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On the spectrum of the one-dimensional Schrödinger equation with a random potential
M. M. Benderskii,
L. A. Pastur
Abstract:
Let
$\mathfrak N(\lambda,a,b)$ be the number of eigenvalues not exceeding
$\lambda$ for the selfadjoint boundary problem
\begin{gather*}
-y''+q(x)y=\lambda y,\\
y(a)\cos\alpha-y'(a)\sin\alpha=0,\quad y(b)\cos\beta-y'(b)\sin\beta=0
\end{gather*}
with random potential
$q(x)$, and let
$$
N(\lambda)=\lim_{L\to\infty}\frac{\mathfrak N(\lambda,0,\,L)}L.
$$
Our problem is to clarify the conditions under which this function will exist and to indicate methods for calculating it.
In the present article we establish the existence of a nonrandom limit
$N(\lambda)$ for a wide class of stationary ergodic potentials. This limit is calculated under the assumption that the potential
$q(x)$ is Markovian, and the argument is based on the well-known theorems of Sturm.
At the end of the article we consider an example in which
$q(x)$ is a Markov process with two states. In this case the calculations can all be carried out completely in a practical way, with the result that we obtain a formula expressing
$N(\lambda)$ by means of integrals of elementary functions.
Bibliography: 9 titles.
UDC:
517.93+530.145
MSC: 81Q05,
60J35,
65F15,
65L15,
65L10,
37A05 Received: 14.07.1969
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