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\begin{document}
\vspace*{5mm}
\noindent
{\Large\bf The absence of the absolutely continuous \\ spectrum for
$\,\delta'\,$ Wannier--Stark ladders}
\vspace{5mm}
\begin{quote}
{\large P.~Exner}
\vspace{10mm}
{\em Nuclear Physics Institute, Academy of Sciences,
25068 \v Re\v z near Prague, \\
and Doppler Institute, FNSPE, Czech Technical University,
B\v rehov{\'a} 7, \\ 11519 Prague, Czech Republic \\
\rm exner@ujf.cas.cz}
\vspace{8mm}
A modification of the Kronig--Penney model consisting of
equidistantly spaced $\,\delta'$--interactions is considered. We
prove that absolutely continuous spectrum of such a system disappears
under influence of an external electric field. The result extends to
periodic lattices of non--identical $\delta'$ interactions and
potentials which are lower unbounded and, up to a bounded term,
asymptotically decreasing with bounded first two derivatives.
\end{quote}
\section{Introduction}
One--dimensional Schr\"odinger operators with potentials composed of
a periodic and an aperiodic, most often linear, part have been
studied by many authors --- see, \eg,
%\cite{BG,CFKS,GMS,Ne}
[1--4] and
references therein --- most attention being paid to the resonance
structure of such systems. A basic ingredient of the typical
scattering picture is that the spectrum of the corresponding
Hamiltonian is absolutely continuous and covers the whole real line
provided the aperiodic part is below unbounded; this property can
be proven under rather weak differentiability requirements on
the potential.
The aim of the present paper is to show that the spectral properties
may change substantially if a smooth periodic potential is replaced
by an array of singular interactions. The best known system of this
type is the Kronig--Penney model, \ie, a sequence of equally spaced
$\,\delta$--interactions; its spectral properties in presence of an
external field remain an open problem.
However, in one--dimensional systems there are singular interactions
different from $\,\delta\,$. Another important class is
represented by the $\,\delta'$--interactions specified by the
boundary conditions (\ref{bc}) below; a detailed discussion of their
properties can be found in \cite{AGHH}. In distinction to $\,\delta$,
they cannot be approximated by fami\-lies of Schr\"odinger operators
with scaled short--range potentials, instead one can use fami\-lies
of rank--one operators \cite{Se1} or velocity--dependent potentials
\cite{Ca,CH}. Moreover, this does not exhaust all possible aspects of
the $\,\delta'$--interaction; elsewhere we have presented a heuristic
argument showing that it can be regarded as a paradigm for geometric
scatterers \cite{AEL}.
An important distinction between the two types of contact
interactions is mani\-fested in Kronig--Penney--type models (without an
external force): if we replace $\,\delta\,$ by $\,\delta'\,$ we
obtain the spectrum in which the band widths are asymptotically
constant while the gaps are widening. To understand this difference,
recall that the transmission probability through a single $\,\delta'\,$
barrier vanishes in the high--energy limit, and the boundary conditions
defining the interaction represent there an effective Neumann
decoupling \cite{AEL}. A propagation through an array of $\,\delta'\,$
barriers is made possible by their conspiracy, but only within narrow
energy windows close to the eigenvalues of the Neumann problem.
If an external potential is added, a useful insight into the problem is
obtained from the heuristic tilted--band picture according to which the
particle moves as quasifree in the regions where its energy finds within
a tilted spectral band of the periodic problem, while the complement
corresponding to the tilted gaps represents a classically forbidden
region which must be tunneled through. This suggests that in the
$\,\delta'\,$ case, where the gaps dominate the ``periodic'' spectrum at
high energies, an unrestricted propagation under influence of the external
force may not be possible.
Our main result confirms this conjecture: using the stability of the
absolutely continuous spectrum with respect to trace--class
perturbations in a way somewhat analogous to \cite{SiS}, we
demonstrate that the spectrum of $\,\delta'$ Wannier--Stark--ladder
Hamiltonian $\,H(\beta,E,a)\,$ is purely singular for any nonzero
``coupling constant'' $\,\beta\,$ and external field strength
$\,E\,$. Moreover, the validity of this result can be extended to
lattices of non--identical $\,\delta'$ interactions and a wide family
of external potentials, as we shall discuss in Section~4.
Let me add the following remark. The problem treated here was
formulated in collaboration with J.E.~Avron and Y.~Last. Together we
devised the mentioned strategy and wrote a proof for the $\,\delta'$
Kronig--Penney model with a linear potential. The result was
announced in Ref.\cite{AEL}; recently the same conclusion has been
reached by a completely different method \cite{MS}. Our original
proof had a flaw, however; in attempt to rectify it we formulated
separately two different arguments. The one published here appeared
to be applicable to a considerably wider class of operators, and my
coauthors insisted that I publish it in my name. While I respect
their decision and appreciate their scrupulous attitude, I want to
state that the credit for the main result of the paper is shared
equally by all three of us.
\section{The main result}
Consider the free Stark Hamiltonian $\,H_E:= \,-\,{d^2\over dx^2}\,
-Ex\,$ on $\,L^2(\R)\,$ with $\,D(H_E):= \,\{\,f\in H^{2,2}(\R):\;
H_Ef\in L^2\,\}\,$, and an equidistant lattice,
$\,\LL:=\{na\}_{n=-\infty}^{\infty}\,$ with a spacing $\,a>0\,$.
