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\begin{document}
\title{ Lower Bounds on the Width of Stark--Wannier Type Resonances}
%%%%%%%%%\titlerunning{Lower Bounds on Stark--Wannier Resonances}
\author{J. Asch, P. Briet\\Centre de Physique Th\'eorique,
CNRS - Luminy, Case 907,\\ F-13288 Marseille Cedex 9, France\\ e-mail:
asch@cpt.univ-mrs.fr}
\maketitle
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\begin{abstract} We prove that the Schr\"odinger operator
$-d^2/dx^2+Fx+W(x)$ on $L^2({\bf R})$ with
$W$ bounded and analytic in a strip has no resonances in a region
${\hbox{\rm Im }} E\ge-\exp{(-C/F)}$.
\end{abstract}
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\def\R{{\bf R}}
\def\boxproof{\par\hskip11cm$\Box$\par}
\def\reminder{\vrule height 10pt width 6pt}
\def\bignorm{\vert\vert\vert f\vert\vert\vert}
\def\poid{\langle x\rangle}
\newtheorem{thm}{Theorem}[section]
\newtheorem{definition}[thm]{Definition}
\newtheorem{lemma}[thm]{Lemma}
\newtheorem{cor}[thm]{Corollary}
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\section{Introduction and Main Result} The resonance problem of a
Schr\"odinger particle subject to an electric field with non-vanishing mean
bears interesting physical and mathematical aspects and has attracted much
activity in both fields. From the point of view of transport in solids not
only fluctuations of short range or Coulomb type but also random,
quasiperiodic and especially periodic potentials, the Wannier case
\cite{wann}, are of interest. For the physics we refer to Avron \cite{avro},
Grecchi and Sacchetti \cite{grecsacc95} and their references. Mathematically
the classical questions are definition, existence and location of resonances.
They are non--trivial even in a one dimensional situation.
For the case of fluctuations which do not decay at infinity the definition
setup for resonances by spectral deformation was essentially given by Herbst
and Howland
\cite{herbhowl}.
The existence of resonances in the Wannier case was discussed by Avron
\cite{avro}. Rigorous results were obtained in the high field regime by
Agler, Froese \cite{aglefroe}; in a small field and semiclassical context by
Combes and Hislop \cite{combhisl}, Bentosela and Grecchi \cite{bentgrec}; for
potentials with a finite number of bands by Buslaev and Dmitrieva
\cite{busldimi}, Grecchi and Maioli and Sacchetti \cite{grecmaiosacc94} who
have also results for the disordered case with large periods
\cite{grecmaiosacc92}. The techniques in
\cite{combhisl,bentgrec,grecmaiosacc92,grecmaiosacc94} were spectral
deformation and perturbation theory; in \cite{busldimi} the complex poles of
the reflection coefficient were studied directly by ODE methods; in
\cite{aglefroe} a Birman--Schwinger technique was employed.
Concerning the location in the Wannier case it is suggested by the Zener
tilted band picture that the width --or imaginary part, or the inverse
lifetime
of the resonances-- is exponentially small in the strength of the
homogeneous part of the field
\cite{avro}. The works on the existence confirm this: upper bounds on the
imaginary part were given in \cite{combhisl,bentgrec,grecmaiosacc94} for the
semiclassical and for the finite band case.
There are several results concerning the notorious problem of lower bounds
on the resonance width. In \cite{6} Bentosela et al. proved for a very
general class of fluctuations that the spectrum is purely absolutely
continuous. In \cite{busldimi} asymptotics of the resonance width in the
field strength were obtained by a detailed study of the wavefunctions in the
Wannier case for finite band potentials; Jensen
\cite{jens} showed for bounded analytic fluctuations that resonances go away
from the real axis with increasing field strength; Ahia
\cite{ahia} gave an exponential bound with an explicit constant for compactly
supported fluctuations.
