Content-Type: multipart/mixed; boundary="-------------0203190740144" This is a multi-part message in MIME format. ---------------0203190740144 Content-Type: text/plain; name="02-135.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="02-135.keywords" Integrated density of states, high energy asymptotics, heat invariants ---------------0203190740144 Content-Type: application/x-tex; name="ids2.tex" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="ids2.tex" %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% % A REMARK ON THE HIGH ENERGY ASYMPTOTICS % OF THE INTEGRATED DENSITY OF STATES % %E. KOROTYAEV and A. PUSHNITSKI % % % 19 March 2002 %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \documentclass[11pt]{article} \usepackage{amssymb} \usepackage{amscd} \usepackage{amsmath} \usepackage{amsthm} \hoffset -1.5cm \voffset -1in \textwidth 16.3truecm \textheight 22truecm %%%%%%%%%%%%%%%%%%%% GREEK LETTERS %%%%%%%%%%%%%%%%%%%%%%%% \renewcommand{\a}{{\alpha}} \renewcommand{\b}{{\beta}} \newcommand{\g}{{\gamma}} \renewcommand{\d}{{\delta}} \newcommand{\e}{{\varepsilon}} \renewcommand{\k}{{\varkappa}} \renewcommand{\l}{{\lambda}} \newcommand{\m}{{\mu}} \newcommand{\f}{{\varphi}} %\renewcommand{\o}{{\omega}} \renewcommand{\t}{{\theta}} \newcommand{\s}{{\sigma}} \renewcommand{\r}{{\rho}} \newcommand{\x}{{\xi}} \newcommand{\z}{{\zeta}} \newcommand{\D}{{\Delta}} \newcommand{\G}{{\Gamma}} \renewcommand{\L}{{\Lambda}} \renewcommand{\O}{{\Omega}} %%%%%%%%%%%%%%%%%%% OPERATOR NAMES AND SUCH %%%%%%%%%%%%%%%%%%%%%%% \DeclareMathOperator{\const}{const} \DeclareMathOperator{\iindex}{index} \DeclareMathOperator{\sign}{sign} \DeclareMathOperator{\meas}{meas} \DeclareMathOperator{\card}{card} \DeclareMathOperator{\vol}{vol} \DeclareMathOperator{\dist}{dist} \DeclareMathOperator{\supp}{supp} \DeclareMathOperator{\rank}{rank} \DeclareMathOperator{\Ran}{Ran} \DeclareMathOperator{\Dom}{Dom} \DeclareMathOperator{\Ker}{Ker} \DeclareMathOperator{\Tr}{Tr} \DeclareMathOperator*{\slim}{s-lim} \DeclareMathOperator{\ord}{ord} \DeclareMathOperator{\Ai}{Ai} \renewcommand\Im{\hbox{{\rm Im}}\,} \renewcommand\Re{\hbox{{\rm Re}}\,} \newcommand{\abs}[1]{\lvert#1\rvert} \newcommand{\norm}[1]{\lVert#1\rVert} \newcommand{\br}[1]{\left(#1\right)} \newcommand{\laplace}{{\Delta}} \newcommand{\loc}{{\hbox{\rm\scriptsize loc}}} %%%%%%%%%%%%%%%%%% BOLD AND CALLIGRAPHIC LETTERS %%%%%%%%%%%%%% \newcommand{\R}{{\mathbb R}} \newcommand{\N}{{\mathbb N}} \newcommand{\Z}{{\mathbb Z}} \newcommand{\C}{{\mathbb C}} \newcommand{\T}{{\mathbb T}} \newcommand{\Q}{{\mathbf Q}} \renewcommand{\H}{{\mathcal H}} \newcommand{\K}{{\mathcal K}} \newcommand{\calB}{{\mathcal B}} \newcommand{\calS}{{\mathcal S}} \newcommand{\calM}{{\mathcal M}} \newcommand{\hatH}{{\widehat H}} \renewcommand{\SS}{{\mathfrak S}} %%%%%%%%%%%%%%%% EQUATIONS %%%%%%%%%%%%%%%%%%%%% %\numberwithin{equation}{section} \newcommand{\erpm}[1]{{$(\ref{#1}\pm)$}} \newcommand{\erp}[1]{{$(\ref{#1}+)$}} \newcommand{\erm}[1]{{$(\ref{#1}-)$}} %\renewcommand{\theequation}{\thesection.\arabic{equation}} %%%%%%%%%%%%%%%% THEOREM ENVIRONMENTS %%%%%%%%%%%%%%%%%% \theoremstyle{plain} \newtheorem{theorem}{\bf Theorem} \newtheorem{lemma}[theorem]{\bf Lemma} \newtheorem{proposition}[theorem]{\bf Proposition} \newtheorem{condition}[theorem]{\bf Assumption} \newtheorem*{corollary}{\bf Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{\bf Definition} \theoremstyle{remark} \newtheorem{remark}[theorem]{\bf Remark} \newcommand{\thm}[1]{{Theorem~\ref{t.#1}}} \newcommand{\lma}[1]{{Lemma~\ref{l.#1}}} \newcommand{\prp}[1]{{Proposition~\ref{p.#1}}} \newcommand{\rmk}[1]{{Remark~\ref{r.#1}}} \newcommand{\crl}[1]{{Corollary~\ref{c.