Suppose that at each point of $\,\LL\,$ we introduce the
$\,\delta'$--interaction of a strength $\,\beta_n\,$, \ie, we define
the operator $\,H(\{\beta_n\},E,a)\,$ in the following way: it acts
as $\,H_E\,$ on the intervals $\,J_n:=(na,(n\!+\!1)a)\,$ and its
domain differs from $\,D(H_E)\,$ by replacing the smoothness
requirement at the points of $\,\LL\,$ by the boundary conditions
\begin{equation} \label{bc}
f'(na+)\,=\, f'(na-)\,=:\, f'(na)\,, \;\quad f(na+)-f(na-)\,=\,
\beta_n f'(na)\;;
\end{equation}
the $\,\beta_n\,$ are real numbers or $\,+\infty\,$ in which case
(\ref{bc}) is replaced by the Neumann condition, $\,f'(na)=0\,$. If
the interaction strength at each point is the same, $\,\beta_n=
\beta\,$, we write $\,H(\{\beta_n\},E,a)=: H(\beta,E,a)\,$; in
particular, $\,H(0,E,a)= H_E\,$ and $\,H(\infty,E,a)\,$ is the
orthogonal sum of the single--interval operators obtained by imposing
the Neumann condition at each point of $\,\LL\,$.
Our main result is the following.
\begin{theorem} \label{ac spectrum theorem}
Let $\,E,\beta\ne 0\;$; then the absolutely continuous spectrum of
the $\,\delta'$ Wannier--Stark Hamiltonian $\,H(\beta,E,a)\,$ is
empty.
\end{theorem}
As mentioned above, we shall demonstrate this property by using known
results about the stability of the absolutely continuous spectrum.
More specifically, the proof will be based on comparing the resolvent
of $\,H(\beta,E,a)\,$ to that of a suitable operator with a pure
point spectrum. The latter is obtained by changing the $\,\delta'\,$
conditions to the Neumann one at points of a sublattice $\,\tilde\LL
\subset\LL\;$; following the heuristic tilted--band picture sketched
above we switch the boundary conditions deep in the tilted--gap regions,
because the wavefunctions are expected to be suppressed there due to
tunneling and the effect of such ``cuts'' is likely to be minimal.
\section{The proof}
To simplify the problem, one can cut the line into two halflines and
to consider each of them separately. Recall that changing the
boundary conditions at a point of $\,\LL\,$ to the Neumann one means
decoupling the Hamiltonian into an orthogonal sum; at the same time
it represents a rank--one perturbation in the resolvent which does
not change the essential spectrum. Furthermore, without loss of
generality we may put $\,a=1\,$, since the character of the spectrum
does not change at scaling transformations; for the sake of simplicity
we drop then $\,a\,$ from the symbols of the operators.
\subsection{The resolvent of $\,H(\{\beta_n\},E)\,$}
We shall start with the more general operator
$\,H(\{\beta_n\},E)\,$ on $\,L^2(\R^+)\,$ defined above and
compare it to a suitable ``decoupled'' operator; we assume the Neumann
boundary at the origin, $\,f'(0+)=0\,$. First we have to introduce
some notation.
Given $\,z=k^2\,$, we have in each interval $\,J_n\,$
just one solution $\,u_n\equiv u_n(\cdot,k)\,$ of the equation
$\,-u''(x)-[Ex+z]u(x)=0\,$ which satisfies the boundary conditions
$\,u_n(n+)=1\,, \;u'_n(n+)=0\,$. Similarly, there is exactly one
solution $\,v_n\,$ of this equation with $\,v_n((n+1)-)=1\,,
\;v'_n((n+1)-)=0\;$;
their Wronskian equals $\,W_n:= W(u_n,v_n)= v'_n(n+)=
-u'_n((n+1)-)\,$. Each of these solutions is of the form $\,f(-E^{1/3}
(\cdot\,+{z\over E}))\,$, where $\,f\,$ is a combination of Airy functions;
we assume for definiteness that $\,E>0\,$.
More generally, let $\,\tilde\LL:=\{n_\ell\}_{\ell=1}^\infty\,$ be an
increasing sequence of integers with $\,n_1=0\,$, which specifies a
sublattice of $\,\LL\,$. We denote
$$
\tilde\beta_n\,:=\, \left\{ \begin{array}{lll} \infty &\quad \dots
&\quad n=n_\ell \\ \beta_n &\quad \dots &\quad {\rm otherwise}
\end{array} \right.
$$
and moreover, $\,\tilde J_\ell:= (n_\ell, n_{\ell+1})\,$.
Then to any $\,z=k^2\,$, there is just one function $\,\tilde u_\ell\,$
which satisfies in $\,\tilde J_\ell\,$ the equation
$\,-u''(x)-[Ex+z]u(x)=0\,$ together with the b.c. (\ref{bc}) at
the points $\,n_\ell\!+1, \dots, n_{\ell+1}\!-1\,$ and
$\,\tilde u_\ell(n_\ell +)=1\,, \;\tilde u'_\ell(n_\ell +)=0\,$.
Similarly, there is a unique $\,\tilde v_\ell\,$ with
$\,\tilde v_\ell(n_{\ell+1}-)=1\,, \;\tilde v'_\ell(n_{\ell+1}-)
=0\,$, and the corresponding Wronskian equals
$$
\tilde W_\ell\,:=\, W(\tilde u_\ell,\tilde v_\ell) \,=\,
\tilde v'_\ell(n_\ell +) \,=\, -\tilde u'_\ell(n_{\ell+1}-)\,.
$$
The functions $\,\tilde u_\ell,\, \tilde v_\ell\,$ can be again
written explicitly in terms of Airy functions but we shall not need
it. Next, we define the operator $\,\Gamma\,$ on the weighted Hilbert
space $\,\ell^2(\beta_{n_\ell}n_\ell^{-1})\,$ by
\begin{eqnarray} \label{operator Gamma}
\Gamma_{\ell\ell} &\!\!:=\!\!& {1 \over \beta_{n_\ell}}\, \left(
\beta_{n_{\ell+1}}\,-\, {\tilde v_{\ell+1}(n_{\ell+1}+) \over
\tilde W_{\ell+1}}\,-\, {\tilde u_\ell (n_{\ell+1}-) \over
\tilde W_\ell} \right)\,, \nonumber \\
\Gamma_{\ell,\ell+1} \!&\!\!\phantom{:}=\!\!&\, \Gamma_{\ell+1,\ell} \,:=\,
{1 \over \sqrt{\beta_{n_\ell} \beta_{n_{\ell+1}}} \tilde W_{\ell+1}}
\end{eqnarray}
for $\,\ell=1,2,\dots\,$. This allows us to express the resolvent
difference.
