta^{{2\over3}-\ve}$ for all $i\ne j$ and some $\ve>0$. Now, due to relation (2.50), $$ E_i\ = \ \beta^{-4} e_i (K_\beta)\ ,\eqno (3.26) $$ where $E_i$, recall, are the eigenvalues of $P$. Taking into account the scaling property of the Thomas-Fermi energy $$ E^\TF (Z,R)\ =\ \beta^{-7} E^\TF (\lambda,y)\ , \eqno (3.27) $$ where $Z=\beta^{-3}\lambda$ and $R=\beta y$, we derive from (3.25) that $$ \sum_{i=1}^N E_i \ = \ E^\TF (Z,R) + {\Sigma Z_i^2\over 4} + D_\TF + O\big(a^{-{1\over3}}|Z|^{{16\over9}+\delta}\big) \ , $$ provided $|R_i-R_j|\ge|Z|^{-{5\over9}+\ve}$ for all $i\ne j$ and some $\ve>0$. This relation together with equations (2.21) and (3.1) implies (1.8). $\square$\hfill \vfill\eject \beginsection 4. Energy Bounds In this section we prove two kinds of estimates: bounds on the momentum in terms of energy and bounds on accessibility of energetically forbidden regions of the phase-space. Both bounds are needed in the following sections. Though the latter bounds have obvious classical meaning the operators involved are not pseudodifferential. In other words we obtain results, which normally follow from symbolic calculus, for non-symbolic operators. In a different context such results were obtained earlier in [SigSof 1987]. In this and the next three sections we consider a self-adjoint operator $$ H_\alpha\ = \ -{\alpha^2\over2} \Delta - W(x) $$ on $L^2(\BR^d)$. Here $\alpha>0$ is a quasiclassical parameter about which we assume only that $\alpha\le 1$. We assume that $W(x)$ is real, is in $L_{\rm loc}^2$ and obeys the Kato inequality $$ \| Wf\|\ \le\ \varepsilon \|\Delta f\| + {C\over\varepsilon^2} \| f\|\eqno (4.1) $$ for any $f\in D(\Delta)$ for any $\varepsilon>0$ and with $C$ independent of $\varepsilon$. (4.1) is satisfied for Kato potentials, i.e. the potentials from $L^p(\BR^d)+L^\infty(\BR^d)$ with $p>{d\over2}$ for $d\ge4$ and $p=2$ for $d<4$ (see e.g. [CFKS 1987]). Under the last restriction $H_\alpha$ is self-adjoint on $D(H_\alpha)=D(\Delta)$. Note that a standard interpolation argument (see e.g. [RSII, thm IX.20] and (4.1) yield that $$ | \langle Wf,f\rangle |\ \le \ \ve \langle -\Delta f , f\rangle + {C\over\ve^2} \| f\|^2\eqno (4.2) $$ (see e.g. [RSII, thm X.18]). \proclaim Lemma 4.1. Let $W$ obey (4.1). Then $$ \|\Delta (H_\alpha +i)^{-1}\|\ \le\ C\max(\alpha^{-2},\alpha^{-6})\ ,\eqno (4.3) $$ $$ \|\nabla (H_\alpha +i)^{-1}\|\ \le\ C\max(\alpha^{-1},\alpha^{-2})\eqno (4.4) $$ and $$ \|\nabla (H_\alpha+i)^{-1}\nabla\|\ \le\ C\max(\alpha^{-2},\alpha^{-4})\ .\eqno (4.5) $$ \par {\bf Proof.} Using $$ \|\Delta f\| \le\ 2\alpha^{-2} (\| H_\alpha f\| + \| Wf\| ) $$ and (4.1) with $\ve = {1\over4}\alpha^2$, we obtain $$ \|\Delta f\|\ \le\ 4\alpha^{-2}\| H_\alpha f\| + C\alpha^{-6}\| f\| $$ which implies (4.3). Next, let $u=(H_\alpha+i)^{-1} f$. Using that $\|\nabla u\|^2=\langle -\Delta u,u\rangle$, we obtain $$ \|\nabla u\|^2\ =\ {2\over\alpha^2} \big(\langle (H_\alpha +i)u,u\rangle + \langle (W-i)u,u\rangle\big)\ . $$ Applying (4.2) with $\ve={\alpha^2\over4}$ to the last term, we obtain $$ \|\nabla u\|^2\ \le\ {2\over\alpha^2} |\langle (H+i)u,u\rangle | + {1\over2} \|\nabla u\|^2 + {C\over\alpha^4}\| u\|^2\ .\eqno (4.6) $$ Since $(H+i)u=f$ and $\| u\|\le \| f\|$, this yields (4.4). Finally, to prove (4.5) we note that (4.6) with $u=(H_\alpha+i)^{-1} \nabla f$ yields $$ \|\nabla u\|^2\ \le\ {4\over\alpha^2} \| f\| \|\nabla u\| + {C\over\alpha^4}\| u\|^2\ . $$ Since by the previous result $\| u\|\le C\alpha^{-2}\| f\|$, this yields (4.6). $\square$ \medskip In what follow $\| A\|_q = ({\rm tr} |A|^q)^{1\over q}$, the $I_q$-trace norm of the operator $A$ (see [RSII, p. 41] for the definition and properties used below). \proclaim Lemma 4.2. Assume $W(x)$ obeys (4.1). Let $\psi\in C_0^\infty$ and $|\partial^\nu \psi(x)|\le C_\nu$. Let $n=\Big[ {d\over 2}\Big] + 1$. Then for $\alpha\le 1$ $$ \|\psi (H_\alpha+i)^{-n}\|_1\ \le\ C\alpha^{-6n}\eqno (4.7) $$ with $C$ independent of $\alpha$.\par {\bf Proof.} We conduct the proof by induction. As a result it is convenient to prove a more general statement: $$ \|\psi (H_\alpha+i)^{-m}\|_{n\over m}\ \le\ C\alpha^{-6m}\ .\eqno (4.8) $$ First we prove this statement for $m=1$. By a property of the trace norms $$ \eqalign{ &\|\psi (H_\alpha +i)^{-1}\|_n\cr \le\ &\|\psi(-\Delta+i)^{-1}\|_n \|(-\Delta+i)(H_\alpha+i)^{-1}\|\cr} \eqno (4.9) $$ Since $n>{d\over2}$, then by a standard result the first factor on the r.h.s. is bounded. By (4.3), the second factor is bounded by ${\rm const}\,\cdot\alpha^{-6}$. Thus (4.8) with $m=1$ follows. Now we assume (4.8) is valid for some $m\ge 1$ and prove it for $m+1$. Let $\psi_1\in C_0^\infty$, $=1$ on ${\rm supp}\,\psi$, so that $\psi\psi_1=\psi$, and obey $|\partial^\nu\psi_1(x)|\le C_\nu$. Using that $$ [\psi, (H_\alpha+i)^{-1}]\ =\ \alpha(H_\alpha+i)^{-1} L_\psi (H_\alpha+i)^{-1}\ ,\eqno (4.10) $$ where $$ L_\psi\ =\ -\alpha\nabla\cdot(\nabla\psi)-{\alpha\over2}(\Delta \psi)\ ,\eqno (4.11) $$ and that $\psi=\psi_1\psi$, we obtain $$ \eqalign{ \psi (H_\alpha+i)^{-m-1}\ =\ &\psi_1 (H_\alpha+i)^{-1}\psi (H_\alpha+i)^{-m}\cr &+ \alpha\psi_1 (H_\alpha+i)^{-1} L_\psi (H_\alpha+i)^{-m-1}\ .\cr} \eqno (4.12) $$ Writing $L_\psi=L_\psi\psi_1$ and repeating this procedure in the last term we arrive at $$ \eqalign{ \psi(H_\alpha+i)^{-m-1}\ =\ &\psi_1(H_\alpha+i)^{-1} [ \psi +\alpha L_\psi (H_\alpha+i)^{-1}\psi_1\cr &+ \alpha^2 L_\psi (H_\alpha+i)^{-1} L_{\psi_1}] \psi_2 (H_\alpha+i)^ {-m}\ ,\cr}\eqno (4.13) $$ where $\psi_2=1$ on ${\rm supp}\, \psi_1$ and obeys $\psi_2\in C_0^\infty$, $|\partial^\nu\psi_2|\le C_\nu$. By lemma 4.1 the expression in the square brackets is bounded. Hence by a property of the trace norms $$ \eqalign{ &\|\psi (H_\alpha+i)^{-m-1}\|_{{n\over m+1}}\cr \le\ &C\|\psi_1(H_\alpha+i)^{-1}\|_n \|\psi_2 (H_\alpha+i)^{-m}\|_{{n\over m}}\ .\cr}\eqno (4.14) $$ By (4.8) with $m=1$, proven above, and by the induction hypothesis the r.h.s. is bounded by $C\alpha^{-6(m+1)}$. This yields (4.8) with $m$ replaced by $m+1$. The induction step is completed. $\square$ \medskip We will also need the following statement: \proclaim Lemma 4.3. Assume $W$ obeys (4.1). Let $\psi\in C^\infty$. Then $$ \| \nabla\psi(H_\alpha+i)^{-1}\|\ \le\ K\alpha^{-1}\ ,\eqno (4.16) $$ where $K=[(\sup\limits_{{\rm supp}\,\psi} W)_+ +1]^{1\over2}\sup|\psi| + \alpha \sup|\nabla\psi|$. \par {\bf Proof.} Let $R=(H_\alpha+i)^{-1}$. We have $$ \eqalign{ &\alpha^2{1\over2}R^*\psi^* \Delta\psi R\cr \le\ &R^*\psi^* H_\alpha \psi R + \sup( W|\psi|^2)\ .\cr} \eqno (4.17) $$ Commuting $H_\alpha$ through $\psi$ and using that $$ \eqalign{ &{\rm Re}(R^*\psi^* [H_\alpha,\psi]R)\cr =\ &\alpha^2 R^*|\nabla \psi|^2 R\cr} $$ and using that $\| H_\alpha R\|\le 1$ we obtain that $$ \eqalign{ &R^*\psi^* H_\alpha\psi R\cr \le\ &\sup |\psi|^2 + \alpha^2 \sup |\nabla\psi|^2\ .\cr} $$ This together with (4.17) yields (4.16). $\square$\hfill\medskip Next we have \proclaim Lemma 4.4. Assume $W$ obeys (4.1). Let $\psi$ be smooth and obey $|\partial^\nu\psi(x)|\le C_\nu$. Then $$ \|\Delta\psi (H_\alpha+i)^{-1}\|\ \le\ C\alpha^{-2} (\sup_{{\rm supp}\,\psi} |W|+1) + 1\ .\eqno (4.18) $$ \par {\bf Proof.} We have $$ \eqalign{ &\|\Delta \psi (H_\alpha+i)^{-1} f\|\cr \le\ &{2\over\alpha^2} \| (H_\alpha+i)\psi (H_\alpha+i)^{-1} f\|\cr &+ {2\over\alpha^2} (\sup\limits_{{\rm supp}\,\psi} |W|+1)\sup |\psi|^2 \|f\|\ .\cr}\eqno (4.19) $$ Commuting $(H_\alpha+i)$ through $\psi$, we obtain $$ \eqalign{ &\| (H_\alpha+i)\psi (H_\alpha+i)^{-1} f\|\cr \le\ &\|\psi f\| + \alpha \| L_\psi (H_\alpha+i)^{-1} f\|\ .\cr} $$ Applying lemma 4.3 to the last term we arrive at $$ \eqalign{ &\| (H_\alpha+i)\psi (H_\alpha+i)^{-1}f\|\cr \le\ &C\big[\big(\alpha (\sup_{{\rm supp}\, \psi} W)_+ + 1\big)^{1\over2} +\alpha^2 +1\big]\|f\|\ .\cr} $$ This together with (4.19) yields (4.18). $\square$ \medskip Proceeding as in the proof of lemma 4.2 but using lemmas 4.3 and 4.4 instead of lemma 4.1, one proves the following \proclaim Lemma 4.5. Assume $W$ obeys (4.1). Let $\psi\in C_0^\infty$ and obey $|\partial^\nu\psi|\le C_\nu$. Then for $n=\Big[ {d\over 2}\Big] + 1$ $$ \| \psi (H_\alpha+i)^{-n}\|_1 \ \le\ C \alpha^{-2n} (\sup_\Omega |W| +1)^n\ ,\eqno (4.20) $$ where $\Omega$ is $\ve({\rm diam}({\rm supp}\, \psi)) -{\rm neighbourhood\ of\ supp}\, \psi$ and $C$ independent of $\alpha$.\par Now we proceed to a less trivial result needed in this and forthcoming sections. This result shows how in the Operator Calculus we can pass from one Hamiltonian to another. Denote by $B(y,r)$ a ball in $\BR^d$ of the radius $r$ and centered at $y$. \proclaim Theorem 4.6. Let (a) $H_\alpha=-{\alpha^2\over2}\Delta- W(x)$ and $H_{0,\alpha}=-{\alpha^2\over2}\Delta-W_0(x)$ with $W(x)$ and $W_0(x)$ obeying (4.1) and $$ W(x)\ =\ W_0(x)\qquad {\rm for}\ x\in B(0,2)\ ,\eqno (4.21) $$ (b) $\psi\in C_0^\infty (B(0,1))$ with $|\partial^\nu\psi(x)|\le C_\nu$ and (c) $\vp$ be smooth and obeying $|\partial^n\vp(\lambda)| \le C_n\langle\lambda\rangle^{m-n}$. Then for any $A\ge\Big[ {d\over2}\Big] +3$ $$ \eqalign{ &\|\psi(x)\big( \vp (H_\alpha) - \vp (H_{0,\alpha})\big)\|_1\cr \le\ &C( \alpha L)^A\alpha^{-3d-12}\|\vp\|_A\ ,\cr} \eqno (4.22) $$ where $\|\vp\|_A=\sup\limits_\lambda (\langle\lambda\rangle^{A} |\partial_\lambda^{A+m} \vp (\lambda)|)$, $L=\sup\limits_ {B(0,2)\backslash B(0,1)} \big(W(x)\big)_+^{1\over2}+1$ and $C$ is independent of $\alpha$.\par {\bf Proof.} We omit the subindex $\alpha$ in this proof: $H=H_\alpha$ and $H_0=H_{0,\alpha}$. Let $$ \eqalign{ U\ &=\ H-H_0\cr &=\ W_0-W\ .\cr} $$ By (4.1) and (4.3) $$ \| U(H-z)^{-1}\|\ \le\ C\alpha^{-6}\langle z\rangle | {\rm Im}\, z|^{-1}\ .\eqno (4.23) $$ We begin with \proclaim Lemma 4.7. Let $\psi\in C_0^\infty (B(0,1))$ and let $\psi_1$ be a smooth and bounded function supported in $\BR^d\backslash B(0,2)$. Then for any $n\ge 0$ and for any $A\ge 0$ $$ \| H^n \psi (H-z)^{-1}\psi_1\|\ \le\ C(\alpha L |{\rm Im}\, z|^{-1})^A \langle z\rangle^n\eqno (4.24) $$ and similarly for $H_0$. Here $C$ is independent of $\alpha$ and $z$.