Localization in general one-dimensional random systems. II: Continuum Schrödinger operators
[For Part I by the second author, see ibid. 102, 327--336 (1985; Zbl 0604.60062).] The authors establish the pure point character of the spectrum of the Schrödinger operator in \(L_ 2({\mathbb R})\) with random potential. Their proofs are based on the extension of ideas which have their roots in the work by \textit{I. Ya. Gol'dshtein}, \textit{S. Ya. Molchanov} and the reviewer [Funct. Anal. Appl. 11, 1--8 (1977; Zbl 0368.34015)], were made explicit by \textit{R. Carmona} [Duke Math. J. 49, 191--213 (1982; Zbl 0491.60058)] and especially by the first author [Stochastic analysis, Proc. Taniguchi Int. Symp., Katata \& Kyoto/Jap. 1982, North-Holland Math. Libr. 32, 225--247 (1984; Zbl 0549.60058)] and developed in powerful techniques by \textit{F. Delyon}, \textit{Y. Levy} and \textit{B. Souillard} [J. Stat. Phys. 41, 375--388 (1985)] and by the second author, \textit{M. Taylor} and \textit{T. Wolff} [Phys. Rev. Lett. 54, 1589--1592 (1985)]. According to these ideas, the main fact, additional to the positivity of the Lyapunov exponent (exponential decay of the Green function in the multidimensional case) which guarantees the pure point character of the Schrödinger operator spectrum is roughly speaking the ``sufficiently smooth'' distribution of the random potential. The efficient form of this fact was given by the first author as the absolute continuity with respect to the Lebesgue measure of the spectral measure of the operator averaged over the potential conditioned outside of a finite region. In discrete case when the potential is i.i.d. random variables it is sufficient to average over one variable, say \(q(0)\). For the continuous case of the operator \(-d^ 2/dx^ 2+q(x,\omega)\) with ergodic \(q(x,\omega)\), \(| q(x)| \leq C(\omega)(1+| x|^ 2)\), the authors prove the following Theorem 2.1: Let \((\Omega,{\mathcal F},\mu)\) be the probability space carrying the potential and \(\phi(x,\lambda,\theta)\) the phase of the Cauchy solution \(y(x,\lambda,\theta)\) of the Schrödinger equation satisfying the initial conditions \(y(0,\lambda,\theta)=\sin \theta\), \(y'(0,\lambda,\theta)=\sin \theta\). Further let \(\Omega_ 0\subset \Omega\), \(\mu (\Omega_ 0)>0\) and for some \(a, C>0\) and \(\Omega_ 1\subset \{q\in {\mathbb R}\setminus [0,a]\}\) with \(\mu\{q\in \Omega_ 0|\) \(q\in {\mathbb R}\setminus [0,a]\}>0\) we have that for each fixed \(\theta\in [0,\pi)\), \(\mu\{\phi(a,\lambda,\theta)\in d\phi\), \(q\in \Omega_ 0| \Omega_ 1\}=G(\phi)d\phi\), \(| G(\phi)| \leq C\), \(| \lambda | \leq B<\infty\). Suppose also that \(\sup\{| q(x)||\) \(q\in \Omega_ 0\), \(x\in [0,a]\}<\infty\). Then with probability 1 the Schrödinger operator has pure point spectrum and the eigenfunctions are exponentially decreasing with a rate which equals to the Lyapunov exponent. The authors also give a number of interesting and important applications of their theorem and in particular prove under more weaker conditions the Gol'dshtein et al. result [which was in essence proved by \textit{S. A. Molchanov} and \textit{H. Seidel}, Math. Nachr. 109, 57--78 (1982; Zbl 0534.60058)]. As a by-product the author also extends \textit{F. Wegner}'s result [Z. Phys. B 44, 9--16 (1981)] on the Lipschitz nature of the integrated density of states for some continuous multidimensional cases.
Schrödinger equation with random potential
scientific article; zbMATH DE number 4038956
scientific article