\begin{theorem} \label{resolvent difference}
For any $\,k\,$ for which the operator $\,\Gamma\,$ is invertible,
the difference $\,C:=
(H(\{\beta_n\},E)-z)^{-1}- (H(\{\tilde\beta_n\},E)-z)^{-1}\,$
is an integral operator with the kernel
\begin{eqnarray} \label{resolvent kernel}
C(x,y) &\!=\!& \sum_{\ell,m}\, \Bigg\lbrace
{\tilde u_\ell(x) \over \sqrt{\beta_{n_\ell}} \tilde W_\ell \tilde W_m}\,
\left( {M_{\ell,m-1} \tilde v_m(y) \over \sqrt{\beta_{n_{m-1}}}} \,-\,
{M_{\ell m} \tilde u_m(y) \over \sqrt{\beta_{n_m}}} \right)
\nonumber \\ \\ &\!+\!&
{\tilde v_\ell(x) \over \sqrt{\beta_{n_{\ell-1}}} \tilde W_\ell \tilde W_m}\,
\left( {M_{\ell-1,m} \tilde u_m(y) \over \sqrt{\beta_{n_m}}} \,-\,
{M_{\ell-1,m-1} \tilde v_m(y) \over \sqrt{\beta_{n_{m-1}}}} \right)
\Bigg\rbrace \,, \nonumber
\end{eqnarray}
where $\,M:=\Gamma^{-1}$.
\end{theorem}
\noindent
{\em Proof:} The resolvent kernel of the ``chopped'' operator
$\,H_{\tilde\beta}:=H(\{\tilde\beta_n\},E)\,$ can be expressed as
\begin{equation} \label{chopped kernel}
G_{\tilde\beta}(x,y)\,=\, \left\lbrace\:
\begin{array}{lll} -\tilde W_\ell^{-1} \tilde u_\ell(x_<)\tilde
v_\ell(x_>) & \quad \dots \quad &
x,y\in \tilde J_\ell \\ \\ 0 & \quad \dots \quad & x,y\;\: {\rm belong\:
to\: different\;} \tilde J_\ell
\end{array}\right.
\end{equation}
where $\,x_<:= \min(x,y)\,,\; x_>:= \max(x,y)\,$. We look for the
resolvent kernel of $\,H_{\beta}:=H(\{\beta_n\},E)\,$ in the form
\begin{eqnarray} \label{Krein Ansatz}
G_\beta(x,y) \,=\, G_{\tilde\beta}(x,y) &\!+\!& \sum_{\ell,m}\, \Big\lbrack
\lambda_{\ell m}^{11} \tilde u_\ell(x)\tilde u_m(y) +\lambda_{\ell m}^{12}
\tilde u_\ell(x)\tilde v_m(y)
\nonumber \\
&\!+\!& \lambda_{\ell m}^{21} \tilde v_\ell(x)\tilde u_m(y) +
\lambda_{\ell m}^{22} \tilde v_\ell(x)\tilde v_m(y)
\Big\rbrack\,,
\end{eqnarray}
where $\,\lambda_{\ell m}^{jk}\,$ are coefficients to be found.
Since $\,H_{\tilde\beta}\,$ and $\,H_\beta:=H(\{\beta_n\},E)\,$ are
self--adjoint extensions of the same symmetric operator and
$\,\tilde u_\ell,\,\tilde v_\ell,\, \ell=1,2,\dots\,$, span its
deficiency subspaces, the last relation may be regarded as an
application of Krein's formula. Of course, the latter is conventionally
formulated for the case of finite deficiency indices
\cite[Appendix~A]{AGHH} and the existing infinite--dimensional
versions require one of the two extensions to be below bounded
\cite{BKN} or at least to have a gap \cite{Ne2} which might not be
the case here (\cf Conjecture~V.1). Fortunately, we do not need it:
since the functions
$\,\tilde u_\ell,\,\tilde v_\ell,\,$ with different indices have
disjoint supports, the \rhs of (\ref{Krein Ansatz}) is a finite sum;
at the same time it is the most general Ansatz for the sought kernel
due to the fact that $\,G_{\beta}(x,\cdot)\,$ and
$\,G_{\beta}(\cdot,y)\,$ have to solve the Schr\"odinger equation
\cite{convergence}.
The coefficients satisfy the obvious symmetry requirements
$$
\lambda_{\ell m}^{jj}=\lambda_{m\ell}^{jj}\;, \quad
\lambda_{\ell m}^{12}=\lambda_{m\ell}^{21}\,.
$$
To find them, notice that by definition $\,(H_\beta\!-z)^{-1}\,$ maps
$\,L^2(\R^+)\,$ into $\,D(H_\beta)\;$; in particular, $\,f:=G_\beta g\,$
must satisfy the boundary conditions (\ref{bc}) for any $\,g\in
L^2(\R^+)\,$ at each point of $\,\tilde\LL\,$. The relations
(\ref{chopped kernel}) and (\ref{Krein Ansatz}) give
\begin{eqnarray*}
f(x) &\!=\!& -\,\tilde W_\ell^{-1}\tilde u_\ell(x) \int_x^{n_{\ell+1}}
\tilde v_\ell(y)g(y)\,dy\,-\, \tilde W_\ell^{-1}\tilde v_\ell(x)
\int_{n_{\ell}}^x \tilde u_\ell(y)g(y)\,dy \\ \\
&& +\, \sum_m \left(\, \lambda_{\ell m}^{11} \tilde u_\ell(x) h_m^u+
\lambda_{\ell m}^{12} \tilde u_\ell(x) h_m^v+
\lambda_{\ell m}^{21} \tilde v_\ell(x) h_m^u+
\lambda_{\ell m}^{22} \tilde v_\ell(x) h_m^v\, \right)
\end{eqnarray*}
for $\,x\in\tilde J_\ell\,$, where
$$
h_m^u\,:=\, \int_{\tilde J_m} \tilde u_m(y)g(y)\,dy\,, \quad
h_m^v\,:=\, \int_{\tilde J_m} \tilde v_m(y)g(y)\,dy\,.