\par {\bf Proof.} Commuting $H^n$ factor-by-factor through $\psi$ and using that $$ [H,f]\ =\ \alpha L_f \ , \eqno (4.25) $$ where $L_f$ is defined in (4.11), we reduce the problem to one of showing that $$ \|\chi_1 H^m (H-z)^{-1}\psi_1\|\ \le\ C(\alpha L |{\rm Im}\, z|^{-1})^A\langle z\rangle ^m\eqno (4.26) $$ for any $m\ge 0$ and any $A\ge 0$ and for $C$ as above, provided $\chi_1\in C_0^\infty \big( B(0,1+\ve)\big)$ and $=1$ on $B(0,1)$ and obeys $\partial^\nu\chi_1=O(1)$ with $\ve={1\over4A}$. Next, representing $H=H-z+z$ and using that $\chi_1\psi_1=0$, we reduce the problem further to showing that $$ \chi_1(H-z)^{-1}\psi_1\ =\ O\big( (\alpha L |{\rm Im}\, z|^{-1})^A\big)\ .\eqno (4.27) $$ Now commuting $\chi_1$ through $(H-z)^{-1}$ and using that $\chi_1\psi_1=0$, we obtain $$ \eqalign{ &\chi_1(H-z)^{-1}\psi_1\cr =\ &\alpha(H-z)^{-1}L_{\chi_1}\chi_2(H-z)^{-1}\psi_1\cr}\eqno (4.28) $$ where $\chi_2\in C_0^\infty \big( B(0,1+2\ve)\big)$ and $=1$ on $B\big( 0,1+\ve\big)$ and $\chi_2$ satisfies $\partial^ \nu\chi_2=O(1)$. Applying the above procedure to $\chi_2(H-z)^{-1}\psi_1$, etc., we arrive at $$ \chi(H-z)^{-1}\psi_1\ =\ \alpha^A \prod_{i=1}^A [(H-z)^{-1} L_{\chi_i}] (H-z)^{-1}\psi_1\ ,\eqno (4.29) $$ where $\chi_k\in C_0^\infty \big( B(0,1+k\ve)\big)$, $=1$ on $B\big( 0,1+(k-1)\ve\big)$ and obey $\partial^\nu\chi_k= O(1)$. Equation (4.16), the estimate $\partial^\nu\chi_i=O(1)$, the fact that $\nabla\chi_i$ are supported in $B(0,2)\backslash B(0,1)$ and the restriction $1\ge\alpha^2$ imply that each factor on the r.h.s. is $O(|{\rm Im}\, z|^{-1}L)$. This yields (4.27). Hence (4.24) follows. $\square$ \medskip Now we proceed directly to the proof of theorem 4.6. There is a smooth function, $\phi$, on $\BR^2$, supported in the strip $\{(\lambda,\mu)\bigm| |\mu|\le 1\}$ and s.t. $\phi(\lambda,0)=\vp(\lambda)$ and for any $A\ge 0$ $$ |\partial_{\bar z}\phi(\lambda,\mu)|\ \le\ \|\vp\|_A |\mu|^{A+m-1}\langle\lambda\rangle^{-A}\ ,\eqno (4.30) $$ where $z=\lambda+i\mu$ and $\partial_{\bar z}=\partial_\lambda + i\partial_\mu$. To prove the existence of such a $\phi$ one uses a partition of unity associated with the length scale $\ell(\lambda)= \langle\lambda\rangle$ (see section 8) and then applies a standard extension theorem to each compactly supported piece of $\vp(\lambda)$. Following [HSj 1989], one can represent $$ \vp(A)\ =\ {1\over 2\pi i} \int\int \partial_{\bar z}\phi (A-z)^{-1} d\lambda d\mu\eqno (4.31) $$ for any self-adjoint operator $A$. Using this representation, the second resolvent equation and the equation $H-H_0=U$, we find $$ \eqalign{ &\vp(H) -\vp(H_0)\cr =\ &-{1\over 2\pi i} \int\int \partial_{\bar z}\phi\cdot (H-z)^{-1} U (H_0-z)^{-1} d\lambda d\mu\ .\cr}\eqno (4.32) $$ Denote $I_n=(H+i)^n\psi \big(\vp(H)-\vp(H_0)\big)$. Using this relation and the fact $U$ is supported in $\BR^d \backslash B(0,4)$, we transform $$ I_n\ =\ - {1\over 2\pi i} \int\int \partial_z \phi A(z) B(z) d\lambda d\mu\ ,\eqno (4.33) $$ where $$ A(z)\ =\ (H+i)^n\psi (H-z)^{-1}\psi_1\ ,\eqno (4.34) $$ with $\psi_1$, a smooth and bounded function supported in $\BR^d\backslash B(0,3)$, and $$ B(z)\ =\ U(H-z)^{-1}\ .\eqno (4.35) $$ By lemma 4.7, $A(z)=O\big( (\alpha L |{\rm Im}\, z|^ {-1})^M\langle z\rangle^n\big)$ for any $M$, and by (4.23), $B(z)=O(\alpha^{-2}\langle z\rangle |{\rm Im}\, z|^{-1})$. This together with (4.30) yields that $$ \| I_n\|\ \le\ C\|f\|_M \alpha^{M-2} L^M \eqno (4.36) $$ for any $M>n+1$. Let now $\psi_2\in C_0^\infty \big( B(0,2)\big)$ and $=1$ on $B(0,1)$. We write $$ \psi\big(\vp(H)-\vp(H_0)\big)\ =\ \psi_2(H+i)^{-n} I_n\ . \eqno (4.37) $$ Equations (4.20), (4.36) and (4.37) yield (4.22). $\square$ \medskip Denote $D_x=-i\alpha\,{\rm grad}_x$ and $D_t=i\alpha{\partial\over \partial t}$. We assume now that in addition to (4.1), $W(x)$ is smooth in $B(0,2)$ and obeys $$ |\partial^\nu W(x)|\ \le\ C_\nu\qquad \forall\nu\quad {\rm on}\ B(0,2)\ .\eqno (4.38) $$ We need the following microlocal estimate saying that operators are quasiclassically small in the classically forbidden region: \proclaim Theorem 4.8. Assume $W$ obeys (4.1) and (4.38). Let $g(\lambda)$ be a measurable function satisfying $\sup({\rm supp}\, g)<\infty$ and $|g(\lambda)|\le C \langle \lambda\rangle^m$ for some $m$ and let $\vp(x,\xi)$ be a smooth function supported in $B(0,1)\times\BR^d$ and obeying $|\partial_x^\alpha\partial_\xi^\beta \vp (x,\xi)|\le C_{\alpha\beta}\langle\xi\rangle^{-|\beta|}$. Assume there is $\ve>0$ s.t. ${\rm supp}\,\vp\cap h^{-1} (\ve\!-\!{\rm supp}\, g)$, where $\ve\!-\!Q=\{\lambda\in\BR\bigm| {\rm dist} (\lambda,Q)\le\ve\langle\lambda\rangle\}$ for $Q\subset \BR$ and, recall, $h={1\over2}|\xi|^2-W(x)$, the symbol of $H_\alpha$. Then $$ \|g(H_\alpha)\vp(x,D_x)\|_1\ =\ O(\alpha^A)\eqno (4.39) $$ for any $A\ge 0$ and any $0\le\alpha\le 1$.\par {\bf Proof.} Pick up $f(\lambda)$ smooth, supported in $\ve - {\rm supp}\, g$ with $\ve$ given in the theorem and obeying $|\partial^n f(\lambda)|\le C_n\langle\lambda\rangle^{m-n}$ and $f(\lambda)\ge |g(\lambda)|$. Using the definition of the trace norm (see e.g. [ReedSim II, p. 42]), we derive that $$ \| g(H_\alpha) \vp(x,D_x) \|_1\ \le\ \| f(H_\alpha) \vp (x,D_x) \|_1\ . $$ Introduce an auxiliary potential $W_0(x)$ as $$ W_0\in C_0^\infty (\BR^d)\qquad {\rm and}\qquad = W (x)\qquad {\rm in}\ B\Big(0,{3\over2}\Big)\ . \eqno (4.40) $$ Let $\psi_1\in C_0^\infty \big( B(0,5/4)\big)$ and $=1$ on $B(0,6/5)$. Then by theorem 4.6 $$ \| \big( f(H_\alpha) - f(H_{0,\alpha})\big)\psi_1(x)\|_1\ \le\ C\alpha^A $$ for any $A\ge 0$, where $H_{0,\alpha}=-{\alpha^2\over 2}\Delta - W_0 (x)$. By the definition of $\alpha$-pseudodifferential operators (see e.g. [Robert 1987]) $$ \big( 1 - \psi_1(x)\big)\vp(x,D_x)\ =\ 0 $$ for any $A\ge 0$. Next, due to the restrictons on $f$, since $H_{0,\alpha}$ is an elliptic $\alpha$-pseudo-differential operator, then so is $f(H_{0,\alpha})$ (see [Robert 1987]). Moreover, the $\alpha$-symbol of $f(H_{0,\alpha})$ is supported in $h^{-1}({\rm supp}\, f)$. Hence using again $\alpha$-pseudo-differential Calculus and trace norm estimates, we derive $$ \|f(H_{0,\alpha})\vp(x,D_x)\|_1\ \le\ C\alpha^A $$ for any $A\ge 0$. The last four estimates yield (4.39). $\square$ \medskip \noindent {\bf Remark 4.9.} We use theorem 4.8 in the following two situations: a) $\vp(x,\xi)$ is supported in $B(0,1)\times\big(\BR^d\backslash B(0,K)\big)$ and $g(\lambda)$ is smooth and supported in $|\lambda|\le K_1$ with $K_1$ and $K$ related as $$ K\ =\ (K_1 +1) [(\sup_{B(0,2)} W)_+ + 2]^{1\over2}\ . $$ b) $\vp(x,\xi)=\psi(x)$ with $\psi(x)\in C_0^\infty\big( B(0,1) \big)$ and $g(\lambda)$ is supported in $(-\infty,-\sup_{B(0,\rho)} W)$ for some $\rho>1$. Finally, we present the following rough estimate \proclaim Theorem 4.10. Assume $W$ obeys (4.1) and (4.38). Let $g$ be a piecewise continuous function on ${\BR}$ obeying $ | g(\lambda) | \le C \langle \lambda \rangle^m $ for some $ m $ and $\sup({\rm supp}\, g)<\infty$. Then for any $\psi\in C_0^\infty \big( B(0,1)\big)$ $$ \|\psi g(H_\alpha)\|_1\ \le\ C\alpha^{-d}\ .\eqno (4.41) $$ \par {\bf Proof.} Without a loss of generality one can assume $g$ to be smooth. Let $W_0$ be a $C_0^\infty$ potential satisfying (4.40). Introduce $H_{0,\alpha}=-{1\over2}\alpha^2\Delta - W_0(x)$. Theorem 4.6 implies that $$ \| \psi\big( g(H_\alpha)-g(H_{0,\alpha})\big)\|_1\ \le\ C_A\alpha^A\eqno (4.42) $$ for any $A\ge 0$. On the other hand, it is straightforward to show that for $n>{d\over2}$ $$ \| \psi(-\alpha^2\Delta+i)^{-n}\|_1\ \le\ C\alpha^{-d}\eqno (4.43) $$ and then for any $n\ge 0$ $$ \| (-\alpha^2\Delta+i)^n(H_{0,\alpha}+i)^{-n}\|\ \le\ C\ .\eqno (4.44) $$ The last two estimates yield $$ \|\psi g(H_{0,\alpha})\|_1\ \le\ C\alpha^{-d}\ .\eqno (4.45) $$ Equations (4.42) and (4.45) yield (4.41). $\square$ \vfill\eject \beginsection 5. Approximate Evolution In this section and the next one we study behaviour of the evolution group $$ U(t)\ = \ e^{-iH_\alpha t/\alpha}\eqno (5.1) $$ for small times. To this end we approximate $U(t)$, in the relevant part of the phase space, by a family of $F(t)$, of the Fourier integral operators in a spirit of geometrical optics and then estimate $F(t)$ by the stationary phase method (see [Lax 1957, Kell 1958, Masl 1965, H\"orm 1968, Kum 1981, Chaz 1980]). In this section we construct $F(t)$ and study its properties. An important point here is that due to simple microlocal estimates of the previous section, $F(t)$ has a $C_0^\infty$ symbol. This makes the investigation of $F(t)$ an exercise in Calculus. We want to construct a Fourier integral operator $$ F(t)u\ = \ \alpha^{-d}\int\int e^{i\big(S(t,x,\xi)-z\cdot\xi\big)/\alpha} a(t,x,\xi,\alpha)u(z) dz d\xi \ , \eqno (5.2) $$ satisfying (possibly, modulo terms supported outside $B(0,{6\over5})$) $$ (D_t-H)F\ = \ O(\alpha^A) $$ for some $A>0$ sufficiently large and $$ F(0)\ = \ \vp(x,D_x)\ ,\eqno (5.3) $$ where $\varphi\in C_0^\infty \big( B(0,{5\over4})\times B(0,K)\big)$, with a fixed $K$ chosen later, and is even in $\xi$. Note that after taking the Fourier integral in $z$, the remaining $\xi$-integral in (5.2) is absolutely convergent (for sufficiently small times, see the paragraph before eqn (5.10)). Let $$ h(x,\xi)\ = \ {1\over2}|\xi|^2 - W(x)\ . $$ Clearly, estimating $(D_t-H)F(t)$ reduces to evaluating $(D_t-H)(ae^{{iS\over\alpha}})$. Taking $S$ independent of $\alpha$ and equating the coefficient at $\alpha^0$ in this expression to $0$ leads to $$ \partial_tS+h(x,\partial_xS)\ = \ 0 \ , \eqno(5.4) $$ the Hamilton-Jacobi equation. The initial condition is $$ S|_{t=0}\ = \ x\cdot\xi\ .