Lower bounds on the resonance width of Schr\"odinger particles in potentials
decaying at infinity are known from the works of Harrell \cite{harr} and of
Fernandez, Lavine \cite{fernlavi} for potentials of compact support and from
Helffer, Sj\"ostrand \cite{helfsjos} for harmonic wells on an island in the
semiclassical limit. The results in \cite{fernlavi,helfsjos} apply in more
than one dimension.
Our contribution are lower bounds which are exponentially small in the field
strength; they are valid for the class of potentials for which they are
expected to be true.
The condition on the potential is:
\begin{description}
\item{({\bf A})} Let $W$ be analytic and bounded in a strip around and real
on the real axis.
\end{description} The result is:
\begin{thm}\label{maintheorem} Let $W$ satisfy {\rm ({\bf A})}. There are
$F_0>0, c>0$ such that the selfadjoint operator
$$-d^2/dx^2+Fx+W,$$ uniquely defined by extension from $C_0^\infty({\bf
R})$, has no resonances in the region
$$\lbrace E\in{\bf C},{\hbox{\rm Im }} E\ge -e^{-c/F}\rbrace\qquad (FFrom the relation $(1+\theta f^\prime)(H(\theta)-E)(1+\theta f^\prime)=
H-E+\theta Pf +O(\theta^2)$ with the differential expression $P$ defined by
$$Pf:=-{f^{\prime\prime\prime}\over2}+2(Fx+W-E)f^\prime+(F+W^\prime)f,$$
it will be deduced that the existence of an $f$ which satisfies
$Pf=-1$ and is slowly oscillating implies (for $Re\theta=0$) that $\
Im(H(\theta)-E)$ is negative for
$-Im E-Im \theta<0$. So there are no resonances with $Im E>-1/\sup\vert
f^\prime\vert$. The problem of bounding $Im E$ is coded in the control of
solutions of $Pf=-1$. Our analysis of these is based on their intimate
relation to the solutions of the Schr\"odinger equation whose asymptotics are
well studied in the case at hand.
The physical background of this is: $Pf$ is the on--shell part of the
commutator of $H$ and the distortion operator:
$$\lbrack{1\over 2}(fD+Df),H\rbrack= -i(Pf+f^\prime(H-E)+(H-E)f^\prime)$$
((D:=$-i\partial_x$)). In this sense $Pf=-1$ means that
${1\over 2}(fD+Df)(t)$ is decreasing along trajectories:
for a wave packet concentrated near the energy shell
$\langle\psi(t),{1\over 2}(fD+Df)\psi(t)\rangle\sim -t$ for
$i\partial_t\psi(t)=H\psi(t)$. Remark that we discuss the full on--shell part
of the commutator and not, as it is usually done, its classical limit
$2(Fx+W-E)f^\prime+(F+W^\prime)f$.
In more than one dimension the on--shell part of the above commutator is not a
function; this is the main obstacle for an extension of our approach to that
case, see also \cite{briet}.
We shall proceed as follows. In Sect. 2 the resonance theory will be given
supposing existence
of a suitable distortion field $f$. In Sect. 3 we shall establish our
analytic result: the existence of a unique, slowly oscillating $f$ with
$\Vert f^\prime\Vert_\infty\le e^{c/F}$ and
$Pf=-1$ on a sufficiently large region and prove Theorem
\ref{maintheorem}.
Finally a remark concerning notation: the quantities which are derived from
$H$ and $f$ depend on the parameters $E,F,\theta$; if we feel that this is
clear from the context we shall not burden the reader with an explicit
notation of the indices. A generic constant not depending on the parameters
will be denoted by $cte.$, the norm of $u\in L^2({\bf R})$ by $\Vert u\Vert$.
$C_b^\infty$ are the functions whose derivatives are bounded.
\section{Resonances}\label{resonances} We shall now define resonances by
spectral deformation and establish a theory of resonance free domains.
Let $W$ satisfy ({\bf A}), $F>0$; denote by
$$H_F:=-\Delta +Fx+W(x)$$ --or simply $H$-- the selfadjoint operator which by
the Faris--Lavine theorem (see Reed and Simon \cite{RS2}) is uniquely defined
by extension from $C_0^\infty({\bf R})$.