$$
We express from here the boundary values, substitute into (\ref{bc}) and
use the fact that $\,h_m^u,\, h_m^v\,$ are independent quantities; this
yields the following infinite system of linear equations
\begin{eqnarray*}
\lambda_{\ell m}^{2,j} \tilde v'_\ell(n_\ell +) - \lambda_{\ell-1,m}^{1,j}
\tilde u'_{\ell-1}(n_\ell -) &\!=\!& 0\,, \\ \\
\lambda_{\ell m}^{11} + \lambda_{\ell m}^{21} \tilde v_\ell(n_\ell +)
- \lambda_{\ell-1,m}^{11} \tilde u_{\ell-1}(n_\ell -)
- \lambda_{\ell-1,m}^{21} +
\delta_{\ell-1,m} \tilde W^{-1}_{\ell-1} &\!=\!& \beta_{n_\ell}
\lambda_{\ell m}^{21} \tilde v'_\ell(n_\ell +)\,, \\ \\
\lambda_{\ell m}^{12} + \lambda_{\ell m}^{22} \tilde v_\ell(n_\ell +)
- \lambda_{\ell-1,m}^{12} \tilde u_{\ell-1}(n_\ell -)
- \lambda_{\ell-1,m}^{22} - \delta_{\ell m} \tilde W^{-1}_{\ell}
&\!=\!& \beta_{n_\ell} \lambda_{\ell m}^{22} \tilde v'_\ell(n_\ell +)\,,
\end{eqnarray*}
where $\,j=1,2\,,\; \ell=2,3,\dots\,$ and $\,m=1,2,\dots\,$, together
with the boundary requirement $\,\lambda_{1,m}^{2,j}=0\,$. This can be
simplified further by choosing
\begin{eqnarray*}
\lambda_{\ell m}^{11} &\!=\!& -\,{M_{\ell m} \over
\sqrt{\beta_{n_\ell}\beta_{n_m}}
\tilde W_\ell \tilde W_m}\;, \qquad
\lambda_{\ell m}^{21} \,=\, {M_{\ell-1,m} \over
\sqrt{\beta_{n_{\ell-1}}\beta_{n_m}} \tilde W_\ell \tilde W_m}\;, \\ \\
\lambda_{\ell m}^{12} &\!=\!& {M_{\ell,m-1} \over
\sqrt{\beta_{n_\ell}\beta_{n_{m-1}}} \tilde W_\ell \tilde W_m}\;, \;\qquad
\lambda_{\ell m}^{22} \,=\, -\,{M_{\ell-1,m-1} \over
\sqrt{\beta_{n_{\ell-1}}\beta_{n_{m-1}}} \tilde W_\ell \tilde W_m}
\end{eqnarray*}
for $\,\ell,m=1,2,\dots\,$. The first two sets of equations are then
satisfied identically, while the third one can be written with the help
of the identitites $\,\tilde W^{-1}_{\ell-1} \delta_{\ell-1,m}=
\tilde W^{-1}_m \delta_{\ell-1,m}\,$ and $\,\sqrt{\beta_{n_{\ell-1}}}\,
\delta_{\ell-1,m}= \sqrt{\beta_{n_m}}\, \delta_{\ell-1,m}\,$ as
\begin{eqnarray*}
\lefteqn{{M_{\ell m} \over \sqrt{\beta_{n_\ell}\beta_{n_{\ell-1}}}
\tilde W_\ell} \,-\,{M_{\ell-1,m} \over \beta_{n_{\ell-1}}}
\left( \beta_\ell -\,{\tilde v_\ell(n_\ell +) \over \tilde W_\ell}
-\,{\tilde u_{\ell-1}(n_\ell -) \over \tilde W_{\ell-1}} \right)} \\ \\ &&
\phantom{\hspace{57mm}} +\,
{M_{\ell-2,m} \over \sqrt{\beta_{n_{\ell-1}}\beta_{n_{\ell-2}}}
\tilde W_{\ell-1}} \,=\,\delta_{\ell-1,m}\,, \phantom{\hspace{13mm}}
\end{eqnarray*}
or $\,(\Gamma M)_{\ell-1,m}= \delta_{\ell-1,m}\,$. The fourth set of
equations leads to the same conclusion; since $\,\Gamma\,$ is invertible
by assumption, we have $\,M=\Gamma^{-1}\,$. \quad \QED
\vspace{3mm}
Before proceeding further, let us comment on the hypothesis of the
theorem. The invertibility of $\,\Gamma\,$ is related naturally to
spectral properties of $\,H_\beta\,$. To see this, we write the solution
to the Schr\"odinger equation in question for a given $\,k\,$ as
$$
f(x)\,=\, -\,\tilde W^{-1}_\ell f'(n_{\ell+1}) \tilde u_\ell(x)\,+\,
\tilde W^{-1}_\ell f'(n_{\ell}) \tilde v_\ell(x)
$$
if $\,x\in\tilde J_\ell\,,\; \ell=1,2,\dots\,$; it is unique up to a
multiplicative constant. A straightforward modification of the argument
from \cite[Sec.III.2.1]{AGHH} shows that $\,f\,$ satisfies the
conditions
\begin{equation} \label{discrete reformulation}
\sum_{m=1}^{\infty} \Gamma_{\ell m} \sqrt{\beta_{n_m}}\, f'(n_{m+1})
\,=\,0
\end{equation}
for any $\,\ell=1,2,\dots\,$. The operator $\,H_\beta\,$ has an
eigenvalue \Iff $\,f\in L^2(\R^+)\,$. In the situation considered below,
\ie, for $\,\im z\,$ large enough and the cutting points chosen in the
tilted gaps, this is further equivalent to $\,\{f'(n_m)\}\in
\ell^2(\beta_{n_m} k^{-2}_{n_m})\,$ --- \cf (\ref{exponential decay})
and (\ref{UV norm}). Hence $\,\Gamma\,$ is invertible in this case;
below we shall demonstrate directly that it is bounded
invertible.