\eqno (5.5) $$ Next, picking $a$ of the form $$ a\ = \ \sum_{j=0}^N \alpha^j a_j (t,x,\xi)\eqno (5.6) $$ with $N=M-1$ and equating the coefficients in $(D_t - H)(ae^{{iS\over\alpha}})$ at $\alpha^j$, $j=1,\ldots,N+1$, to zero leads to the transport equations $$ \partial_t a_j + \partial_x S \cdot\partial_x a_j + {1\over 2} \Delta_x S a_j \ = \ - {i\over 2} \Delta_x a_{j-1} \eqno (5.7) $$ for $j\ge 0$ and with $a_{-1}=0$ and with the initial conditions $$ a_j|_{t=0}\ = \ \delta_{j,0}\vp\ .\eqno (5.8) $$ The initial conditions yield $$ ae^{iS/\alpha}|_{t=0}\ = \ \varphi e^{ix\cdot\xi/\alpha}\ ,\eqno (5.9) $$ which guarantees, due to the relation $$ \alpha^{-d}\int\int e^{i(x-z)\cdot\xi/\alpha} \varphi(x,\xi) u(z) dz d\xi\ = \ \varphi(x,D_x)u(x)\ , $$ that $F(0)=\vp(x,D_x)$. The Hamilton-Jacobi equations (5.4)--(5.5) and the transport equations (5.7)--(5.8) can be solved by the method of characteristics (see e.g. [Arn 1989, Chaz 1980, Kum 1981]). For instance, the unique solution of the Hamilton-Jacobi equations is given for $|t|$ sufficiently small by the action function $$ S(t,x,\eta)\ = \ z\cdot\eta + \int (\xi\cdot dx-h) $$ along the classical trajectory for the Hamiltonian $h(x,\xi)$ with the momentum $\eta$ at time $s=0$ and the position $x$ at time $s=t$. Here $z$ is the position at time $s=0$ as a function of $t$, $x$ and $\eta$. Fix the degree of approximation $N$. Let $T_1>0$ (depending of $K$) be such that the Hamiltonian flow for $h(x,\xi)$ starting in $B(0,{3\over2})\times B(0,K+2)$ (resp. $B(0,{5\over4})\times B(0,K)$) exists for $|t|\le T_1$ and stays during this period inside of $B(0,2)\times \BR^d$ (resp. $B(0,{4\over3})\times B(0,K+1)$). Then decreasing $T_1$ depending on $N$, if necessary, we conclude that (5.4)--(5.5) and (5.7)--(5.8) with $j\le N$ have unique solutions in $B(0,{3\over2})$ for $\xi\in B(0,K+2)$ and $a_j\in C_0^\infty\big( B(0,{4\over3})\times B(0,K+1)\big)$ for $|t|\le T_1$. Hence $a\in C_0^\infty\big( B(0,{4\over3})\times B(0,K+1)\big)$ for $|t|\le T_1$. Thus $F(t)$ is well defined and obeys (5.3). The expression $(D_t-H)F(t)$ will be estimated in the next section. In this section we study the asymptotic behaviour of $F(t)$. In what follows we assume that $T\le T_1$. Let $\hat\chi\in C_0^\infty ([-T,T])$. Then the operator $$ F_\chi(\lambda)\ = \ \int F(t)\hat\chi(t) e^{i\lambda t/\alpha} dt\eqno (5.10) $$ is well-defined and, due to the fact that $a\in C_0^\infty$, is trace class. \proclaim Theorem 5.1. If $\lambda<-\sup\limits_{B(0,2)}W-1$, then for any $A\ge 0$ and uniformly in $\lambda$ $$ \| F_\chi(\lambda)\|_1\ =\ O(\alpha^A)\ .\eqno (5.11) $$ Let $\eta$ be smooth and bounded on $\BR^d$ and supported in $|\xi|\ge K_0$, where $K_0\ge 2(\sup\limits_{B(0,2)} W+\lambda+1)_+^ {1\over2}$, and let $T\le \min(\delta K_0^{{1\over2}},T_1)$ with $\delta$ sufficiently small and independent of $K_0$ and $K$. Then for all $\lambda$ $$ \| F_\chi(\lambda)\eta(D_x)\|_1\ =\ O(\alpha^A)\eqno (5.12) $$ for any $A\ge 0$. \par {\bf Proof.} The integral kernel of $F_\chi(\lambda)\eta (D_x)$ is $$ \int\int\hat\chi a\eta e^{i\phi_1/\alpha}d\xi dt $$ where $\phi_1=S-z\cdot\xi+\lambda t$ and the remaining abbreviation is obvious. The Hamilton-Jacobi equation for $S$, the relation $\partial_x S= \xi + O(t)$ and the condition $T\le \delta K_0^{1\over2}$ imply $$ -\partial_t S\ \ge\ {1\over4} K_0^2 - \sup_{B(0,2)} W $$ and therefore $-\partial_t\phi_1 \ge 1$ on ${\rm supp}(\hat\chi a\eta)$. Integrating by parts using the relation $$ Le^{i\phi_1/\alpha}\ =\ e^{i\phi_1/\alpha}\ , $$ where $$ L\ = \ -(\partial_t S + \lambda)^{-1} D_t\ , $$ we obtain (5.12). Equation (5.11) is proven similarly if one observes that $$ -\partial_t\phi_1\ \ge\ -\lambda - \sup_{B(0,2)} W $$ on ${\rm supp}\, a$. $\square$\hfill\medskip Let $\chi\in C_0^\infty ([-T,T])$ and $\psi\in C^\infty ({\Bbb R}^d)$ and consider the function $$ I(\lambda,\alpha)\ = \ {\rm tr}\,\psi\int F(t)\hat\chi (t) e^{i\lambda t/\alpha} dt\ .\eqno (5.13) $$ Due to theorem 5.1, for any $A\ge 0$ $$ I(\lambda,\alpha)\ = \ O\biggl( \bigl({\alpha\over|\lambda|}\bigr)^A \biggr)\qquad {\rm if}\quad \lambda <-\sup_{B(0,1)} W-1\ .\eqno (5.14) $$ Using the Fourier integral representation of $F(t)$, we rewrite $I(\lambda,\alpha)$ as $$ I(\lambda,\alpha)\ = \ \alpha^{-d} \int\int\int e^{i\phi/\alpha} b\, dtdxd\xi\ ,\eqno (5.15) $$ where $$ b\ = \ \psi\hat\chi a $$ and $$ \phi(t,x,\xi)\ = \ S(t,x,\xi) - x\cdot\xi + \lambda t\ .\eqno (5.16) $$ In the rest of this section $y$ stands for a phase-space point $(x,\xi)\in\BR^{2d}$ and $dy=dxd\xi$. Let $\cE_\lambda=\{ y| h(y)=\lambda\}$, the energy shell at the energy $\lambda$. If $\nabla h\ne 0$ on $\cE_\lambda\cap{\rm supp}\,\vp$, then there is $\tau>0$ (depending, in general, on the restriction of $|\nabla h|^{-1}$ to $\cE_\lambda\cap{\rm supp}\,\vp$) s.t. no periodic orbit of $h$ with a period in $(0,\tau]$ passes through points of ${\rm supp}\,a$. This well-known statement follows from the beginning of the proof of theorem 5.2 below (see the paragraph containing eqn (5.23)). The next result is rather standard, we give its proof for the sake of completeness. \proclaim Theorem 5.2. Let $\hat\chi$ be a smooth function, supported in $[-\tau,\tau]$ and obeying $\hat\chi(0)=1$ and $\hat\chi^{(k)}(0)=0$ for $k=1,2$ and let $\varphi\in C_0^\infty \big( B(0,{5\over4})\times B(0,K)\big)$. Let $\lambda$ be a regular value of $h$ on $\supp a$. Then $$ I(\lambda,\alpha)\ = \ 2\pi \alpha^{1-d} \int_{\cE_\lambda}\psi\vp dS_\lambda + O(\alpha^{3-d})\ ,\eqno (5.17) $$ where $dS_\lambda$ is the $h$-induced area element of the surface $\cE_\lambda$, i.e. $dS_\lambda= |\nabla h|^{-1}\times$ (Riemannian surface measure on $\cE_\lambda$).\par {\bf Proof.} In order to prove (5.17) we use the method of stationary phase. We begin with a study of the critical manifold of $\phi$. First, we claim that if $\nabla h\ne0$ on $\cE_\lambda\cap{\rm supp}\, a$ and if $t$ is sufficiently small, then the critical manifold of $\phi$ is $$ C_\lambda\ = \ \{0\}\times\cE_\lambda\ . \eqno (5.18) $$ Indeed, since $$ \phi|_{t=0}\ = \ 0\ ,\eqno (5.19) $$ we have that $$ \partial_y\phi|_{t=0}\ =\ 0\ .\eqno (5.20) $$ Next, by the Hamilton-Jacobi equation $$ \eqalign{ \partial_t\phi\ &=\ \partial_t S+\lambda\cr &=\ -h(x,\partial_xS)+\lambda\ .\cr} $$ This and the initial condition $S|_{t=0}=x\cdot\xi$ yield $$ \partial_t\phi |_{t=0}\ = \ -h + \lambda\ .\eqno (5.21) $$ Hence $$ \partial_t\phi\ = \ 0\quad {\rm on}\quad C_\lambda\ . $$ Using (5.21), the fact that $\phi$ is smooth in all its variables on ${\rm supp}\, a$ and that $\supp a$ is compact, we obtain $$ \partial_t\phi\ =\ -h + \lambda + O(t) $$ on ${\rm supp}\, a$. Here and below $O(t^n)$ stands for a smooth symbol of the order $|t|^n$ on ${\rm supp}\, a$. Moreover, (5.21) yields $$ \partial_{yt}^2\phi\bigm|_{t=0}\ =\ -\nabla h\ , \eqno (5.22) $$ which together with (5.20) implies that $$ \partial_y\phi\ = \ -\nabla ht + O(t^2) $$ on ${\rm supp}\, a$. Hence if $|t|$ is sufficiently small, then (5.18) exhausts all the critical points of $\phi$ on ${\rm supp}\, a$. To see how large $|t|$ can be taken, we study the critical points of $\phi$ more carefully. Due to (5.16), the critical points $(t,x,\xi)$ of $\phi$ obey the equations $$ \eqalign{ \partial_tS \ &=\ -\lambda\cr \partial_xS \ &=\ \xi\cr \partial_\xi S\ &=\ x\ .\cr}\eqno (5.23) $$ Since $S$ satisfies the Hamilton-Jacobi equation with the Hamiltonian function $h$ and the initial condition $S|_{t=0}=x\cdot\xi$, we have that $$ \nabla_{(x,\xi)} S(t,x,\xi)\ = \ (\eta,z)\ ,\quad \hbox{provided}\ \phi_t(z,\xi)\ = \ (x,\eta)\ , $$ where $\phi_t$ is the flow generated by the Hamiltonian function $h$. Hence if $(t_0,y_0)$ is a solution of (5.23), then $t_0$ is a period of a periodic orbit of $\phi_t$ passing through $y_0$with the energy $$ h(y_0)\ = \ -(\partial_tS)(t_0,y_0)\ = \ \lambda \ . $$ Thus, we have shown that (i) the critical manifold of $\phi$ is the union of $C_\lambda$ and the set of all points $(t,y)$ s.t. $t\ne 0$, $y\in\cE_\lambda$ and there is a periodic orbit through $y$ which has the period $t$ and (ii) if $\lambda$ is not a critical value of $h$ on ${\rm supp}\, a$, then there is an interval $I$ around $t=0$ (depending on ${\rm supp}\, a$) s.t. the critical manifold of $\phi$ on $I\times{\rm supp}\, a$ is $C_\lambda\cap(I\times{\rm supp}\, a)$. In particular, this implies that under the above condition on $h$ and $\lambda$ there are no periodic orbits of $\phi_t$ through ${\rm supp}\, a$ with sufficiently small periods and therefore $\tau$ defined in the paragraph before theorem 5.2 is positive and $C_\lambda$ is the critical manifold of $\phi$ on ${\rm supp}\,\hat\chi\times {\rm supp}\, a$. Next, we compute the Hessian, $\phi''$, of $\phi$ on the critical manifold (5.18). First, we note that $$ \partial_t^2\phi|_{t=0}\ =\ -\xi\partial_xW\ .\eqno(5.24) $$ Indeed, using twice that $S$ is a solution to the Hamilton-Jacobi equation, we obtain $$ \eqalign{ \partial_t^2S\ &=\ -\partial_t\left( {1\over2}(\partial_xS)^2-W(x) \right)\cr &=\ -\partial_xS\partial_{xt}^2 S\cr &=\ \partial_xS\partial_x\left( {1\over2}(\partial_xS)^2-W(x)\right)\cr &=\ \partial_xS({\rm Hessian}_xS)\partial_xS-\partial_x S\partial_x W\ .\cr} $$ Applying now the initial condition $S|_{t=0}=x\cdot\xi$ and remembering that $\partial_t^2\phi=\partial_t^2S$, we arrive at (5.24). Next, since $\phi|_{t=0}=0$, we get that $$ \partial_{yy}^2\phi |_{t=0}\ = \ 0\ . $$ This relation together with (5.22) and (5.24) yield $$ \phi''|_{t=0}\ = \ - \pmatrix{ \xi\cdot\nabla W&\nabla h\cr (\nabla h)^T&0\cr}\ ,\eqno (5.25) $$ where $\nabla h$ and $(\nabla h)^T$ stand for row and column vectors, respectively, and $0$, for the $2d\times 2d$ zero matrix. A simple computation shows that $\phi''(\sigma)$ with $\sigma\in C_ \lambda$ is non-degenerate on $N_\sigma\equiv T_\sigma\BR^d \ominus T_\sigma C_\lambda$. Finally, the determinant of the restriction of $\phi''|_{t=0}$ to $N_\sigma$ is $$ \det_N(\phi|_{t=0})\ = \ -|\nabla h|^2 \eqno (5.26) $$ and the signature of this restriction, which is the difference between the number of positive and negative eigenvalues, is $$ \sign_N(\phi''|_{t=0})\ = \ 0\ . $$ Since the restriction of $\phi''(\sigma)$, $\sigma\in C_\lambda$, to $N_\sigma$ is non-degenerate, a stationary phase theorem is applicable to the integral (5.15) and it produces (see Appendix) $$ \eqalign{ &I(\lambda,\alpha)\ = \ 2\pi\alpha^{1-d} \sum_{k=0}^{N-1} k!