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Integrated density of states, high energy asymptotics,  
heat invariants
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%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% A REMARK ON THE HIGH ENERGY ASYMPTOTICS 
% OF THE INTEGRATED DENSITY OF STATES
%
%E. KOROTYAEV and A. PUSHNITSKI
%
% 
% 19 March 2002
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%


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\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. Acknowledgments} The authors are grateful to A.~Sobolev
for useful discussions.




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