\subsection{A trace--class estimate}
As we have said, the proof of Theorem~\ref{ac spectrum
theorem} can be split into two halfline problems. The
growing--potential case is easy. The Kronig--Penney operator without
an electric field is below bounded, $\,H(\beta,0)\ge -c\,$ for some
$\,c\ge 0\,$. Changing the boundary condition to Neumann at a point
$\,-n\,$ does not change the essential spectrum and a finite interval
does not contribute to it, so
$$
\inf \sigma_{ess}(H(\beta,E))\,\ge\, -c+En\,.
$$
Since $\,n\,$ can be chosen arbitrarily, the spectrum is pure point.
Let us turn to the more difficult case when the potential decreases
along the halfline. If the b.c. (\ref{bc}) are changed to the Neumann
one at infinitely many points, the corresponding operator
$\,H(\{\tilde\beta_n\},E)\,$ has certainly a pure point spectrum.
Our aim is to show that for a suitably chosen sequence
$\,\{n_\ell\}\,$ and some $\,z\in\C\,$, the resolvent difference of
Theorem~\ref{resolvent difference} is a trace--class operator, in
which case the sought result follows from the Birman--Kuroda theorem
\cite[Sec.XI.3]{RS}.
We denote $\,z=\rho+i\nu\,$, where both $\,\rho,\,\nu\,$ will be
chosen large positive. If $\,\rho\,$ is large, the asymptotic
properties of the Airy functions \cite[Chap.10]{AS} show that the
elementary solutions introduced above are in the intervals $\,J_n\,$
of the form
\begin{equation} \label{WKB}
u_n(x)= \cos k_n(x\!-\!n)\,(1+\OO(k_n^{-1}))\,, \quad
v_n(x)= \cos k_n(x\!-\!n\!-\!1)\,(1+\OO(k_n^{-1}))\,,
\end{equation}
where $\,k_n:=\sqrt{z+E\overline x_n}\,$ for some
$\,\overline x_n \in J_n\,$. The momentum values behave
asymptotically as
\begin{equation} \label{momentum}
k_n\,=\, \left( \sqrt{E}\,n^{1/2} +{z\over 2\sqrt E}\,n^{-1/2} \right)
\,(1+\OO(n^{-1}))
\end{equation}
for $\,n\to\infty\,$. The Wronskian is then $\,W_n= k_n \sin k_n
\,(1+\OO(n^{-1/2}))\;$; since $\,z\,$ is complex, it is never zero and
one can estimate it by
\begin{equation} \label{Wronskian}
|W_n|\,\geq\, {\nu\over 2}\,\left(1+\OO(n^{-1/2})\right)\,.
\end{equation}
We also need a bound for
$$
{W_{n-1}\over W_n}\,=\, \left(1+\, {\sin k_{n-1} -\sin k_n
\over \sin k_n} \right) \,\left(1+\OO(n^{-1/2})\right)\;;
$$
it is clear that one should pay attention only to the vicinity of the
points where $\,\re k_n\approx \pi\ell\,$. There we have
$$
\left| {\sin k_{n-1} -\sin k_n \over \sin k_n} \right|
\,\leq\, 2\left({E\over\nu}+{1\over n}\right)\,
\left(1+\OO(n^{-1})\right)\,,
$$
which yields the bound
\begin{equation} \label{Wronskian ratio}
\left| {W_{n-1}\over W_n}\,-\,1 \right| \,\leq\, {2E\over\nu}\,+\,
\OO(n^{-1/2})\,.
\end{equation}
The functions $\,\tilde u_\ell\,,\; \tilde v_\ell\,$ can be estimated
using ``transfer matrices'' relating the elementary solutions in
neighboring intervals. A general solution of the equation
$\,-f''(x)-[Ex+z]f(x) =0\,$ in $\,J_n\,$ has the form $\,f_n(x)=
\xi_n u_n(x)+ \eta_n v_n(x)\;$; we look for $\,T_n\,$ such that
$$
\left( \begin{array}{c} \xi_n \\ \eta_n \end{array} \right) \,=\,
T_n \left( \begin{array}{c} \xi_{n-1} \\ \eta_{n-1} \end{array} \right)\,.
$$
Using the b.c. (\ref{bc}), one proves easily
\begin{eqnarray*}
T_n &\!=\!& \left( \begin{array}{cc} -\beta_n W_{n-1}+
\, {W_{n-1}\over W_n} v_n(n+) +u_{n-1}(n-)\; & 1 \\ \\
-\,{W_{n-1}\over W_n} & 0 \end{array} \right)\,, \\ \\
T_n^{-1} &\!=\!&\; \left( \begin{array}{cc} 0 & -\,{W_n\over W_{n-1}}
\\ \\ 1\;\; & -\beta_n W_n + v_n(n+)+ \,{W_n\over W_{n-1}} u_{n-1}(n-)
\end{array} \right)\;;
\end{eqnarray*}
in our case $\,\beta_n=\beta\,$. Using now
(\ref{WKB})--(\ref{Wronskian ratio}), we get
\begin{eqnarray*}
|(T_n)_{11}| &\!\geq\!& |\beta|\, |W_{n-1}| -|u_{n-1}(n-)+v_n(n+)|
-|v_n(n+)|\,\left( {2E\over\nu}+\OO(n^{-1/2})\right) \nonumber \\
&\!\geq\!& \left( {\beta\nu\over 2}\, -2 -{2E\over\nu}\,
\right) \,\left(1+\OO(n^{-1/2})\right)\,,
\end{eqnarray*}
so choosing $\,\nu\,$ large enough, the \rhs can be made positive
and sufficiently large as $\,n\to\infty\,$. At the same time,
$$
|(T_n)_{21}|\,\leq\, 1\,+\,{2E\over\nu}\,+\, \OO(n^{-1/2})\,.