#1}}} \newcommand{\cnd}[1]{{Assumption~\ref{condition.#1}}} \newcommand{\dfn}[1]{{Definition~\ref{d.#1}}} %%%%%%% MISCELLANEOUS NOTATIONS %%%%%%%%%%%%% \renewcommand{\qed}{\vrule height7pt width5pt depth0pt} %%%%%%%%%%%% LOCAL DEFINITIONS (FOR THIS TEXT ONLY) %%%%%% \newcommand{\p}{{V}} \DeclareMathOperator{\spec}{spec} %%%%%%%%%%%%%%%%%%%% END OF DEFINITIONS %%%%%%%%%%%%%%%%%%%%%%%% \begin{document} \title{A remark on the high energy asymptotics of the integrated density of states} \date{19 March 2002} \author{E.~Korotyaev\thanks{ Institut f\"ur Mathematik, Humboldt Universit\"at zu Berlin, Rudower Chaussee 25, 12489, Berlin, Germany. e-mail: ek@mathematik.hu-berlin.de}\mbox{ } and A.~Pushnitski\thanks{ Department of Mathematical Sciences, Loughborough University, Loughborough, LE11 3TU, U.K. e-mail: a.b.pushnitski@lboro.ac.uk}} \maketitle \begin{abstract} Assuming that the integrated density of states of a Schr\"odinger operator admits a high energy asymptotic expansion, explicit formulae for the coefficients of this expansion are given in terms of the heat invariants. \end{abstract} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\section{Introduction and Main Results} %\label{sec.a} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{1. Introduction} Let $H=-\Delta+\p(x)$ in $L^2(\R^d)$, $d\geq1$, where $\p\in C^\infty(\R^d)$ is a real-valued potential such that $\p$ and all the derivatives of $\p$ are uniformly bounded in $\R^d$. Let $\O_L=[-L/2,L/2]^d\subset\R^d$ and let $\chi_L$ be the characteristic function of $\O_L$. One says that $H$ \emph{has a density of states measure} (see e.g. \cite{Simon} or \cite{Kirsch}) if for all $g\in C_0^\infty(\R)$ the quantity $L^{-d}\Tr(\chi_L g(H))$ has a limit as $L\to\infty$. If the above limit exists for all $g$, then it can be represented as an integral \begin{equation} \lim_{L\to\infty}L^{-d}\Tr(\chi_L g(H))= \int_{-\infty}^\infty g(\l)dk(\l), %............................................................................(a.00) \label{a.00} \end{equation} where the Borel measure $dk(\l)$ is by definition the density of states measure. It is well known that in the case of periodic and almost-periodic potentials $\p$, the density of states measure exists. The function $$ k(\l):=\int_{-\infty}^\l dk(\l),\quad \l\in\R, $$ is called the \emph{integrated density of states}. The asymptotics of $k(\l)$ as $\l\to+\infty$ has been attracting considerable attention --- see \cite{ShenkShubin,HelfferM,Karpeshina} and references therein. For $\p\equiv0$, one has $k(\l)=(2\pi)^{-d}\omega_d\l_+^{d/2}$, where $\omega_d=\pi^{d/2}/\G(1+\frac{d}2)$ is the volume of a unit ball in $\R^d$ and $\l_+=(\abs{\l}+\l)/2$. For an arbitrary bounded $\p$ by a simple variational argument \cite{Shubin} one obtains \begin{equation} k(\l)=(2\pi)^{-d}\omega_d\l^{d/2}(1+O(\l^{-1})), \quad \l\to\infty. %............................................................................(a.0) \label{a.0} \end{equation} Moreover, if $d=1$ and $\p$ is periodic, an asymptotic expansion of $k(\l)$ is known \cite{ShenkShubin} (see also related results in \cite{Marchenko}): \begin{equation} k(\l)=(2\pi)^{-d}\omega_d\l^{d/2}\biggl( \sum_{j=0}^{N-1}Q_j\l^{-j}+O(\l^{-N})\biggr), \quad \l\to\infty, %............................................................................