^{-1} \int_{\cE_\lambda} \left[ |\nabla h|^{-1}\left( {i\over2} \alpha L\right)^k (b\rho e^{i\theta/\alpha}) \right]_{C_\lambda} dS_\lambda\cr &\qquad\qquad + O(\alpha^{N+1-d})\ ,\cr} \eqno (5.27) $$ where, with $\phi''(\sigma)^{-1}$ standing for the inverse of the restriction of $\phi''(\sigma)$ to $N_\sigma$, $\sigma\in C_\lambda$, $$ L\ = \ \langle \phi''(\sigma)^{-1} \nabla,\nabla\rangle\ ,\eqno (5.28) $$ with $\nabla$, the gradient in $t$ and $y$, $\rho$ is a smooth function of $x$ and $|\xi|$ obeying $$ \rho\ = \ |\nabla h|\qquad {\rm on}\qquad \cE_\lambda\ , \eqno (5.29) $$ $dS_\lambda$ is the $h$-induced surface measure on $\cE_\lambda$ ($=|\nabla h|^{-1} \times$ the natural surface measure on $\cE_\lambda$), $$ \theta\ = \ \phi(z)-\phi(\sigma)-{1\over2}\langle z-\sigma ,\phi'' (\sigma)(z-\sigma)\rangle \eqno (5.30) $$ with $z=(t,y)\in N_\sigma$ and $\sigma\in C_\lambda$. Here we have identified $C_\lambda$ with $\cE_\lambda$. We will use the expansion above for $N=2$ and we compute explicitly its first two terms. Since $$ b|_{t=0}\ = \ \psi\varphi\eqno (5.31) $$ and $\rho=|\nabla h|$ on $\cE_\lambda$ and $$ \theta |_{C_\lambda}\ = \ 0\ , $$ we find for the $k=0$ term $$ \rho be^{i\theta/\alpha} |_{C_\lambda}\ = \ \psi\varphi |\nabla h|\ .\eqno (5.32) $$ Next we compute the $k=1$ term. Since $$ \theta\ = \ O\big( (z-\sigma)^3\big)\ ,\eqno (5.33) $$ for $\sigma\in C_\lambda$, we have that $$ L(b\rho e^{i\theta/\alpha})\ = \ L(b\rho)\qquad {\rm on}\qquad C_\lambda \ . \eqno (5.34) $$ Since $\hat\chi^{(k)}(0)=0$ for $k=1,2$, we find $$ L(b\rho)\ = \ L(\psi a\rho) \qquad {\rm at}\qquad t=0\ . $$ Next, we use that $$ a\ = \ a_0 + O(\alpha)\ , $$ where $a_0$ obeys the first of transport equations (5.7)--(5.8) and $O(\alpha)$ stands for a smooth, compactly supported symbol of order $O(\alpha)$ (see equation (5.6)). The last two equations yield $$ L(b\rho)\ = \ L(\psi a_0\rho) + O(\alpha) \eqno (5.35) $$ at $t=0$. Of course, this estimate remains valid after restriction to $\cE_\lambda$ and integration over it. Next we use the transport equation for $a_0$ $$ \partial_t a_0 + \partial_x S\cdot\partial_x a_0 + {1\over2} \Delta_x S a_0\ = \ 0 \eqno (5.36) $$ and the initial condition $$ a_0|_{t=0}\ = \ \vp \ .\eqno (5.37) $$ These two relations imply $$ \partial_t a_0|_{t=0}\ = \ -{1\over2} \Delta_x S|_{t=0}\vp\ . $$ This together with $S|_{t=0}=x\cdot\xi$ gives $$ \partial_t a_0|_{t=0}\ = \ 0\ .\eqno (5.38) $$ Now we need the explicit form of $L$. It is easy to check that $$ \phi''(\sigma)^{-1}\ =\ \pmatrix{ 0&-\nabla h |\nabla h|^{-2}\cr -(\nabla h)^T|\nabla h|^{-2}&\xi\cdot\nabla W|\nabla h|^{-4}\nabla h\otimes \nabla h\cr} $$ where $\nabla h\otimes\nabla h$ stands for the matrix $(\partial_{y_i} h\partial_{y_j} h)$. Plugging this into (5.28) gives $$ L\ = \ -2 {\nabla h\over |\nabla h|^2}\cdot\partial_y\partial_t + {\xi\cdot\nabla W\over |\nabla h|^4} \langle (\nabla h\otimes \nabla h)\nabla,\nabla\rangle \ .\eqno (5.40) $$ Using this expression together with (5.37) and (5.38), we find $$ L(\psi\rho a_0)\bigm|_{t=0}\ =\ {\xi\cdot\nabla W\over |\nabla h|^4} \langle (\nabla h\otimes\nabla h)\nabla,\nabla\rangle (\psi\varphi\rho) \ . \eqno (5.41) $$ Combining equations (5.34), (5.35) and (5.41), we obtain $$ L(b\rho e^{i\theta/\alpha})\ = \ \xi\cdot\nabla Wj + O(\alpha) \eqno (5.42) $$ on $C_\lambda$, where $$ j\ =\ |\nabla h|^{-4} \langle (\nabla h \otimes\nabla h)\nabla,\nabla\rangle (\psi\varphi\rho)\ .\eqno (5.43) $$ Note now that $$ j(x,-\xi)\ = \ j(x,\xi)\eqno (5.44) $$ and the same property holds for $|\nabla h|$ and $h$. Hence $$ \int_{\cE_\lambda} (\xi\cdot\nabla W j|\nabla h|^{-1})_{\cE_\lambda} dS_\lambda \ = \ 0 \ .\eqno (5.45) $$ Plugging (5.32) and (5.42) into (5.27) and taking into account (5.45), we find (5.17). $\square$\hfill \vfill\eject \beginsection 6. Estimates of the Evolution In this section we estimate the difference $U(t)-F(t)$, between the true evolution operator $U(t)$ and its approximation $F(t)$. After that, using estimates on $F(t)$, derived in the previous section, we estimate $U(t)$. The estimates on $U(t)$ obtained in this section are used in the next section in order to find asymptotic behaviour of the spectral projections $E(\lambda,H_\alpha)$. Our arguments follow closely those of [Chaz 1980]. Recall that $\vp\in C_0^\infty\big( B(0,2)\times B(0,K)\big)$ is the cut-off function entering the construction of $F(t)$ and $N$ is the order of approximation used in this construction (see eqn (5.6)). \proclaim Lemma 6.1. Let $\vp_1\in C_0^\infty$ and be supported in $\{ \vp=1\}$ and let ${\rm supp}\,\vp_1$ be disjoint from ${\rm supp}(\nabla_x\vp)$. Then $$ \sup_{|t|\le T} \| \big( F(t)-U(t)\big)\phi_1\|_1\ \le\ C\alpha^{N+1-d}\ , \eqno (6.1) $$ where $\phi_1=\varphi_1(x,D_x)$, provided $T$ is sufficiently small.\par {\bf Proof.} Introduce $$ r\ \equiv\ e^{-iS/\alpha} (D_t - H) (e^{iS/\alpha} a)\ , \eqno (6.2) $$ where $S$ and $a$ obey (5.4)--(5.8). Performing differentiations and using equations (5.4) and (5.7), we arrive at $$ r\ = \ {1\over2} \alpha^{N+2} \Delta_x a_N\ .\eqno (6.3) $$ Let $$ R(t) u\ = \ \alpha^{-d}\int\int e^{i(S-z\cdot\xi)/\alpha} ru dz d\xi\ . \eqno (6.4) $$ Since $a_N$ is smooth and compactly supported, we have $$ \sup_{|t|\le T} \|\nabla_x^n R(t)\|_1\ \le\ C\alpha^{N+2-d-n}\eqno (6.5) $$ where $T$ is the time till which the Hamilton-Jacobi and transport equations were solved. Now observe that, due to (6.2), $F(t)$ obeys $$ (D_t - H) F(t)\ = \ R(t)\ .\eqno (6.6) $$ Moreover, $$ F(0)\ = \ \vp(x,D_x)\ .\eqno (6.7) $$ Consequently, the family $G(t)=F(t)-U(t)$ obeys the equation $$ (D_t-H)G(t)\ = \ R(t)\eqno (6.8) $$ and $$ G(0)\ = \ -\bar\phi \ , \eqno (6.9) $$ where $\bar\phi={\bf 1}-\vp(x,D_x)$. Integrating out these equations, we obtain $$ G(t)\ = \ {1\over\alpha i}\int_0^t U(t-s) R(s)ds - U(t) \bar\phi \ , $$ which together with (6.5) for $n=0$ and the relation $\overline\phi\cdot\phi_1=O(\alpha^M)$ for any $M$, which follows from $(1-\vp)\cdot\vp_1=0$, gives (6.1). $\square$ \proclaim Theorem 6.2. Let $\vp=1$ on $B(0,{3\over2})\times B(0,K-1)$ and let $\nabla_x\vp$ be supported in $|x|\ge{3\over2}$. Let $K=2(\sup\limits_{B(0,2)}W)_+^{1\over2}+3$ and let $\psi\in C_0^\infty\big( B(0,1)\big)$. Then $$ \eqalign{ &\|\int\theta(t)\big( F(t)-U(t)\big) dt\psi\|_1\cr \le\ &C\alpha^{N-2d}\cr} \eqno (6.10) $$ with the constant independent of $\alpha$, provided $\theta\in C_0^\infty([-T,T])$ with $T$ sufficiently small. \par {\bf Proof.} In this proof we omit the argument $t$ and replace the symbol $\int\limits_{-\infty}^\infty dt$ by $\int$. Let $\eta\in C_0^\infty\big( B(0,K-{3\over2})\big)$ and $=1$ on $B(0,K-2)$. By (5.12) $$ \| \int\theta F\bar\eta (D_x)\|_1\ \le\ C\alpha^A\ , \eqno (6.11) $$ where $\bar\eta =1-\eta$, for any $A\ge 0$. By lemma 6.1 $$ \| \int\theta (F-U) \eta(D_x)\psi \|_1\ \le\ C\alpha^{N-d}\ . \eqno (6.12) $$ Now we consider the term $\int\theta U\bar\eta (D_x)$. If $\check\theta^{\rm normal}(\lambda)={1\over2\pi} \int e^{it\lambda}\theta(t)dt$ is the standard inverse Fourier transform of $\theta(t)$ (to distinguish from the $\alpha$-Fourier transform used in the next section), then $$ \int\theta U\ = \ \check\theta^{\rm normal} (-H_\alpha/\alpha)\ . \eqno (6.13) $$ Since $|\check\theta^{\rm normal}(\lambda)|\le C\langle\lambda\rangle^ {-A}$ for any $A\ge 0$, we have that $\bigl({H_\alpha\over\alpha}\bigr)^ A\check\theta^{\rm normal}\bigl( -{H_\alpha\over\alpha}\bigr)$ is bounded for any $M\ge 0$. On the other hand, for $\bar g(\lambda)$, smooth, bounded and $=0$ for $|\lambda|\le{1\over2}$, $H_\alpha^{-M}\bar g(H_\alpha)$ is bounded for any $M\ge 0$. The last two relations together with (4.5) yield that $$ \|\int\theta U\bar g(H_\alpha)\psi\|_1\ \le\ C\alpha^A\eqno (6.14) $$ for any $A\ge 0$. Let $\bar g(\lambda)=1$ for $|\lambda|\ge 1$ and set $g(\lambda)=1-\bar g(\lambda)$. By theorem 4.3 and by our choice of $K$ we have $$ \|\psi(x)\bar\eta(D_x)g(H_\alpha)\|_1\ = \ O(\alpha^A) $$ for any $A>0$. This together with (6.14), implies $$ \|\psi \bar\eta (D_x) \int \theta U\|_1\ = \ O(\alpha^A) $$ for any $A\ge 0$. This together with (6.11) and (6.12) yields (6.10). $\square$ \bigskip Replacing in theorem 6.2 $H_\alpha$ by $H_\alpha-\lambda$ and remembering definition (5.13), we obtain \proclaim Corollary 6.3. Let $\psi\in C_0^\infty\big( B(0,1)\big)$ and $\varphi(x,\xi)=\psi_1(x)\eta(\xi)$ in (5.8) with $\psi_1=1$ on ${\rm supp}\,\psi$, $\eta\in C_0^\infty\big( B(0,K)\big)$ and $K\ge2(\sup\limits_{B(0,2)}W+\lambda)_+^{1\over2} +3$. Let $\hat\chi$ be a smooth function supported in a sufficiently small neighbourhood of $t=0$. Then $$ \eqalign{ &|{\rm tr}\big(\psi\int\hat\chi(t)U(t) e^{it\lambda/\alpha}dt\big) - I(\lambda,\alpha)|\cr &\qquad\le\ \const\,\alpha^{N-d}\ ,\cr} $$ where $N$ is the same as in (5.6) and $I(\lambda,\alpha)$ is defined with the $\psi,\vp$ and $\hat\chi$ as above.