For $f\in C^\infty({\bf R},{\bf R})\quad \sup_{x\in\bf R}\vert
f^\prime(x)\vert0$:
$\bignorm\le e^{c/F}$.\\ Then there exists $c^\prime>c, F_0 >0$ such that
for $F\le F_0$,
$H_F(\theta)$ can be extended to a Type (A) analytic family in
$\lbrace\theta\in{\bf C};\vert\theta\vert\le e^{-c^\prime/F}\rbrace$.
\end{thm} {\it Proof.}\par Since $C_0^\infty({\bf R})$ is a core for $H_F$
it is sufficient to show equivalence of the graph norms of $H_F(\theta)$ for
all $\vert\theta\vert$ small enough and analyticity of $\theta\mapsto
H_F(\theta)u$ for $u\in C_0^\infty({\bf R})$.
Denote $J(x,\theta):=1+\theta f^\prime(x)$, by $B$ a generic bounded
operator and by $cte.$ a generic positive real number. It holds:
\begin{eqnarray} H(\theta)-H & = & \nabla
g_1\nabla-J^{-1/2}(J^{-1}(J^{-1/2})^\prime)^\prime
\nonumber\\
& &+\theta F f + W(x+\theta f(x))-W(x)
\nonumber\\
& = & g_1\Delta+g_1^\prime\nabla +g_2 +B
\nonumber
\end{eqnarray}
with $g_1:=1-J^{-2}, g_2:=-J^{-1/2}(J^{-1}(J^{-1/2})^\prime)^\prime$. Let
$\chi^+$ be a real $C^\infty$ characteristic function of $(r,\infty)$ for an
$r>0$,
$\chi^-:=1-\chi^+$, $g^\pm:=\chi^\pm g$ for a function $g$. From the
assumption on
$f$ it follows with
$$\Vert g_1^+\Vert_{\bf R}+\Vert\langle x\rangle^{} g_1^-\Vert_{\bf R}
+\Vert\langle x\rangle^{-1/2} (g_1^\prime)^+\Vert_{\bf R}+$$
$$\Vert \langle x\rangle^{1/2} (g_1^\prime)^-\Vert_{\bf R} +\Vert \langle
x\rangle^{-1}g_2^+\Vert_{\bf R}+\Vert g_2^-\Vert_{\bf R}
\le cte. \vert\theta\vert e^{c/F}$$
Combined with the following four estimates (see also below):
$$\Vert\langle x\rangle^{-1}\Delta u\Vert+
\Vert\langle x\rangle^{-1/2}D u\Vert+
\Vert\chi^+\Delta u\Vert+\Vert\chi^+\langle x\rangle u\Vert
\le cte.(\Vert H_F u\Vert+\Vert u\Vert),$$ this gives
$\Vert (H_F(\theta)-H_F)u\Vert\le cte.\vert\theta\vert e^{c/F}(\Vert H_F
u\Vert+\Vert u\Vert) $ and so there is a $c^\prime>0$ and an $\alpha<1$ such
that for
$\vert\theta\vert\le e^{-c^\prime/F},$
$$\Vert (H_F(\theta)-H_F)u\Vert
\le \alpha(\Vert H_F u\Vert+\Vert u\Vert),$$ which implies equivalence of
the $H_F(\theta)$ graph--norms. Analyticity of \\$\theta\mapsto H_F(\theta)u$
is immediate. We finish the proof giving one of the operator estimates used
above:
\begin{eqnarray}
\Vert\langle x\rangle^{-1/2}Du\Vert^2&=& -\langle u, \nabla \langle
x\rangle^{-1}\nabla u\rangle
\nonumber\\ &=& -{1\over 2}\langle u, (\langle x\rangle^{-1}\Delta +\Delta
\langle x\rangle^{-1})u\rangle
+{1\over 2}\langle u, (\langle x\rangle^{-1})^{\prime\prime} u\rangle
\nonumber\\ &\le&{1\over 2}\vert\langle u, \langle x\rangle^{-1}H_F
+H_F\langle x\rangle^{-1}\langle u\rangle\vert+\vert\langle u, Bu\rangle\vert
\nonumber\\ &\le& cte.(\Vert\langle x\rangle^{-1}u\Vert\Vert H_Fu\Vert+\Vert
u\Vert^2)
\le cte.(\Vert H_F\Vert^2+\Vert u\Vert^2).