(a.16) \label{a.16} \end{equation} where $Q_j\in\R$ are some coefficients and $N>0$ can be taken arbitrary large. However, in the case $d\geq2$, $\p$ periodic, only a two-term asymptotic formula for $k(\l)$ is known \cite{HelfferM,Karpeshina} and the proof of this formula appears to be quite difficult. The purpose of this note is to discuss explicit formulae for the asymptotic coefficients $Q_j$ in \eqref{a.16}. We use the following simple observation which is probably known to some specialists but does not seem to have appeared explicitly in the literature. Consider the Laplace transform $L(t)=\int_{-\infty}^\infty e^{-t\l}dk(\l)$, $t>0$ of the density of states measure. It appears that for a wide class of potentials $\p$ including the periodic ones, a complete asymptotic expansion of $L(t)$ as $t\to+0$ can be easily obtained and the coefficients of this expansion can be explicitly computed in terms of the \emph{heat invariants} of the operator $H$. This expansion does not, of course, directly imply the asymptotics \eqref{a.16} of $k(\l)$. However, if the expansion \eqref{a.16} holds true with some (unknown) coefficients $Q_j$, then we immediately obtain explicit formulae for these coefficients. Proving the validity of the asymptotics \eqref{a.16} is, of course, a difficult analytic problem. However, it is often the case that the proof does not readily yield explicit formulae for the coefficients $Q_j$. We feel therefore that an independent simple method of computing these coefficients is of some value. \textbf{2. Heat invariants} Consider the operator $e^{-tH}$ and its integral kernel $e^{-tH}(x,y)$. It is well known that the following asymptotic expansion holds true as $t\to+0$: \begin{equation} e^{-tH}(x,x)\sim(4\pi t)^{-d/2} \sum_{j=0}^\infty t^j a_j(x), %............................................................................(a.2) \label{a.2} \end{equation} locally uniformly in $\R^d$. Here $a_j$ are polynomials (with real coefficients) in $\p$ and the derivatives of $\p$. The coefficients $a_j(x)$ are called \emph{local heat invariants} of the operator $H$. Explicit formulae for $a_j$ are given in \cite{HitrikP} (see also \cite{ColindV,Polt2}): \begin{equation} a_j(x)=\sum_{k=0}^{j-1} \frac{(-1)^j\G(j+\frac{d}{2})}{4^k k! (k+j)!(j-1-k)!\;\G(k+\frac{d}{2}+1)} (-\Delta_y+\p(y))^{k+j} (\abs{x-y}^{2k})\mid_{y=x}. %............................................................................(a.2a) \label{a.2a} \end{equation} In particular, $$ a_0=1,\quad a_1=-\p,\quad a_2=\tfrac12 \p^2-\tfrac16\Delta\p,\quad a_3=-\tfrac16 \p^3+\tfrac16 \p\Delta \p+\tfrac1{12}(\Delta\p)^2 -\tfrac16\Delta^2\p. $$ \textbf{3. Laplace transforms of $dk(\l)$} We start with a formal computation which explains the heart of the matter. Assume that all the limits \begin{equation} \lim_{L\to\infty}L^{-d} \int_{\O_L}a_j(x)dx=:M(a_j), \quad j=0,1,2\dots %............................................................................(a.4) \label{a.4} \end{equation} exist (note that this is obviously the case for periodic $\p$). By \eqref{a.00} and \eqref{a.2}, we obtain (formally!) \begin{equation} \int_{-\infty}^\infty e^{-t\l}dk(\l)=\lim_{L\to\infty}L^{-d}\Tr(\chi_L e^{-tH}) \sim(4\pi t)^{-d/2} \sum_{j=0}^\infty t^j M(a_j). %............................................................................(a.3) \label{a.3} \end{equation} The above formal computation can be easily justified: \begin{theorem} \label{t.a.1} Let $\p\in C^\infty(\R^d)$ and suppose that $\p$ and all the derivatives of $\p$ are uniformly bounded. Assume that the density of states measure $dk(\l)$ for $H=-\Delta+\p(x)$ exists and that the limits \eqref{a.4} exist for $j=0,1,2,\dots,N-1$. Then \begin{equation} \int_{-\infty}^\infty e^{-t\l}dk(\l)=(4\pi t)^{-d/2}\bigl( \sum_{j=0}^{N-1} t^j M(a_j)+O(t^N)\bigr), \quad t\to+0. %............................................................................