\par \vfill\eject \beginsection 7. Estimates of the Local Traces In this section we estimate $\tr\big(\psi(x) g(H_\alpha)\big)$, where $\psi\in C_0^\infty\big(B(0,1)\big)$ and $g$ is a smooth function on $\BR\backslash\{0\}$ with a compact support. Due to the presence of a cut-off function $\psi$ we call such a trace a local trace. In our approach we express $g(H_\alpha)$ in terms of the evolution $U(t)$ and use the information about the latter derived in the previous section. Since we have a good control of $U(t)$ for $|t|\le T$ and $T=O(1)$ we can estimate smooth functions of $H_\alpha$ supported on the scale $O(\alpha)$ (uncertainty principle). Using this and a more or less standard Tauberian technique, we estimate non-smooth functions of $H_\alpha$ with an error term related to the degree of their non-smoothness. The main result of this section is theorem 7.1. In what follows we will use the following $\alpha$-dependent Fourier transform $$ \hat f(t)\ = \ \int e^{-it\lambda/\alpha} f(\lambda)d\lambda\ . $$ For the standard Fourier transform $(\alpha=1)$ we reserve the notation $$ \hat f^{\rm normal}(t)\ =\ \int e^{-it\lambda} f(\lambda) d\lambda\ . $$ We will use the following Fourier representation for functions of $H_\alpha$: $$ g(H_\alpha)\ =\ {1\over 2\pi\alpha} \int \hat g(-t)U(t)dt\ , \eqno (7.1) $$ where $\hat g(t)$ is the $\alpha$-Fourier of $g(\lambda)$. $E(\lambda,H_\alpha)$ will stand for the spectral projection of $H_\alpha$ corresponding to the interval $(-\infty,\lambda)$. We define also the local counting function $$ e(\lambda,\psi,H_\alpha)\ = \ {\rm tr}\big( \psi E(\lambda, H_\alpha)\big)\ .\eqno (7.2) $$ In the rest of this section we keep the subindex $\alpha$ at $H_\alpha$ in the theorems and omit it in the proofs. Consider the following class of functions: $g(\lambda)$ is smooth on $\BR\backslash\{0\}$, has a compact support and obeys for some $s\in [0,1]$, for some $\mu>0$ and for all $\nu>0$ $$ b(\nu,g)\ \equiv \ \int_{|\sigma|\le3\nu} |g'(\sigma)|d\sigma + \nu \int_{|\sigma|\ge\nu} |g''(\sigma)|d\sigma\ \le\ (M\nu)^s\ . \eqno (7.3) $$ Let $b(g)=\sup\limits_{\nu>0}\big( \nu^{-s} b(\nu,g)\big)$. We begin with \proclaim Theorem 7.1. Let $g$ be as specified above with $0\le s\le 1$ and $M>0$ (see (7.3)). Let $\psi\in C_0^\infty\big( B(0,1)\big)$ and let $0$ be a regular value of the function $h$ restricted to $\supp\psi\times\BR^d$. Then for $\alpha\le 1$ $$ \eqalign{ \tr\big(\psi g(H_\alpha)\big)\ = \ &\alpha^{-d}\int\int \psi(x)g\big( h(x,\xi)\big)dx d\xi\cr &+ O(b(g)\alpha^{s+1-d})\ .\cr}\eqno (7.4) $$ \par {\bf Proof.} Let $\chi$ be a symmetric function, $\chi(-\lambda)=\chi(\lambda)$ whose $\alpha$-Fourier transform, $\hat\chi$, is smooth and obeys $$ \hat\chi(0)=1\qquad {\rm and}\qquad {\rm supp}\,\hat\chi\subset [-T;T]\eqno (7.5) $$ with $T\le\tau$. (For the definition of $\tau$ see the paragraph after eqn (5.16).) Function $\chi$ will serve as a $\delta$-approximation supported on the scale $O(\alpha)$. It is of the form $\chi(\lambda)={1\over\alpha}\chi_1\Big( {\lambda\over\alpha}\Big)$, where $\chi_1$ is a function whose standard ($\alpha$-independent) Fourier transform satisfies (7.5). To probe different energies we use $$ \eqalign{ {1\over 2\pi\alpha}\int\hat\chi(t) U(t)e^{i\lambda t/\alpha}dt \ &=\ \chi(\lambda-H_\alpha)\cr &=\ \chi * dE\ .\cr}\eqno (7.6) $$ This equation, Corollary 6.3 and Theorem 5.2 yield $$ \chi * de - de_0 \ = \ O(\alpha^{2-d})\ ,\eqno (7.7) $$ at a non-critical value of $h$ on ${\rm supp}\,\psi\times B(0,K)$. Here $K\ge 2(\sup\limits_{B(0,2)}W+\lambda)_+^{1\over2}+3$ and $$ e_0(\lambda,\psi,H_\alpha)\ = \ \alpha^{-d}{\int\int}_ {h(x,\xi)\le\lambda} \psi(x) dx d\xi\ .\eqno (7.8) $$ Integrating (7.7) from $-\sup\limits_{B(0,1)}W-1$ to $\lambda$ and using theorem 4.9, we obtain $$ \chi *e - e_0 \ = \ O(\alpha^{2-d})\ ,\eqno (7.9) $$ provided $(-\infty,\lambda)$ contains no critical values of $h$ on ${\rm supp}\,\psi\times B(0,K)$. Note here that making an argument slightly longer we could use a special case of simple inequality (6.14) instead of the more sophisticated theorem 4.4. \proclaim Lemma 7.2. Let $\chi$ and $g$ be as above and let $\varphi (\lambda)=-g'(-\lambda)$. Then $$ \varphi * (\chi * e - e) \ = \ O(\alpha^{1+s-d})\ . \eqno (7.10) $$ \par {\bf Proof.} It suffices to prove (7.10) for $\psi\ge 0$. Let $\theta$ be a real smooth positive function with $\int\theta=1$ whose $\alpha$-Fourier transform $\hat\theta(t)$ is supported in $(-\tau,\tau)$. As before define $\theta_1$ by $\theta(\lambda)={1\over\alpha}\theta_1 \left( {\lambda\over\alpha}\right)$. (4.41) implies $$ \theta * d_\lambda e\ \le\ C\alpha^{-d}\ . \eqno (7.11) $$ By the definition of $\theta$, there are $C$ and $\ve>0$ s.t. $$ C\alpha \theta(\lambda)\ \ge\ \eta(\lambda) -\eta (\lambda-\ve\alpha)\ , \eqno (7.12) $$ where $\eta(\lambda)$ is the characteristic function of the interval $(-\infty,0]$. The last two inequalities together with (7.6) imply $$ e(\mu)-e(\mu-\ve\alpha)\ \le\ C\alpha^{1-d}\ .\eqno (7.13) $$ Here and in the rest of the proof we use the shorthand $e(\mu)=e(\mu,\psi,H)$. Let $[\nu,\mu]$ contain no critical values of $h$. Decomposing the interval $[\nu,\mu]$ into ${|\mu-\nu|\over\ve\alpha}$ subintervals of the length $\ve\alpha$ and applying (7.13) to each of these subintervals, we obtain $$ \eqalign{ &|e(\mu)-e(\nu)|\cr &\qquad\le\ C\alpha^{1-d}\left( {|\mu-\nu|\over\alpha}+1\right)\ .\cr}\eqno (7.14) $$ Let $e_\varphi=\varphi *e$. Then $$ \eqalign{&\varphi *(\chi *e-e)\cr &\quad=\ e_\varphi - \chi * e_\varphi \ .\cr}\eqno (7.15) $$ Since $\chi(\lambda)$ is even we can write $\chi * e_\vp$ as $$ (\chi * e_\vp)(\mu)\ = \ \int_{-\infty}^\infty \chi(\lambda) \big( e_\vp (\mu+\lambda) + e_\vp (\mu-\lambda)\big) d\mu\ . \eqno (7.16) $$ Next, using that $\int\chi =1$ and that $\chi$ is even, we find $$ \eqalign{ &e_\vp (\mu) - (\chi * e_\vp) (\mu)\cr =\ &\int_0^\infty \chi(\lambda) \big( 2e_\vp(\mu) - e_\vp (\mu +\lambda) - e_\vp (\mu-\lambda)\big) d\lambda\ .\cr} \eqno (7.17) $$ Introduce the difference Laplacian $$ (\Delta_h f) (\mu)\ = \ 2f(\mu) - f(\mu+h) -f(\mu-h)\ , $$ the difference gradient $$ (\nabla_h f) (\mu)\ = \ f(\mu + h) - f(\mu) $$ and the adjoint of the latter $$ (\nabla_h^* f) (\mu)\ = \ f(\mu-h) -f(\mu)\ . $$ Then $$ \eqalign{ \Delta_h \ &=\ \nabla_h^* \nabla_h\cr &=\ \nabla_h \nabla_h^*\ .\cr} $$ Using obvious properties of $\nabla_h$, we obtain $$ \Delta_h e_\vp \ =\ (\nabla_h^* \vp ) * \nabla_h e\ . \eqno (7.18) $$ By (7.14) $$ |\nabla_h e |\ \le\ C\alpha^{1-d} \left( {|h|\over\alpha} +1\right) \ . \eqno (7.19) $$ Next, we estimate $$ \eqalign{ &\int_{-\infty}^\infty |\vp (\sigma + h) - \vp (\sigma) |d\sigma\cr \le\ &2 \int_{|\sigma|\le 3|h|} |\vp (\sigma)|d\sigma\cr &+ 2|h| \int_{|\sigma|\ge|h|} |\vp' (\sigma)| d\sigma\ .\cr} $$ Using condition (7.3), we obtain $$ \eqalign{ &\int_{-\infty}^\infty |\vp (\sigma+h)-\vp (\sigma)|d\sigma\cr \le\ &2b(|h|,g)\ .\cr} $$ This together with relations (7.18) and (7.19) yields $$ |\Delta_h e_\vp|\ \le\ C\alpha^{1-d} \left( {|h|\over\alpha} + 1\right) b(|h|,g)\ . $$ Remembering now (7.17) and using that $\chi(\lambda)={1\over\alpha}\chi_1\bigl( {\lambda\over\alpha}\bigr)$, where $\chi_1$ is the standard inverse Fourier transform of $\hat\chi$, we conclude that $$ \eqalign{ &|e_\vp - \chi * e_\vp|\cr \le\ &C\alpha^{1-d} \int_0^\infty {1\over\alpha} \left| \chi_1 \left( {\lambda\over\alpha}\right)\right| \left( {\lambda\over\alpha} +1\right) b(\lambda,g) d\lambda\cr =\ &C\alpha^{1-d+s} b(g)\ .\cr} $$ This inequality together with (7.15) yields (7.11). $\square$\hfill \medskip Now observe that $$ \varphi *(\chi *e)\ =\ \chi *(\varphi *e) $$ and $$ \eqalign{ \varphi *e\big|_{\lambda=0}\ &=\ g_- * de\big|_{\lambda=0}\cr &=\ \tr\big(\psi g (H)\big)\ ,\cr} $$ where $g_-(\lambda)=g(-\lambda)$. Moreover, with $e_0$ defined in (7.9), $$ \eqalign{ \varphi * e_0\big|_{\lambda=0}\ &=\ g_- *de_0\big|_{\lambda=0}\cr &=\ \alpha^{-d}\int\int \psi(x)g\big(h(x,\xi)\big)dxd\xi\ .\cr} $$ These relations together with equations (7.3) and (7.10) and lemma 7.2 yield \proclaim Lemma 7.3. Assume, in addition to conditions of theorem 7.1, that ${\rm supp}\, g$ contains no critical values of $h$ restricted to ${\rm supp}\, \vp$. Then (7.4) holds.\par \proclaim Remark 7.4. Lemma 7.3 would suffice for $d>1$ due to a result of the next section removing the restriction on the critical points of $h$ in this case.\par Results in the spirit of lemma 7.3 but in a smooth case were obtained in [Chaz 1980, HelffRob 1990, Hux 1988, Ivrii 1986, Rob 1987]. Now we estimate $\tr\big(\psi(x)g(H_\alpha)\big)$ in a different, more elementary way. The restrictions on $g$ are stronger now (in particular, step functions $(s=0)$ are not allowed) and the result below is weaker than (7.4), however, it has no restrictions concerning critical points of $h$. Combining this result with lemma 7.3 we will arrive at theorem 7.1. \proclaim Theorem 7.5. Let $g$ be a smooth function obeying $|\partial^n g(\lambda)|\le C_n(1+\lambda_+)^{-d} \langle\lambda\rangle^{-n}$. Let $\psi\in\BC_0^\infty \big( B(0,1)\big)$. Then $$ \eqalign{ {\rm tr}\big( \psi g(H_\alpha)\big)\ =\ &\alpha^{-d}\int\int \psi(x) g\big( h(x,\xi)\big) dxd\xi\cr &+ O(\alpha^{2-d})\ .\cr}\eqno (7.20) $$ \par {\bf Proof.} Introduce an auxiliary potential $W_0(x)$ as $W_0\in C_0^\infty(\BR^d)$ and $=W(x)$ in $B(0,2)$. Let $H_{0,\alpha}=-{\alpha^2\over 2}\Delta - W_0(x)$. Then by theorem 4.6 $$ \|\psi\big( g(H_\alpha)-g(H_{0,\alpha})\big)\|_1\ \le\ C \alpha^A\eqno (7.21) $$ for any $A$. By a standard $\alpha$-pseudo-differential Calculus $g(H_ {0,\alpha})$ is an $\alpha$-pseudo-differential operator with the symbol of the form $$ g(h_0) - {1\over2}\alpha\xi\cdot\nabla W_0 g''(h_0) + \alpha^2 r_\alpha\eqno (7.22) $$ where $h_0={1\over2}|\xi|^2 - W_0(x)$ and the symbol $r_\alpha$ obeys $$ \int\int |\psi(x) r_\alpha(x,\xi)| dxd\xi\ \le\ C\eqno (7.23) $$ uniformly in $\alpha$. Writing out the trace in terms of the symbol of $\psi(x) g(H_{0,\alpha})$ and using the fact above, we obtain $$ \eqalign{ {\rm tr} \psi g(H_{0,\alpha})\ =\ &\alpha^{-d}\int\int \psi g(h_0)\cr &- {1\over2}\alpha^{1-d}\int\int\xi\cdot\nabla W_0 g''(h_0) dxd\xi + O(\alpha^{2-d})\ .\cr}\eqno (7.24) $$ Since $h_0(x,-\xi)=h(x,\xi)$, we have $$ \int\xi\cdot\nabla W_0 g''(h_0) d\xi \ =\ 0\ .\eqno (7.25) $$ Equations (7.21), (7.24) and (7.25) and the relation $h_0=h$ on ${\rm supp}\, \psi$ yield (7.20). $\square$ \medskip \noindent {\bf Remark 7.6.