$$
Since the coefficients of $\, {\xi_n \choose \eta_n}\,$
corresponding to $\,\tilde u_\ell(x)\,$ start from $\,{1
\choose 0}\,$ at $\,x=n_\ell\,$, the above bounds tell us
that for all sufficiently large $\,n\,$ there is a number
$\,d>1\,$ independent of $\,n\,$ such that
\begin{equation} \label{exponential decay}
|\xi_n|\,\ge\, d^j|\xi_{n-j}|\,, \qquad |\xi_n|\,\ge\, d|\eta_n|\,.
\end{equation}
The norm of $\,\tilde u_\ell\,$ can be then estimated as follows,
$$
\|\tilde u_\ell\|^2\,\le\, 2
\sum_{n=n_\ell}^{n_{\ell+1}-1}\, (|\xi_n|^2+|\eta_n|^2) \,\le\, 2\,
|\xi_N|^2\,(1+d^{-2})\, \sum_{j=0}^{N'} d^{-2j}\,,
$$
where $\,N:= n_{\ell+1}-1\,$ and $\,N':=n_{\ell+1}-n_{\ell}-1\,$.
Hence there is a positive $\,d'\,$ independent of $\,\ell\,$ such
that $\,\|\tilde u_\ell\| \le d'|\xi_{n_{\ell+1}-1}|\,$. The
coefficients of $\,\tilde v_\ell\,$ behave similarly if we change the
direction and switch the roles of $\,\xi_n,\,\eta_n\;$; this yields
the inequality $\,\|\tilde v_\ell\| \le d'|\eta_{n_{\ell}}|\,$.
For simplicity, we introduce $\,U_\ell:= -\, {\tilde u_\ell(\cdot)\over
\tilde u'_\ell(n_{\ell+1}-)}\,$ and $\,V_\ell:=
{\tilde v_\ell(\cdot)\over \tilde v'_\ell(n_\ell +)}\,$. The
denominators depend clearly on the choice of the sequence
$\,\{n_\ell\}\,$. We pick its points roughly in ``the middle of the
gaps'' assuming, \eg,
\begin{equation} \label{subsequence}
\left|\, k_{n_{\ell}}\,-\, \pi\left(\ell+{1\over 2}\right)\,\right|
\,\leq\,{\pi\over 4}\,,
\end{equation}
so that $\,n_\ell= {\pi^2\ell^2\over E}\,\left(1+\OO(\ell^{-1})\right)\,$.
Since $\,\tilde v_{n_{\ell+1}-1}(n_{\ell+1}-)=0\,$ by
assumption, we have in view of (\ref{WKB}) and (\ref{subsequence})
$$
\left| \tilde u'_\ell(n_{\ell+1}-)\right| \,\ge\, {1\over\sqrt 2}\,
k_{n_{\ell+1}-1} |\xi_{n_{\ell+1}-1}|\,\left(1+\OO(\ell^{-1})\right)\,.
$$
In the same way, $\,|\tilde v'_\ell(n_\ell +)|\,$ has a bound
proportional to $\,|\eta_{n_\ell}|\,$; using the above inequalities
for the norms of these functions, we get
\begin{equation} \label{UV norm}
\left\| U_\ell \right\| \,\leq\, {\sqrt{2}\,d'\over
k_{n_{\ell+1}-1}}\,\left(1+\OO(\ell^{-1})\right)\,, \qquad
\left\| V_\ell \right\| \,\leq\, {\sqrt{2}\,d'\over
k_{n_{\ell}}}\,\left(1+\OO(\ell^{-1})\right)\,.
\end{equation}
To estimate the difference of the resolvents, we need also a bound for
the coef\-fi\-cients $\,M_{\ell m}\,$ in (\ref{resolvent kernel}).
Since all the $\,\beta_n\,$ are the same by assumption, the operator
(\ref{operator Gamma}) can be written as $\,\Gamma= I+N\,$ with
$$
N_{\ell\ell}\,:=\, -\beta^{-1} (U_\ell(n_{\ell+1}-)
+V_{\ell+1}(n_{\ell+1}+))\,, \qquad
N_{\ell,\ell+1}\,=\, N_{\ell+1,\ell}\,:=\, \beta^{-1}\tilde W_\ell^{-1}\,.
$$
The above estimates then yield
\begin{eqnarray*}
|N_{\ell\ell}| &\!\leq\!& {2\over\pi\ell\beta}\,
\left(1+{\sqrt{2}\over d}\, \right)\,\left(1+\OO(\ell^{-1})\right)\,, \\
|N_{\ell,\ell+1}| &\!\leq\!& {\sqrt{2}\over\pi\ell\beta}\, d^{-N'}
\,\left(1+\OO(\ell^{-1})\right)\,\le\, {2\over\pi\ell\beta}\,
e^{-2\ell\pi^2 E^{-1}\ln d}\,.
\end{eqnarray*}
Recall again that a contribution from a finite interval cannot affect
the essential spectrum. Hence we may consider $\,\{n_\ell\}\,$ with
$\,\ell\,$ starting at a sufficiently large value, so that $\,\|N\|<1\,$.