\nonumber
\end{eqnarray}
\boxproof We are now coming to the announced theory of resonance free
domains: Technically the simple idea to prove absence of resonances
explained in the introduction does not work for large values of $x$ where we
shall use ellipticity instead. We introduce a family of helper functions which
satisfy the following assumptions:
\begin{description}
\item{({\bf T})} For $F>0$ let $a\in C^\infty_b({\bf R},{\bf R})$,
$a(x)\in\lbrack -1,0\rbrack$,\\
$a(x)=-1\quad(x\in(-\infty, (1+\Vert W\Vert_{\bf R}+E)/ F))$,\\
$a(x)=0\quad(x\in((2+\Vert W\Vert_{\bf R}+E)/ F,\infty))$.
\end{description} The result is:
\begin{thm}\label{resonancefree} Assume $({\bf T})$ holds for $a$. Let $f$
be a solution of
$Pf=a$ such that $\exists c>0\forall E\in {\bf R}$ : $\bignorm\le e^{c/F}$.
Define $b$ by
$a+b=-1$.Then \\ there exists $c^\prime>2c, F^\prime >0$ such that for $F\le
F^\prime$it holds for
$\vert\theta\vert-{\beta\over 2}\rbrace \subset
\rho(H_F(\theta)),$$ where $\rho$ denotes the resolvent set.
\end{thm} {\it Proof.}\par A real shift in $\theta$ leads to a unitary
equivalent $H(\theta)$, so it is sufficient to treat the case
$\theta=i\beta$. Because of technicalities we treat the region of
ellipticity separately, this is accounted for by the function
$b$. Denote \hbox{$\mu(\theta):=(1+\theta b)(1+\theta f^\prime)$}. Let
$E\in{\bf R}$. It is known (cf.: \cite{briet,briecombducl87}, see also
\cite{briecombducl89}) that
$$\lbrace z\in {\bf C};Im \langle u,
\mu(\theta)(H(\theta)-z)\mu(\theta)u\rangle\le0\quad(u\in C_0^\infty({\bf
R})\rbrace
\subset \rho(H_F(\theta)).$$ So all we have to do in order to prove the
assertion is to show
$$Im (\mu(\theta)(H(\theta)-E)\mu(\theta))\le -\beta/2$$ in the quadratic
form sense on
$C_0^\infty({\bf R})$.
Denoting $V(x):= Fx+W(x)-E, J(x,\theta):=(1+\theta f^\prime(x))$ and using
the identity $g\nabla h\nabla g=\nabla g^2h\nabla+g(hg^\prime)^\prime$ we
have
$$(\mu(H-E)\mu)(\theta) =(1+\theta b)(-\Delta+J^2V(x+\theta f)-{1\over
2}J^{1/2}(J^{-3/2}J^\prime )^\prime)(1+\theta b).$$ Using $b<0$ this implies
$$Im(\mu(H-E)\mu)(\theta)\le Im(
\mu^2V(x+\theta f)-\theta b^{\prime\prime}(1+\theta b) -(1+\theta
b)^2({1\over2J}J^{\prime\prime}-{3\over 4J^2}J^\prime{}^2)).$$
A calculation using Taylor expansion in $\theta$ shows that the right-hand
side of this inequality is
$$Im(\theta (Pf+2bV+b^{\prime\prime}(1+\theta b)+
\theta(O(1)+O(1)f^{\prime\prime}{}^2+O(1)f^{\prime\prime\prime}f^\prime)))$$
where, using $\Vert f\Vert+\Vert f^\prime\Vert\le e^{c/F}$, one sees that
$\theta O(1)$ is uniformly bounded in $F,x$ for $\theta\epsilon>0$ there exists
$c^\prime>2c, F^\prime >0$ such that for $F\le F^\prime$ it holds for
$0\le\vert\theta\vert0,W\in C_b^\infty({\bf R},{\bf R})$.