} In the supplement we give a direct and elementary proof of (7.20). We complete now the proof of theorem 7.1. Let $g$ be the same as in theorem 6.1. We write it as $$ g\ = \ g_1 + g_2 $$ where $g_1$ has the same properties as $g$ with addition that it is supported in a small neighbourhood of $0$ so that ${\rm supp}\, g_1$ contains no critical values of $h$ restricted to ${\rm supp}\, \psi\times\BR^d$ except, possibly, $\lambda=0$, and $g_2\in C_0^\infty$. Applying lemma 7.3 to $g_1$ and theorem 7.5 to $g_2$, we arrive at the result of theorem 7.1. $\square$\hfill \bigskip \noindent {\bf Remark 7.7.} In order to obtain the next term in expansion (7.4) one would have to improve lemma 7.2. One way of doing this is by exercising a better control of $\tau$, the maximal time during which the classical trajectories beginning in $B(0,1)\times\BR^d$ neither hit one of the singularities nor return to their starting points. It is easy to trace the explicit dependence of the r.h.s. of (7.11) on $\tau$. Namely, we have $$ \vp * (\chi * e-e)\ = \ O\left( \alpha^{-d} \bigg( {\alpha\over \tau}\bigg)^{1+s}\right)\ . $$ Thus if $\tau=O(\alpha^{-\ve})$ for some $\ve>0$, one can obtain the second $-\alpha^{2-d}$-term in the expansion for ${\rm tr}\big(\psi g (H_\alpha)\big)$. An equation extending (7.22) to a higher order in which $g(\lambda)$ is replaced by $g(\lambda \alpha^{-\mu})$ shows that it suffices to study the classical trajectories in the energy interval $[-\alpha^\mu,0]$ with $\mu<{1\over6}$. (The latter condition is not sharp: the uncertainty principle suggests that it should suffice to take $\mu<1$.)\hfill \vfill\eject \beginsection 8. Multiscale Analysis The core of this section is a multiscale analysis which allows us to relax the condition in theorem 7.1 concerning critical values of the Hamiltonian $h$ (or the potential $W(x)$) and to extend this theorem to singular potentials. There are three scales in the problem: the momentum scale $\beta^{-1}$ determined by the quasiclassical parameter $\beta$ entering the definition of the kinetic energy, the length scale $\ell(x)$ determined by the behaviour of the potential near critical points or near singularities and the energy scale, $f(x)^2$, determined by the size of potential. The first scale is constant while the other two depend on $x$. Scaling the coordinate and energy appropriately, we reduce the original problem to a model one, treated in the previous section, but with the effective quasiclassical parameter $$ \alpha_{\rm eff} (x)\ = \ {\beta\over\ell(x)f(x)}\ , $$ which depends on all the scales. Applying to the latter problem theorem 6.1 and rescaling the result back we obtain the desired quasiclassical expansion for the original problem. One of the consequences of this is a quasiclassical expansion for a singular potential outside a small neighbourhood of singularities. Decoupling of the latter neighbourhood and estimation within it is done in the next two sections, respectively. We consider the Schr\"odinger operator $$ \Kb\ = \ -{1\over2}\beta^2\Delta - \phi(x)\quad {\rm on}\quad \BR^d\ . $$ Its symbol is denoted by $$ k(x,\xi)\ =\ {1\over2}|\xi|^2-\phi(x)\ . $$ We assume that $\phi(x)$ is real and obeys the Kato inequality: $$ \|\phi u \|\ \le\ \ve \| \Delta u \| + {C\over\ve^2} \| u\| \eqno (8.1) $$ for all $u\in {\cal D}(\Delta)$ and for all $\ve>0$ and with $C$ independent of $\ve$ and $u$. We impose, in addition, the following conditions on the potentials $\phi(x)$: there are differentiable functions $\ell(x)$ and $f(x)$ obeying $$ \ell(x)>0\quad {\rm a.e.\ and}\quad |\nabla\ell(x)|\le L \eqno (8.2) $$ for some $L>0$ and $$ f(x)>0\quad {\rm a.e.\ and}\ c^{-1}\le {f(x)\over f(y)}\le c\quad {\rm on}\ B\big( y,\ell(y)\big) \eqno (8.3) $$ for $1< c<\infty$, i.e. $f$ is slowly varying on the scale of $\ell$, and s.t. $$ |\partial^{\nu}\phi(x)|\ \le\ C_\nu f(x)^2 \ell(x)^{-|\nu|} \ .\eqno (8.4) $$ In the rest of this section $\int\int$ stands for the $(x,\xi)$-integral over the phase-space (remember the normalization $d\xi=(2\pi)^{-d} \times$ Lebesgue measure). \proclaim Theorem 8.1. Assume conditions (8.1)--(8.4) are obeyed and let $\psi$ be smooth and obey $|\partial^\nu\psi(x)|\le C_\nu\ell(x)^{-|\nu|}$ for any $\nu$. Let $g$ be smooth on $\BR\backslash\{0\}$ and satisfy (7.3) and $|g(\lambda)|\le C(-\lambda)_+^s$ for some $s\in[0,1]$. Then $$ \eqalign{ &|\tr \big( \psi g(K_\beta)\big) - \beta^{-d} \int\int \psi g (k) |\cr &\quad \le\ C\beta^{2s} \int_{\Omega_\psi} \max\Big[\Big({\beta\over f(x)\ell(x)}\Big)^{\rho-s-d},1\Big] \ell(x)^{-2s-d} dx\ ,\cr} \eqno (8.5) $$ where $\Omega_\psi=\bigcup\limits_{y\in{\rm supp}\, \psi} B\big( y,\ell(y)\big)$ and where with $C$ independent of $\beta$. Here $\rho=1$ if either $d\ge 2$ or $d=1$ and $\phi$ obeys $$ |\phi(x)| + \ell(x) |\nabla\phi(x)|\ \ge\ \ve f(x)^2\eqno (8.6) $$ on $\{ x\bigm| \ell(x) f(x)\ge \beta\}$ and with some $\ve>0$ independent of $\beta$ and $\rho={1\over2}$ otherwise. \par {\bf Proof.} First we demonstrate this statement under the additional restrictions that $\psi$ is supported in $\{ x\bigm| \ell(x) f(x)\ge \beta\}$ and that (8.6) holds on $\Omega_1=\{ x\bigm| \ell(x) f(x)\ge \delta\beta\}$ for some sufficiently small $\delta>0$ (e.g. $\delta=c^{-1}(1-L)$ after choosing $L<1$). Then we use the obtained result to remove this restriction. Note first that by rescaling $\ell(x)$ we can assume $L<{1\over2}$ in (8.2). In this case (8.2) implies $$ {1\over 1+L}\ell (y)^{-1}\ \le\ \ell(x)^{-1}\ \le\ {1\over 1-L} \ell(y)^{-1}\quad {\rm on}\quad B\big( y,\ell(y)\big)\ . \eqno (8.7) $$ Next, using (8.3), we derive from (8.4) and (8.6) $$ |\partial^\gamma \phi(x)|\ \le\ C_\gamma f(y)^2 \ell (y)^{-|\gamma|} \eqno (8.8) $$ and $$ |\phi(x)| + \ell(y)|\nabla\phi(x)|\ \ge\ \ve_1 f(y)^2\eqno (8.9) $$ for some $\ve_1>0$ independent of $y$ and $\beta$, for all $x$ in $B\big(y,\ell(y)\big)\cap\Omega_1$. Next, we need the following \proclaim Lemma 8.2. Assume that for some constants $f,\ell>0$, s.t. $f\ell\ge\beta$, $\phi(x)$ obeys on $B(y,2\ell)$ the estimates $$ |\partial^\gamma \phi(x)|\ \le\ C_\gamma f^2\ell^{-|\gamma|} \eqno(8.10) $$ and $$ |\phi(x)| + \ell|\nabla\phi(x)|\ \ge\ \ve f^2\eqno (8.11) $$ for some $\ve >0$. Let $\psi\in C_0^\infty \big( B(y,\ell)\big)$ with $|\partial^\gamma \psi|\le C_\gamma \ell^{-|\gamma|}$. Then $$ \eqalign{ &|\tr\big(\psi g(K_\beta)\big) -\beta^{-d} \int\int \psi g (k)|\cr &\qquad \le\ C\beta^{s+1-d} \ell^{d-s-1} f^{s+d-1} \ ,\cr}\eqno(8.12) $$ where $C$ depends only on the $C_\gamma$'s above. \par {\bf Proof.} The idea of the proof is to scale the given problem to one in the unit ball and with a potential whose bounds are independent of $f$ and $\ell$. To this end we define the unitary transformation $$ U(\ell)\,:\ \ \psi(x)\ \to\ \ell^{d\over2} \psi (y+\ell x)\ , $$ scaling $x$ into $\ell x$, and use it to map $\Kb$ into $$ U(\ell) \Kb U(\ell)^{-1}\ = \ \ell^{-2} \beta^2 D_x^2 - \phi(y+\ell x)\ . $$ Introduce the new potential $$ W(x)\ =\ f^{-2} \phi (y+\ell x)\ , $$ the new quasiclassical parameter $$ \alpha\ =\ {\beta\over f\ell}\eqno(8.13) $$ and the new Schr\"odinger operator $$ H_\alpha\ = \ -{1\over2}\alpha^2 \Delta - W(x)\ . $$ Note that the new Schr\"odinger operator is related to the original one as $$ U(\ell)\Kb U(\ell)^{-1}\ =\ f^2 H_\alpha\quad {\rm with}\quad \alpha = {\beta\over f\ell}\ . \eqno (8.14) $$ Moreover, differentiating the new potential $$ \partial^\gamma W(x)\ = \ f^{-2} \ell^{|\gamma|} (\partial^\gamma\phi) (\ell x) $$ and using estimate (8.10), we find $$ |\partial^\gamma W(x)| \ \le\ C_\gamma\qquad {\rm on}\qquad B(0,2) $$ with $C_\gamma$ independent of $f$ and $\ell$. Moreover, we derive from (8.11) that $$ |W(x)| + |\nabla W(x)|\ \ge\ c\eqno (8.15) $$ on $B(0,2)$ for some $c>0$. Of course, $W$ obeys the Kato inequality (with a constant depending on $f$ and $\ell$). Next, due to (8.14) $$ g(f^2H_\alpha)\ = \ U(\ell)g(K_\beta)U(\ell)^{-1}\ .\eqno (8.16) $$ Using that $U(\ell)\psi U(\ell)^{-1}$ is the multiplication operator by $\vp(x)\equiv\psi (\ell x)$ and using the invariance of the trace under similarity transformations, we obtain $$ \tr\big( \psi g(K_\beta)\big)\ = \ \tr\big(\varphi g (f^2H_\alpha)\big) \eqno (8.17) $$ with $\alpha=\beta/f\ell$. Observe now that $\vp\in C_0^\infty \big( B(0,1)\big)$, $|\partial^\gamma\vp|\le C_\gamma$ independently of $\ell$ and $0$ is not a critical value of $h={1\over2} |\xi|^2-W(x)$ on $\supp\varphi\times\BR^d$. Since $g(f^2\lambda)$ obeys (7.3), theorem 7.1 with $g(\lambda)$ replaced by $g(f^2\lambda)$ is applicable to $H_\alpha$ and it yields $$ \eqalign{ &|\tr\big(\varphi g(f^2H_\alpha)\big)-\alpha^{-d} \int\int \varphi g(f^2h)| \cr &\qquad \le \ Cf^{2s}\alpha^{s+1-d} \ ,\cr}\eqno (8.18) $$ where we have used that $b(g_f)=f^{2s} b(g)$ with $g_f(\lambda)= g(f^2\lambda)$. Remembering that in the first case (8.16) and the relation $\int\int\vp(x)g(f^2h)=0$ yield an even stronger estimate we conclude that (8.18) holds in both cases. Substituting (8.18) into (8.17) and using that $\alpha^{-d} \int\int\vp g(f^2 h)=\beta^ {-d} \int\int \psi g(k)$, we arrive at (8.11). $\square$\hfill \bigskip Now we return to the proof of theorem 8.1 and recall that we have shown that $\phi$ obeys (8.8) and (8.9) on $B\big( y,\ell(y)\big)$. Hence lemma 8.2 is applicable on this ball and with $f=f(y)$ and $\ell=\ell(y)$, provided $\ell(y)f(y)\ge\beta$, which gives $$ \eqalign{ &|\tr\big(\psi_y g(K_\beta)\big)-\beta^{-d} \int\int \psi_y g (k) |\cr &\qquad \le\ C\beta^{s+1-d}f(y)^{d-1} \ell(y)^{d-s-1}\ ,\cr} \eqno (8.19) $$ where $\psi_y\in C_0^\infty \Big( B\big(y,\ell(y)\big)\Big)$ and is s.t. (8.6) holds on $\supp \psi_y \times\BR^d$. Using now (8.3) and (8.7) in order to estimate the r.h.s. of (8.19), we obtain $$ \eqalign{ &|\tr \big( \psi_yg(K_\beta)\big) -\beta^{-d}\int\int \psi_y g (k)|\cr &\qquad\le\ C\beta^{s+1-d} \int_{B\big( y,\ell(y)\big)} f(x)^{d+s-1} \ell(x)^{-s-1} dx\ .