The inverse is then easily computed as a geometric series,
$$
M_{\ell m} \,=\, \delta_{\ell m}\,-\, N_{\ell m} \,+\,
\sum_{s=2}^\infty\,(-1)^s \,\sum_{r_2,\dots,r_s} N_{\ell r_2} N_{r_2
r_3} \dots N_{r_s m}\,.
$$
Since the elements of the $\,r$--th side diagonal contain at least
$\,r\,$ off--diagonal elements of $\,N\;$ (recall that the latter
is tridiagonal!), there are positive $\,C,\,d''\,$ such that
\begin{equation} \label{off-diagonal decay}
|M_{\ell,\ell+r}|\,\leq\, C\, e^{-d''|r|}\,.
\end{equation}
Putting the estimates (\ref{UV norm}) and (\ref{off-diagonal decay})
together in combination with (\ref{momentum}) and
(\ref{subsequence}), we get
\begin{eqnarray*}
\Tr|C| &\!\leq\!& {1\over\beta}\: \sum_{\ell,m} \Big\lbrace
|M_{\ell m}|\, \|U_\ell\|\, \|U_m\|\,+ |M_{\ell,m-1}|\,
\|U_\ell\|\, \|V_m\| \\
&\!+\!& |M_{\ell-1,m}|\, \|V_\ell\|\, \|U_m\|\,+
|M_{\ell-1,m-1}|\, \|V_\ell\|\, \|V_m\|\, \Big\rbrace \\ \\
&\!\leq\!& {C(2d')^2\over\pi^2\beta}\: \sum_{\ell} \Bigg\lbrace\:
{1\over\ell}\: \sum_{r\geq 0} \left( {1\over\ell+r} \,+\,
{1\over\ell+1+r}\right)\, e^{-d''r} \\ \\
&\!+\!& {1\over\ell-1}\: \sum_{r\geq 0} \left( {1\over\ell+r} \,+\,
{1\over\ell-1+r}\right)\, e^{-d''r} \Bigg\rbrace\; <\,\infty\,,
\end{eqnarray*}
what we wanted to prove. \quad \QED
\section{Generalizations}
The presented proof of Theorem~\ref{ac spectrum theorem} extends easily
at least in two directions:
\begin{description}
\item{(i)} {\em the background potential need not be linear:} let
$\,H_V:=\,-\,\frac{d^2}{dx^2}\,+V(x)\,$ be the ``free'' Schr\"odinger
operator with a potential $\,V\,$ to be specified below; then we
can define $\,H(\{\beta_n\},V,a)\,$ as in Section~2, \ie, through the
b.c.(\ref{bc}).
\item{(ii)} one may consider a lattice of {\em non--identical
$\,\delta'$ interactions}, \ie, a general sequence $\,\{\beta_n\}\,$.
\end{description}
For simplicity, let us consider again only the halfline problem with
a decreasing potential and the lattice spacing $\,a=1\,$. The key
observation is that the asymptotics (\ref{WKB}) of the elementary
solutions comes in fact from the WKB expansion, and therefore it is
valid for any potential $\,V\,$ which is decreasing and regular enough
provided we define $\,k_n:=\sqrt{z-V(\overline x_n)}\,$. Assume, \eg,
that
\begin{description}
\item{(a)} $\;V\,$ is locally bounded and there is $\,x_0\,$ such that
for all $\,x>x_0\,$ one has $\,V(x)=-U(x)+W(x)\,$, where
\begin{description}
\item{(a1)} $\;U\,$ is nondecreasing with $\,\lim_{x\to\infty}
U(x)=\infty\,$,
\item{(a2)} $\;U\,$ is $\,C^2$ smooth with $\,|U'(x)|\le c\,$ and
$\,|U''(x)|\le\tilde c U(x)\,$ for some $\,c,\tilde c>0\,$,
\item{(a3)} $\;W\,$ is piecewise continuous and bounded,
$\,\|W\|_{\infty}<\infty\,$.
\end{description}
\end{description}
Using the standard WKB error estimates \cite[Sec.6.2]{Ol} we can replace
(\ref{WKB}) by
$$
u_n(x)= \cos k_n(x\!-\!n)\,(1+\OO(U(x)^{-1/2}))
$$
and an analogous expression for $\,v_n(x)\;$; the momentum behaves
asymptotically as
$$
k_n\,=\, \left(\, \sqrt{U(\overline x_n)}\,+\, {z-W(\overline x_n)\over
2\sqrt{U(\overline x_n)}}\,\right) \,(1+\OO(U(\overline x_n)^{-1}))\,.
$$
Since $\,U'\,$ is bounded by (a2), the key estimate (\ref{Wronskian
ratio}) is replaced by
$$
\left| {W_{n-1}\over W_n}\,-\,1 \right| \,\leq\, \left\lbrace\,
{4(c+\|W\|_{\infty})\over\nu}\,+\, {4c\over |U(n-1)|} \right\rbrace
\,(1+\OO(U(n)^{-1}))\,,
$$
so the ``transfer--matrix'' analysis may be repeated. As for the
``coupling constants'', we adopt the following assumptions:
\begin{description}
\item{(b)} $\;|\beta_n| \geq\beta>0\;$ for all $\,n\,$,
\item{(c)} there is a monotonic sequence
$\,\{n_\ell\}\subset \Z_+\,$ such that $\,\re k_{n_\ell}=
\pi(n_\ell+ \epsilon_\ell)\,$ with $\,\epsilon_\ell
\in ({1\over 4},{3\over 4})\,$, and $\,\beta_{n_\ell}
\beta_{n_{\ell+1}}^{-1}\,$ remains bounded as $\,n_\ell\to\infty\,$,
\end{description}
The matrix elements (\ref{operator Gamma}) of the operator
$\,\Gamma\,$ can be now written as
$$
\Gamma_{\ell m}\,=\, {1\over\sqrt{\beta_{n_\ell}}}\, (B+S)_{\ell m}\,
{1\over\sqrt{\beta_{n_m}}}\,,
$$
where $\,B:={\rm diag\,} \{\dots,\beta_{n_{\ell+1}}, \dots\}\,$,
and $\,S_{\ell\ell}:= -U_\ell(n_{\ell+1}-) -V_{\ell+1}(n_{\ell+1}+)\,$,
$\,S_{\ell,\ell+1}= S_{\ell+1,\ell}:= \tilde W_\ell^{-1}\,$. If the
sequence $\,\{n_\ell\}\,$ starts with $\,\ell\,$ large enough,
$\,\|B^{-1}N\|<1\,$ and the inverse is given by
\begin{eqnarray*}
M_{\ell m} &\!=\!& \beta_{n_\ell}\beta_{n_{\ell+1}}^{-1}
\delta_{\ell m}\,-\, \sqrt{\beta_{n_\ell}}\beta_{n_{\ell+1}}^{-1}
S_{\ell m} \beta_{n_{m+1}}^{-1} \sqrt{\beta_{n_m}} \\ \\
&\!+\!& \sum_{s=2}^\infty\,(-1)^s \,\sum_{r_2,\dots,r_s}
\sqrt{\beta_{n_\ell}}\beta_{n_{\ell+1}}^{-1} S_{\ell r_2}
\beta_{n_{r_2+1}}^{-1} \dots S_{r_s m}
\beta_{n_{m+1}}^{-1} \sqrt{\beta_{n_m}}\,.