Denote $\hbox{V(x):=Fx+W(x)-E}$. There exists a function
$G\in C^{\infty}({\bf R}^2\setminus\lbrace x=y\rbrace,{\bf R})\cap C^1({\bf
R}^2,{\bf R})$ such that for
$a\in C_b^\infty({\bf R},{\bf R}),
a(x)=0\quad(x>r>0 \hbox{ suitable})$ the function
$$f(x):=\int_{\bf R}G(x,y)a(y)dy\qquad(x\in{\bf R})$$ is in $C^\infty({\bf
R},{\bf R})$ and is the unique solution of
$$Pf=(-\partial^3/2+V\partial+\partial V)f=a$$ such that $\exists c>0 \forall
F>0$
$$\bignorm \le e^{c/F},$$ where
$$\bignorm:=\sum_{k=0}^3\Vert\langle x\rangle^{(1-k)\over2}\partial^k
f\Vert_{\bf R^+}+\Vert f\Vert_{\bf R^-} +
\Vert\langle x\rangle f^\prime\Vert_{\bf R^-} +
\Vert\langle x\rangle^{1\over2} f^{\prime\prime}\Vert_{\bf R^-}+
\Vert f^{\prime\prime\prime}\Vert_{\bf R^-}.$$ Furthermore, $c$ can be chosen
uniformly in $E$.
\end{thm} {\it Proof.}\par We shall give an explicit construction of the
Green's function in terms of solutions of (Sch), then verify the assertions.
Let us establish the rules of the game:
\begin{lemma}\label{lemma1} There exist two real, independent solutions
$\varphi_+,\varphi_-\in C^\infty$ of (Sch) with the following properties:
Choose $x_l1\quad(x\notin(x_l,x_r))$ and
$\vert V(x)+V^{\prime\prime}/V(x)\vert>1\quad(x\in(-\infty,x_l))$, define\\
$s(x):=\vert V(x)+
{V^{\prime\prime}/ V}(x)\vert^{1/2}\quad(x\in(-\infty,x_l))$. Then
\medskip
\noindent(1)\quad There exist $\alpha_+,\alpha_-\in{\bf C}\setminus(0)$;
$r_+,r_-,l\in C^\infty({\bf R}), c>0$ such that for $F>0$:
$$\varphi_\pm(x)={1\over
V(x)^{1/4}}\exp{(\pm\int_{x_r}^x\sqrt{V})}(1+r_\pm(x))\quad(x>x_r),$$
$$\varphi_\pm(x)={1\over
s(x)^{1/2}} (\alpha_\pm\exp{(i\int_{x_l}^xs)}+
\overline{\alpha_\pm}\exp{(-i\int_{x_l}^xs)})(1+l(x))\quad(x0, \forall F>0$:
$$\sum_{k=0}^1(\Vert\partial^k\varphi_-^2\Vert_{\bf R}+
\Vert\partial^k\varphi_+^2\Vert_{(-\infty,x_r)})+
\Vert\partial^2\varphi_\pm^2\Vert_{(x_l,x_r)}\le e^{c/F}.$$ For $g\in C_b^2$
it holds:
\noindent(3) for $x_r0,F^\prime>0$
such that
\begin{eqnarray} &&\lbrace z\in{\bf C}; Imz\ge e^{-c/F}\rbrace \subset
\nonumber\\
& &\bigcup_{E\in{\bf R}}\lbrace z\in {\bf C};\inf_{x\in{\bf
R}}
Im( ((1+i\beta b)(1+i\beta f^\prime))^2(z-E))>-{\beta\over 2}\rbrace,
\nonumber
\end{eqnarray}
\noindent Theorem \ref{resonancefree} implies absence of resonances in
$$\lbrace E\in{\bf C}; ImE\ge e^{-c/F}\rbrace.$$
\boxproof
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\end{document}
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