(a.5) \label{a.5} \end{equation} \end{theorem} Note that the hypothesis of \thm{a.1} obviously holds true (with any $N>0$) for any periodic $\p\in C^\infty(\R^d)$. In order to justify the formal computation \eqref{a.3}, one only has to check that under our assumptions on $\p$, the asymptotic expansion \eqref{a.2} holds true uniformly in $x\in\R^d$. For periodic $\p$, this is quite obvious; in general case, this is also not difficult to prove by repeating the arguments of the papers \cite{AgmonKannai,Polt2,HitrikP} and keeping track of the remainder estimates in the asymptotic formulae. For completeness, below we give the proof. \textbf{4. Corollary} \thm{a.1} immediately gives explicit formulae for the coefficients $Q_j$ of the asymptotics \eqref{a.16}: \begin{corollary} \label{c.5} Assume the hypothesis of \thm{a.1}. Suppose that the integrated density of states $k(\l)$ has the asymptotics \eqref{a.16}. Then the coefficients $Q_j$ are given by \begin{equation} Q_0=1,\quad Q_j=\tfrac{d}{2}(\tfrac{d}2-1)\dots(\tfrac{d}{2}-j+1) M(a_j),\quad j=1,\dots, N-1. %............................................................................(a.17) \label{a.17} \end{equation} \end{corollary} Note that for $d$ even and $j\geq\frac{d}2+1$, one has $Q_j=0$. In all other cases, formula \eqref{a.17} can be recast as $$ Q_j=\frac{\G(\tfrac{d}2+1)}{\G(\tfrac{d}2-j+1)}M(a_j). $$ The proof of the Corollary is obtained by a direct application of Lemma 5.2 of \cite{ColindV}. Note that in the one-dimensional case, formulae for $Q_j$ were given in \cite{ShenkShubin}, although not as explicit as \eqref{a.17}: the coefficients $Q_j$ are computed as integrals of a sequence of functions defined by some recurrence relation. In the case $d\geq2$, formulae for $Q_0$ and $Q_1$ are given in \cite{HelfferM}. %\pagebreak %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% %\section{Proof of \protect\thm{a.1}} %\label{sec.b} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \textbf{5. Proof of \thm{a.1}} Essentially, we repeat the arguments of \cite{AgmonKannai} with combinatorial simplifications due to I.~Polte\-rovich \cite{Polt2,Polt3}. However, our proof of \eqref{a.2} is perhaps somewhat simpler than the proofs of \cite{AgmonKannai,Polt2,Polt3}; this is due to the fact that we use the iterated resolvent identity \eqref{b.1a} (discovered in \cite{Kantorovitz}), which gives a simple explicit form for the error term in the asymptotic formulae. The connection between the iterated resolvent identity and the the expansion \eqref{a.2} has been pointed out in \cite{HitrikP}. Denote $H_0=-\Delta$ in $L^2(\R^d)$. Below we use the notation $R_0(z)=(H_0-z)^{-1}$, $R(z)=(H-z)^{-1}$. 1. On the domain $\cap_{n\geq0}\Dom(H_0^n)$ define the operators $X_m$, $m\geq1$, recursively by \begin{equation} X_0=I, \quad X_{m+1}=X_m H_0-H X_m. \label{b.1} %...............................................................................................................(b.1) \end{equation} The operators $X_m$ are differential operators of the form \begin{equation} X_m=\sum_{\abs{a}\leq m-1} b_{m\a}(x)D^\a, \label{b.1b} %...............................................................................................................(b.1b) \end{equation} where $D^\a\equiv (\partial/\partial x_1)^{\a_1}\dots(\partial/\partial x_d)^{\a_d}$ and $b_{m\a}$ are polynomials in $\p$ and the derivatives of $\p$. The following identity holds true \cite{Kantorovitz} for any $M\geq1$: \begin{equation} R(z)=\sum_{m=0}^M X_m R_0^{m+1}(z) +R(z)X_{M+1}R_0^{M+1}(z). \label{b.1a} %...............................................................................................................