\cr}\eqno (8.20) $$ Finally, we cover $\supp\psi$ with balls $B\big( y,\ell(y)\big)$, $y\in{\rm supp}\, \psi$. Since $\ell(x)$ is slowly varying these balls can be chosen so as to have finite intersection property, i.e. there is a constant $M_1$ s.t. the intersection of more than $M_1$ balls is empty. Moreover, there is a partition of unity $\{ j_y\}$ associated with this covering s.t. $j_y$ is supported in $B\big( y,\ell(y)\big)$, $\sum j_y=1$ on ${\rm supp}\,\psi$ and $\partial^\alpha j_y = O\big( \ell(y)^ {-|\alpha|}\big)$ (see [H\"orm I. theorem 1.4.10]). Using this partition of unity, we decompose $$ \psi\ = \ \sum \psi_y $$ with $\psi_y$ supported in $B\big( y,\ell(y)\big)$ and obeying $\partial^\nu \psi_y = O\big( \ell(y)^{-|\nu|}\big)$. Using now (8.21) for each of the $\psi_y$'s, we obtain (8.6) with $\rho=1$. Note that additional restriction (8.6) is equivalent to the condition that $0$ is not a critical value of the rescaled Hamiltonian $h$ on $B(0,2)\times\BR^d$. Thus we have proven theorem 8.1 with $r=1$ and with the additional restriction (8.6) on $\Omega_1$. Now we use this result in order to strengthen the key theorem 7.1 used in the proof of this result. Namely, for $d\ge 2$ we remove from the latter theorem the condition that $0$ is not a critical value of $h$ restricted to $B(0,1)\times\BR^d$, which in turn will allow us to remove this condition from the proof of (8.5) given above! In the one-dimensional case by the order of a critical point $x_0$ of $\phi$ we understand the order of the first non-vanishing derivative of $\phi$ at $x_0$ minus $1$. \proclaim Theorem 8.3. Let $W$ obey the conditions given in the beginning of section 4. Let $\psi\in C_0^\infty \big( B(0,1)\big)$ and let $g$ be the same as in theorem 8.1. Then $$ \eqalign{ \tr\big( \psi g (H_\alpha)\big) &- \alpha^{-d}\int\int \psi g (h)\cr &=\ O\big( \alpha^{s+1-d}\vp (\alpha)\big)\ ,\cr} \eqno (8.21) $$ where $\vp (\alpha)\equiv 1$ if either $d\ge 2$ or $d=1$ and $W$ has no critical points in $B(0,1)\cap W^{-1} ({\rm supp}\, g)$, $\vp(\alpha)=|{\rm ln}\,\alpha|$ if $d=1$ and $n=1$ and $\vp(\alpha)=\alpha^{-{1\over2}}$ if $d=1$ and $\infty\ge n\ge 2$. Here $n$ is the maximal order of critical points of $W$ on $B(0,1)\cap W^{-1}(0)$.\par {\bf Proof.} We define the length scale as $$ \eqalign{ \ell(x)\ &=\ M_1^{-1} |\nabla h| |_{h=0}\cr &=\ {1\over M_1} (|W| + |\nabla W |^2 )^{1\over2}\ ,\cr} \eqno (8.22) $$ where $M_1$ is given by $$ \eqalign{ M_1\ &=\ 1 + 2\sup_x \| \hbox{Hessian}\ W(x)\|\cr &\ge\ 2\,\sup_x\, |\nabla (|\nabla h| \big|_{h=0})|\ .\cr} $$ Since $\ell(x)\le\const$ on $B(0,2)$, we have that $$ |\partial^\gamma W(x)|\ \le\ C_\gamma\ell^{2-|\gamma|} \ . $$ This forces us to set the energy scale to be $$ f(x)\ = \ \ell(x)\ . $$ The definition of $\ell(x)$ and $f(x)$ implies that $W(x)$ obeys (8.2)--(8.4) and (8.6) with those $f(x)$ and $\ell(x)$. We have shown above that under these restrictions (8.5) with $r=1$ is true, which in the present case yields $$ \eqalign{ &|{\rm tr}\big(\psi g(H_\alpha)\big) - \alpha^{-d} \int\int\psi g (h)|\cr \le\ &C\alpha^{s+1-d}\int_{\Omega_\psi} \ell(x)^{d-2} dx\cr} \eqno (8.23) $$ for any $\psi\in C_0^\infty$ supported in $$ \eqalign{ &\{ f(x)\ell(x)\ \ge\ 2\alpha\}\cr =\ &\{\ell(x)\ \ge\ \sqrt{2\alpha}\}\ .\cr} $$ Now we consider $$ \eqalign{ &\{ f(x)\ell(x)\ \le\ 2\alpha\}\cr =\ &\{\ell(x)\ \le\ \sqrt{2\alpha}\}\ .\cr} \eqno (8.24) $$ On this domain we pick the length and energy scales to be $\sqrt\alpha$ and $\alpha$, respectively. Consider $H_\alpha$ on the ball $B(y,\sqrt\alpha)$. Scaling $$ x\ \to\ y + \sqrt\alpha x $$ maps $H_\alpha$ into $\alpha \widetilde H$, where $$ \widetilde H\ = \ -{1\over2}\Delta - \widetilde W(x) $$ with $$ \widetilde W(x)\ = \ {1\over\alpha}W(y+\sqrt\alpha x) $$ obeying (8.16) on $B(0,1)$. Then theorem 7.1 with $\alpha=1$ and with $g(\lambda)$ replaced by $g(\alpha\lambda)$ implies $$ \tr\big(\psi g(\alpha\widetilde H)\big)\ - \ \int\int \psi g (\alpha\tilde h)\ =\ O(\alpha^s)\ , $$ where $\tilde h=|\xi|^2-\widetilde W(x)$, provided $\psi\in C_0^\infty \big( B(0,1)\big)$. (This is a trivial estimate if $g(\alpha\lambda)=\alpha^s g(\lambda)$ for $\alpha>0$.) Rescaling this back to the original variables and using that, as above, $$ \tr\big(\psi g(\alpha\widetilde H)\big)\ = \ \tr \big(\vp g (H_\alpha)\big) $$ and $$ \int\int\psi g (\alpha\tilde h)\ = \ \alpha^{-d}\int\int \vp g(h)\ , $$ where $\vp(x)=\psi\left( {x-y\over\sqrt\alpha}\right)$, we obtain $$ \eqalign{ \tr\big(\vp g (H_\alpha)\big) &- \alpha^{-d}\int\int \vp g (h)\cr &=\ O(\alpha^s)\ .\cr}\eqno (8.25) $$ Now let $$ \ell_1(x)\ =\ \max\big(\ell(x),\sqrt\alpha\big)\ . $$ Then (8.25) can be rewritten as $$ \eqalign{ |\tr\big(\vp g (H_\alpha)\big) &- \alpha^{-d} \int\int \vp g (h) |\cr &\le\ C\alpha^{1+s-d} \int_{B(y,3\ell_1(y))} \ell_1 (x)^{d-2} dx\ ,\cr} \eqno (8.26) $$ provided $\vp$ is supported in $B\big( y, \ell_1(y)\big)$ and $B\big( y,3\ell_1 (y)\big)$ lies in (8.24). On the other hand equation (8.23) and the observation that $\Omega_\vp\subset B\big(y,3\ell_1 (y)\big)$ for $\vp\in C_0^\infty\big( B (y,3\ell(y)\big)$ imply (8.26) for any ball $B\big( y,3\ell_1(y)\big)$ lying in $$ \{ x | \ell(x)\ \ge\ \sqrt \alpha \}\ . $$ Thus (8.26) holds for any ball $B\big( y, 3\ell_1(y)\big)$ (provided $\vp$ is supported in $B\big(y,\ell_1(y)\big)$ and obeys the corresponding estimates). Using, as above, a partition of unity associated with the length scale $\ell_1(x)$, we derive $$ \eqalign{ &|\tr\big(\psi g(H_\alpha)\big) - \alpha^{-d} \int\int \psi g(h)|\cr &\qquad\le\ C\alpha^{s+1-d} \int_{B(0,3)} \ell_1(x) ^{d-2} dx\cr} \eqno (8.27) $$ for any $\psi\in C_0^\infty \big(B(0,1)\big)$ and with $C$ independent of $\alpha$ and of $|\nabla h|^{-1}$. For $d\ge 2$, the integrand on the r.h.s. is bounded. For $d=1$ it is $O\big((|x-x_0|+\sqrt\alpha)^{-1}\big)$, if the order of the critical point $x_0$ is equal to $1$ and is $O(\alpha^{-{1\over2}})$ otherwise. Hence the r.h.s. of (8.27) can be bounded by $\const\,\alpha^ {s+1-r}\vp(\alpha)$ and this inequality can be extended to $\Omega= B(0,1)$. This yields (8.21). $\square$\hfill \bigskip Now redoing the above proof of (8.5) but using theorem 8.3 instead of theorem 7.1 we conclude that (8.5) holds under the conditions of theorem 8.1 without additional restriction (8.6) in the $d\ge 2$ case, provided $\psi$ is supported in $\{ x\bigm| f(x)\ell(x)\ge\beta\}$. Now we analyze the region $\{ x\bigm| f(x)\ell(x)\le{1\over2}\beta\}$. \proclaim Theorem 8.4. Assume $\phi$ obeys (8.1)--(8.4). Let $C\lambda_-^s\le g(\lambda)\le 0$ for $0\le s\le 1$. Let $\psi$ be a smooth and bounded function supported in $\{ x\bigm| f(x)\ell(x)\le \beta\}$. Then $$ |{\rm tr}\big(\psi g(K_\beta)\big)|\ \le\ C\beta^{2s}\int_ {\Omega_\psi} \ell(x)^{-2s-d} dx\ .\eqno (8.28) $$ \par {\bf Proof.} Denote $g_s(\lambda)=\lambda_-^s$. Without the restriction on generality we can assume $\psi\ge 0$ and $|\nabla\ell(x)|\le{1\over2}$. $\psi\ge 0$ implies that $$ C\, {\rm tr}\big(\psi g_s(K_\beta)\big)\ \le\ {\rm tr} \big( \psi g(K_\beta)\big)\ \le\ 0\ . $$ Let $y$ obey $f(y)\ell(y)\le \beta$. Let $\varphi\in C_0^\infty\big( B(y,\ell)\big)$, where $\ell=\ell(y)$, and satisfy $|\partial^\nu\varphi (x)|\le C_\nu\ell^{-|\nu|}$. Rescaling the problem as $x\to y+\ell x$, we obtain $$ \eqalign{ &{\rm tr}\, \varphi(x) g_s(K_\beta)\cr =\ &\beta^{2s}\ell^{-2s} {\rm tr}\,\varphi_1(x) g_s (H)\ ,\cr}\eqno (8.29) $$ where $\varphi_1(x)=\varphi(y+\ell x)$ and $H=-{1\over2}\Delta -\phi_1(x)$ with $$ \phi_1(x)\ =\ \beta^{-2}\ell^2\phi(y+\ell x)\ .\eqno (8.30) $$ Note that $\varphi_1\in C_0^\infty\big( B(0,1)\big)$ and obey $|\partial^\nu\varphi_1|\le C_\nu$. Moreover, estimates (8.4) on $\phi(x)$ and inequality $\beta^{-1}f\ell\le 1$ imply that $\phi_1(x)$ obeys $$ |\partial^\nu\phi_1 (x)|\ \le\ C_\nu $$ on $B(0,2)$. for a fixed $\delta>0$. Hence, e.g. by theorem 4.10, $$ |{\rm tr}\,\varphi_1 (x) g_s (H)|\ \le\ C\ , $$ This together with (8.29) and (8.30) yields $$ \eqalign{ |{\rm tr}\, \varphi(x) g(K_\beta)|\ &\le\ C\beta^{2s}\ell^{-2s}\cr &\le\ C_1\beta^{2s}\int_{B(y,\ell)} \ell(x)^{-2s-d}dx\ .\cr} $$ Covering ${\rm supp}\,\psi$ by balls $B\big( y,\ell(y)\big)$ with $y\in{\rm supp}\,\psi$ and proceeding as in the proof of theorem 8.1, we arrive at (8.29). $\square$ In the classically forbidden region estimate (8.5) can be considerably improved. \proclaim Theorem 8.5. Assume $ \phi (x) $ obeys (8.1)--(8.4). Let $\psi$ be a bounded function and let $g(\lambda)$ satisfy $|g(\lambda)|\le C(-\lambda)_+^s\langle\lambda \rangle^m$ for some $m$ and $s\ge 0$ and let $\sup\limits_{\Omega_\psi} (f^{-2}\phi)<0$. Then $$ |\tr\,\psi (x) g(K_\beta)|\ \le\ C\int_{\Omega_\psi} \Big({\beta\over f(x) \ell(x)}\Big)^A f(x)^{2s} \langle f(x)\rangle^ {2m} \ell(x)^{-d} dx\eqno (8.31) $$ for any $A\ge 0$. Here, recall, $\Omega_\psi$ is defined in theorem 8.1.\par {\bf Proof.} Let $z\in\supp\,\psi$, $\ell=\ell(z)$ and $f=f(z)$ and let $\vp\in C_0^\infty \big( B(y,{1\over2}\ell)\big)$ and obey $|\partial^\nu \vp(x)|\le C_\nu \ell^{-|\nu|}$ rescaling $x\to z+\ell x$ and energy $\to f^{-2}$ $\times$ energy maps $K_\beta$ into $f^2 H_\alpha$, where $$ H_\alpha\ =\ - {\alpha^2\over 2}\Delta_x - \phi_0(x)\ , $$ with $\alpha = {\beta\over f\ell}$ and $\phi_0 (x)=f^{-2} \phi(z+\ell x)$. Observe that $ | \partial^\nu \phi_0 (x) | \le C_\nu $ on $ B(0,2) $. Since the trace is invariant under similarity transformations, we have $$ \tr\, \vp (x) g(K_\beta)\ =\ \tr\, \vp_0 (x) g(f^2 H_\alpha)\ , \eqno (8.32) $$ where $\vp_0\in C_0^\infty \big( B(0,{1\over2})\big)$ and obey $|\partial^ \nu \vp_0(x)|\le C_\nu$. By the restriction on $g(\lambda)$ we have that $|g(f^2\lambda)|\le Cf^{2s} \langle f\rangle^{2m}(-\lambda)_+^s\langle\lambda\rangle^m$. Hence there is a function $\widetilde g (\lambda)$ supported in $(-\infty, \sup\limits_{B(0,1)}\phi_0)$ and obeying $|\partial^n \widetilde g (\lambda)|\le C_n \langle \lambda\rangle^{m-n}$ and $\widetilde g (\lambda)\ge f^{-2s} \langle f\rangle^{-2m} |g(f^2\lambda)|$. Hence by a property of trace norms (see [Reed Sim II, p. 42]) $$ \eqalign{ &|\tr\, \vp_0 (x) g(f^2 H_\alpha)|\cr \le\ & f^{2s} \langle f\rangle^{2m} | \tr\, \vp_0 (x)\widetilde g (H_\alpha)|\ .