\end{eqnarray*}
Since the numbers $\,\beta_{n_\ell}\beta_{n_{\ell+1}}^{-1}\,$
remain bounded at the cutting points in view of (c), the estimate
(\ref{off-diagonal decay}) holds again.
In this way, we arrive at the following result.
\begin{theorem} \label{generalization}
Let $\,H(\{\beta_n\},V,a)\,$ be the operator on
$\,L^2(\R^+)\,$ defined above, with $\,V\,$ and
$\,\{\beta_n\}\subset\R\,$ satisfying the assumptions (a)--(c).
Then $\,\sigma_{ac}(H(\{\beta_n\},V,a)=\emptyset\,$.
\end{theorem}
The result extends easily to operators on the whole real line with
potentials growing in the other direction, because for the other
halfline we have used just the fact that the spectrum is purely
discrete when the potential tends to infinity. A more involved
argument is required, however, if the sequence $\,\{\beta_n\}\,$ may
approach zero, since then the point--interaction Hamiltonian without
the presence of an external field need not be below bounded.
\section{Concluding remarks}
The assumptions used here are clearly not optimal, but we are not
going to push the argument further. Let us remark instead that the
mentioned splitting of the problem on the line into two halflines
together with Proposition~\ref{generalization} shows that the
absolutely continuous spectrum is void not only for $\,\delta'$
Wannier--ladder ``slopes'' but for ``hills'' with the potential
decreasing in both directions as well.
In addition to finding weaker restrictions on the potential decay,
other generalizations are possible. For instance, one can treat
similarly arrays of $\,\delta'$--interactions, where the spacing
assumes a finite number of different values. Furthermore,
$\,\delta\,$ and $\,\delta'$ are extreme cases in the general
four--parameter class of one--dimensional point interactions
\cite{Ca,CH,Se2}. The behavior of bands and gaps in the absence of
the external force suggests that the result will remain valid at
least as long as there is a nonzero $\,\delta'\,$ component in such
an interaction, \ie, a discontinuity of the wavefunction at the
lattice points which depends on the one--sided derivatives.
Having excluded the absolutely continuous spectrum, one asks
naturally how the other parts of the spectrum look like. Consider
again our basic model, \ie, the operator $\,H(\beta,E,a)\,$ for
nonzero $\,E,\,\beta\,$. The above proof shows, in particular, that
the essential spectrum of $\,H(\beta,E,a)\,$ does not change if the
system is ``chopped'' by imposing the Neumann condition at a properly
chosen sequence $\,\{n_{\ell}a\}\,$. The discrete spectra of the
corresponding ``finite sections'' of $\,H(\beta,E,a)\,$ can be found
numerically \cite{A}; in this way we arrive at the following
\begin{conjecture} \label{ess spectrum}
For nonzero $\,E,\,\beta\,$,
$$
\sigma_{ess}(H(\beta,E,a))\,=\, \left\lbrace\, {4\over \beta a}\,+\,
\left(m\pi\over a\right)^2- E\left(n+{1\over 2}\right)a\::\;\,
m,n\in\Z\: \right\rbrace\,.
$$
\end{conjecture}
In other words, the essential--spectrum points are sums of three
terms: the energy step of the ladder, the eigenvalues of
$\,H(\infty,E,a)\,$, and the asymptotic {\em halfwidth} of a band.
Recall that $\,\sigma_{ess}((H(\beta,E,a))\,$ consists of all
accumulation points of the spectrum, since a second--order
differential operator cannot have eigenvalues of infinite
multiplicity. If the conjecture is true, the spectrum exhibits an
intriguing dependence on the number--theoretic properties of the
external field, namely
\begin{description}
\item{\em (a)} for $\,\gamma:= \left(a\over\pi\right)^2Ea\,$ is
rational, the spectrum is nowhere dense, and therefore automatically
pure point.
\item{\em (b)} on the other hand, if $\,\gamma\,$ is irrational,
$\,\sigma((H(\beta,E,a))= \sigma_{ess}((H(\beta,E,a))= \R\,$.
\end{description}
In this way, $\,\delta'$ Wannier--Stark systems still represent a
challenge.
\section*{Acknowledgement}
As mentioned above the problem as well as the strategy of its
solution resulted from collaboration with J.E.~Avron and Y.~Last to
whom I am deeply indebted. I am grateful for the hospitality extended
to me at the Israel Institute of Technology where the work started,
and to the E.~Schr\"odinger Institute in Vienna, where later parts
were done. I am obliged to M.~Tater for assistance in the numerical
analysis supporting Conjecture~\ref{ess spectrum} and to the referee
for helpful comments. The research has been supported in part by the
AS CR Grant No.~148409.
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\end{document}