(b.1a) \end{equation} In \cite{Kantorovitz}, the above identity has been proven in the context of Banach algebras, so strictly speaking, the proof applies only to bounded operators $H_0$, $H$. However, under our assumptions on $\p$, the identity \eqref{b.1a} can be easily proven directly by induction in $M$. Let us fix $c<0$, $c<\inf\spec(H)$, and $t>0$. Multiplying the identity \eqref{b.1a} by $e^{-tz}$ and integrating over $z$ from $c-i\infty$ to $c+i\infty$, one obtains \cite{Kantorovitz}: \begin{equation} e^{-tH}=\sum_{m=0}^M \frac{t^m}{m!}X_m e^{-tH_0}+ \frac1{2\pi i}\int_{c-i\infty}^{c+i\infty} R(z)X_{M+1}R_0^{M+1}(z)e^{-tz}dz, \quad t>0. \label{b.2} %...............................................................................................................(b.2) \end{equation} Multiplying \eqref{b.2} by $\chi_L$ and taking traces, one obtains: \begin{equation} \begin{split} \Tr(\chi_L e^{-tH})&=\sum_{m=0}^M \frac{t^m}{m!}\Tr(\chi_L X_m e^{-tH_0})+I(t), \\ \text{where }\quad I(t)&=\frac1{2\pi i}\int_{c-i\infty}^{c+i\infty} \Tr\bigl(\chi_L R(z)X_{M+1}R_0^{M+1}(z)\bigr)e^{-tz}dz. \end{split} \label{b.3} %...............................................................................................................(b.3) \end{equation} 2. Let us first estimate the remainder term $I(t)$. As $\ord(X_{M+1})\leq M$, the operator \linebreak $X_{M+1}\abs{R_0(z)}^{M/2}$ is bounded. Applying standard trace class estimates (see e.g. \cite{SimonTraceIdeals}), one gets \begin{multline} \abs{\Tr(\chi_L R(z) X_{M+1}R_0^{M+1}(z))} \\ \leq \norm{X_{M+1}\abs{R_0(z)}^{M/2}} \norm{\abs{R_0(z)}^{M/2+1}\chi_L R(z)}_{\SS_1} \leq C L^d\abs{z}^{\frac{d}2-\frac{M}2-2}, \quad \Re z\leq c, \label{b.4} %...............................................................................................................(b.4) \end{multline} where $\norm{\cdot}_{\SS_1}$ is the trace norm. It follows that for sufficiently large $M$, the integral $I(t)$ converges absolutely for all $t\geq0$. Moreover, as the trace in the integrand is analytic in $z$ for $\Re z\leq c$, by the deformation of contour argument it follows that $I(0)=0$. The same reasoning can be applied to the derivatives of $d^kI(t)/dt^k$ of the order $k<\frac{M-d}2+1$. Thus, \begin{equation} I(t) =O(L^d t^k),\quad t\to+0, \label{b.4a} %...............................................................................................................(b.4a) \end{equation} for any $k\in\N$, $k<\frac{M-d}2+1$. 3. Next, using \eqref{b.1b} and explicit formula for the integral kernel of $e^{-tH_0}$, one easily computes the $m$'th term in the sum in \eqref{b.3}: \begin{equation} t^m\Tr(\chi_L X_m e^{-tH_0})= t^{-d/2}\sum_{j=[m/2]+1}^m t^j\int_{\O_L}f_{mj}(x)dx, \quad t>0, \label{b.5} %...............................................................................................................(b.5) \end{equation} where $f_{mj}$ are some polynomials in $V$ and the derivatives of $V$. 4. Substituting \eqref{b.4a} and \eqref{b.5} into \eqref{b.3} and taking $M$ large (in fact $M=2N-2$ is sufficient), one obtains \begin{equation} \Tr(\chi_L e^{-tH})= (4\pi t)^{-d/2}\biggl(\sum_{j=0}^{N-1}t^j \int_{\O_L} a_j(x)dx+O(L^d t^N)\biggr), \quad t\to+0. \label{b.4b} %...............................................................................................................(b.4b) \end{equation} A detailed combinatorial analysis \cite{Polt2,HitrikP} of the coefficients $a_j(x)$ gives explicit formulae \eqref{a.2a}. Finally, recall (see e.g. \cite[Proposition C.7.2]{Simon}) that formula \eqref{a.00} holds true with $g(\l)=e^{-t\l}$ (although $g$ is not compactly supported). Thus, multiplying \eqref{b.4b} by $L^{-d}$ and taking $L\to\infty$, we arrive at \eqref{a.5}. \qed \textbf{6. 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