\cr}\eqno (8.33) $$ Applying theorem 4.8 (see Remark 4.9b) to $\tr\, \vp_0 (x)\widetilde g (H_\alpha)$, we obtain $$ \tr\, \vp_0 (x)\widetilde g (H_\alpha)\ =\ O(\alpha^A) \eqno (8.34) $$ for any $A\ge 0$. Remembering (8.32) and (8.33) and remembering that $\alpha= {\beta\over f\ell}$, we find $$ |\tr\,\vp(x) g(K_\beta)|\ \le\ C\Big( {\beta\over f\ell} \Big)^A f^{2s} \langle f\rangle^{2m}\ .\eqno (8.35) $$ Now, like in the end of the proof of theorem 8.1, covering $\supp\,\psi$ by the balls $B\big( z,\ell(z)\big)$ with $z\in \supp\,\psi$, associating with this covering a partition of unity, splitting $\tr\,\psi g(K_\beta)$ with the help of this partition of unity and applying (8.32) to each of the resulting sum, we obtain (8.28). $\square$ \bigskip \noindent {\bf Remark 8.6.} Theorem 8.5 can be derived from a natural generalization of theorem 8.1 to arbitrary $s\ge 0$. This generalization expresses $\tr\,\psi g(K_\beta)$ as a sum of $[s]+1$ Weyl-type local terms, as given by a standard quasiclassical pseudo-differential calculus, plus the error of order $O(\alpha^{1+s-d})$, where $\alpha={\beta\over f\ell}$. In the classically forbidden region the local, Weyl-type terms vanish, so the result follows. We combine now theorems 8.1 and 8.5. \proclaim Theorem 8.7. Assume the conditions of theorem 8.1 are obeyed and let $\psi,\Omega_\psi$ and let $\rho$ be the same as in theorem 8.1. Then for any $A\ge 0$ and any $\mu\le 0$ $$ \eqalign{ &|{\rm tr}\,\psi g(K_\beta-\mu)-\beta^{-d}\int\int\psi g(k-\mu)|\cr \le\ &C\beta^{2s} \int_{\Omega_\psi \cap Q_{{1\over2}\beta}} \Big( {\beta\over f\ell}\Big)^{\rho-s-d} \ell^{-2s-d} dx\cr &+ C\beta^{2s}\int_{\Omega_\psi\cap Q_\beta^c} \min\Big( {\beta/\sqrt{-\mu}\over\ell},1\Big)^A \ell^{-2s-d} dx\ ,\cr}\eqno (8.36) $$ where $C$ depends only on $d$ and on the constants in (8.3) and (8.4) and where $Q_\beta=\{ x\bigm| f(x)\ell(x)\ge\beta\}$ and $Q_\beta^c=\BR^d\backslash Q_\beta$.\par {\bf Proof.} Let $f_1(x)$ be a positive function satisfying eqn (8.3) and obeying $f_1(x)\ge f(x)$. Define the domains $$ R_1\ =\ \Big\{ x\bigm| \phi (x)\ge -{\mu\over 3} - 3C_0 c^3 f_1(x)^2\Big\} $$ and $$ R_2\ =\ \Big\{ x \bigm|\phi (x)\le -{1\over2}\mu - 2C_0 c^3 f_1(x)^2\Big\}\ , $$ where $C_0$ is the constant entering (8.4) for $\nu=0$ and $c$ is the same as in (8.3). We begin with \proclaim Lemma 8.8. With the conditions and notation of theorem 8.1 we have for any $A$ and for any $\mu\le 0$ $$ \eqalign{ &|\tr\, \psi g(K_\beta-\mu)-\beta^{-d} \int\int \psi g(k-\mu)|\cr \le\ & C\beta^{2s} \int_{\Omega_\psi\cap R_1} \max\Big[ \Big( {\beta\over f_1(x) \ell(x)}\Big)^{\rho-s-d},1\Big] \ell (x)^{-2s-d} dx\cr &+ C\beta^{2s}\int_{\Omega_\psi\cap R_2} \Big( {\beta\over f_1(x) \ell(x)}\Big)^A \ell(x)^{-2s-d} dx\ .\cr} \eqno (8.37) $$ \par {\bf Proof.} On $R_1$ the potential $\phi (x)+\mu$ obeys (8.1)--(8.4) but with $f(x)$ replaced by $f_1(x)$. Hence theorem 8.1, with $f_1(x)$ instead of $f(x)$, is applicable to $K_\beta-\mu$ and yields that the l.h.s. of (8.37) is bounded by $$ C\beta^{2s}\int_{\Omega_\psi} \max\Big[\Big( {\beta\over f_1(x) \ell(x)}\Big)^{\rho-s-d},1\Big] \ell (x)^{-2s-d} dx $$ for any $\psi$ supported in $R_1$. On $R_2$ the potential $\phi(x)+\mu$ obeys (8.1)--(8.4) but with $f(x)$ replaced by $\max\big( f_1(x),\sqrt{-\mu}\big)$. Let $R_3=\{ x\bigm| \phi(x)\le -{1\over2}\mu -C_0 c f_1(x)^2\}$. Using that $\phi(x)\le\phi(y)+|\phi(x)-\phi(y)|$ and that $|\phi(x)-\phi(y)|\le C_0 cf(x)^2$ on $B\big( y,\ell(y)\big)$, we derive that $\Omega_\psi\subset R_3$, provided that ${\rm supp}\, \psi\subset R_2$. Furthermore, since $$ \sup_{R_3} \big\{ \big[\max \big( f_1(x),\sqrt{-\mu}\big)\big]^{-2} \big(\phi(x)+\mu\big)\big\}\ \le\ -\min \Big( {1\over2},cC_0\Big)\ , $$ we have that $$ {\rm supp}\, g\subset \big(-\infty,-\sup_{R_3}\big\{ \big[ \max \big( f_1(x),\sqrt{-\mu}\big)\big]^{-2} \big(\phi(x)+\mu\big)\big\}\big) \ . $$ Hence theorem 8.4 with $f(x)$ replaced by $\max\big( f_1(x), \sqrt{-\mu}\big)$ is applicable to $K_\beta-\mu$ and yields that for any $\mu\le 0$ and for any $A$ $$ |{\rm tr}\, \psi g(K_\beta-\mu)|\ \le\ C\beta^{2s} \int_ {\Omega_\psi} \Big( {\beta\over f_1(x)\ell(x)}\Big)^A \ell(x)^{-2s-d} dx $$ provided $\psi$ is supported in $R_2$ and is bounded. Since, moreover, $\int\int \psi g(k-\mu)=0$ for such $\psi$'s, we conclude that the l.h.s. of (8.37) is bounded by $$ C\beta^{2s} \int_{\Omega_\psi} \Big( {\beta\over f_1(x)\ell(x)}\Big)^A \ell(x)^{-2s-d} dx $$ for any $A$, provided $\psi$ is bounded and is supported in $R_2$. This together with the conclusion of the previous paragraph yields (8.37). $\square$ \medskip Now we return to the proof of theorem 8.7. We pick up $f_1(x)$ as $$ f_1(x)\ =\ f(x) + {\sqrt{-\mu}\over\sqrt{20C_0 c^3}} + \beta\ell (x)^{-1}\ .\eqno (8.38) $$ Then for all $x$ $$ f_1(x)\ell(x)\ \ge\ \beta\ .\eqno (8.39) $$ Moreover, $$ R_1\cap Q_\beta^c\ \subset\ \Big\{ x\bigm| \ell(x)\le C_1 {\beta\over\sqrt{-\mu}}\Big\} $$ and $$ R_2\cap Q_\beta^c\ \subset\ \Big\{ x\bigm| \ell(x)\ge C_2 {\beta\over\sqrt{-\mu}}\Big\}\ , $$ where $C_1=6\sqrt{C_0(1+6c^3)+2}$ and $C_2=\sqrt{C_0(1+4c^2)+2}$. Using this we derive $$ {f_1(x)\ell(x)\over \beta}\ \le\ {f(x)\ell(x)\over\beta} + C_3 $$ on $R_1\cap Q_\beta^c $ and $$ {\beta\over f_1(x)\ell(x)}\ \le\ C_4{\beta\over \sqrt{-\mu}} \ell(x)^{-1} $$ on $R_2\cap Q_\beta^c$, where $C_3$ and $C_4$ depend only on $C_0$ and $c$. These inequalities together with lemma 8.4 imply (8.36), provided that $\psi$ is supported in $Q_\beta^c$. Since the case of $\psi$ supported in $Q_{{1\over2}\beta}$ is covered by theorem 8.1, eqn (8.36) is proven. $\square$ \medskip Now we apply theorems 8.1 and 8.4 to the operator $K_\beta$ defined in (2.51). Recall that the potential $\phi(x)=\phi_{\lambda} (x,y)$ for this operator obeys (8.4) (see (2.9)) with $f(x)$ and $\ell(x)$ given in (2.7)--(2.8). The latter equations yield that these $f(x)$ and $\ell(x)$ obey (8.2)--(8.3). \proclaim Theorem 8.9. Let $K_\beta$ be the operator defined in (2.51). Let $g$, $\psi$ and $\rho$ be the same as in theorem 8.1 and let, besides, $\psi$ be supported in $\{ x\bigm| \ell(x)\ge r\}$ with $r\ge 0$. Then for any $\mu\le 0$ $$ \eqalign{ &|\tr\big(\psi g (K_\beta-\mu)\big) -\beta^{-d} \int\int \psi g (k-\mu)|\cr \le\ & C\beta^{s+\rho-d} \max (r^{1\over2},\beta)^{-(3s+\rho-d)_+} + C \delta_s \Big| \ln \Big( {\beta\over\sqrt{-\mu}}\Big)\Big| \ ,\cr} \eqno (8.40) $$ where the constant is independent of $\beta$, of the $\lambda_i$'s and of the $y_i$'s and $\delta_s=1$ if $s=0$ and $=0$ if $s\ne 0$.\par {\bf Proof.} We derive theorem 8.9 from theorem 8.7. Remembering that the potential of $K_\beta$ obeys (8.4) with $f(x)=\ell(x)^{-{1\over2}}\langle \ell(x)\rangle^{-{3\over2}}$, we find that $$ \Big\{ x\bigm| f(x)\ell(x)\ge {1\over2}\beta\Big\}\ \subset\ \{ x\bigm| \beta^2\le\ell(x)\le 2\beta^{-1}\}\ .\eqno (8.41) $$ This and the fact that $\ell(x)=\min\limits_j |x-y_j|$ yields that the first integral on the r.h.s. of (8.36) is bounded by the first term on the r.h.s. of (8.40). To estimate the second integral on the r.h.s. of (8.36) we observe that $$ \{ x\bigm| f(x)\ell(x)\le\beta\}\ \subset\ \{ x\bigm| \ell(x) \le 4\beta^2\}\ \cup\ \{ x\bigm| \ell(x)\ge\beta^{-1}\}\ . \eqno (8.42) $$ The part of this integral over $\{ x\bigm| \ell(x)\ge\beta^{-1}\}$ is bounded, clearly, by the second term on the r.h.s. of (8.40). The part over $\{ x\bigm|\ell(x)\le 4\beta^2\}$ is, due to lemma 8.10 below, bounded by the first term on the r.h.s. of (8.40). Thus the second integral is bounded by the r.h.s. of (8.40). This together with the result of the previous paragraph implies (8.40). $\square$ \proclaim Lemma 8.10. Let $g(\lambda)$ obey $|g(\lambda)|\le -C\lambda_-^s$. Let $\psi$ be smooth, supported in $\ell(x)\le r$ and obey $|\partial^\nu\psi (x)|\le C_\nu r^{-|\nu|}$. Then $$ \|\psi (x) g(K_\beta)\|_1\ \le\ C\Big( {\max(r,\beta^2)\over\beta^2}\Big)^{ 3([d/2]+1)}\max(r,\beta^2)^{-s}\ ,\eqno (8.43) $$ provided either $r\le\beta^2$ and $0\le s$ or $\beta^2\le r\le {1\over3}\min\limits_{i\ne j} |y_i-y_j|$ and $s=0$. \par {\bf Proof.} Due to the definition of $\ell(x)$ and the restriction $r\le {1\over3}a$ $$ \psi(x)\ =\ \sum \psi_i (x)\ ,\eqno (8.44) $$ where the functions $\psi_i$ are smooth, supported in $|x-y_i|\le r$ and obey $|\partial^\nu\psi_i(x)|\le C_\nu r^{-|\nu|}$. Next, rescaling $x\to y_i + rx$ maps $K_\beta$ into ${1\over r} K_{0,\beta}$, where $$ K_{0,\beta}\ =\ - {\beta^2\over 2r} \Delta - \phi_0 (x)\ , \eqno (8.45) $$ with $\phi_0(x) = r\big(\phi ( y_i + rx)+\mu\big)$. Note that the r.h.s.'s of eqns (2.10) and (2.11) yields that $\phi(x)\le (\Sigma\lambda_j |x-y_j|^{-{3\over2}})^{2\over3}\le (\Sigma\lambda_j |x-y_j|^{-2})^{1\over2}$. This together with the uncertainty principle $-\Delta\ge (4|x|^2)^{-1}$ implies that $\phi_0$ obeys the Kato-type inequality $$ \| \phi_0 u\|\ \le\ \ve \|\Delta u\| + {C\over \ve} \| u\| \eqno (8.46) $$ for any $\ve>0$, $u\in D(\Delta)$ and with $C$ independent of $\ve$, $\beta$, $M$, $y$, $\lambda$ and $u$. By the unitary invariance of the trace norm $$ \eqalign{ &\|\psi_i (x)g_s(K_\beta)\|_1\cr =\ & r^{-s}\| \psi_0 (x)g_s(K_{0,\beta})\|_1\ ,\cr}\eqno (8.47) $$ where $g_s(\lambda)=\lambda_-^s$ and $\psi_0(x)=\psi_i( y_i + rx)$. Note that $\psi_0$ is smooth, supported in $|x|\le 1$ and obeys $|\partial^\nu\psi(x)|\le C_\nu$. We claim now that $$ \|\psi_0(x)g_s(K_{0,\beta})\|_1\ \le\ C\Big( {r\over\beta