×

On the surface spectrum in dimension two. (English) Zbl 0939.60074

The authors carry out a program which is concerned with the spectral and scattering theory of discrete Laplacians \(H_\omega\) on the half space \(l^2(Z^{d+1}_+)\) with random boundary conditions on appropriate probability spaces. The work is connected to the theory of one-dimensional, random Schrödinger operators. The spectrum of these Schrödinger operators can be explicitly computed (decomposed). It is a pure point one outside certain intervals with exponentially decaying eigenfunctions under some conditions (e.g. the distribution function \(p(x)\) has compact support in \(L^\infty\)). This paper aims at improving some known results in this respect with dimension \(d=1\) for random potential \(V_\omega\) with discrete sets as topological boundaries and \(p \in L^\infty (\mathbb R)\). The obtained estimates give some control on the exponentially decaying eigenfunctions of operator \(H_\omega\) outside certain intervals. The optimality and decay control near edges appears to be a highly difficult problem. Their main result is related to the surface spectrum of operator \(H_0 + V\) as the closure of a certain set of energies (eigenvalues of \(H_0 + V\) belonging to some decaying, generalized eigenfunctions). The proof involves very technical details. The arguments are based on the Simon-Hopf theorem and geometric approaches with long-range Laplacians (particles have to tunnel sufficiently long infinite barriers for the energy). The paper represents a continuation of the work of the authors and L. Pastur [in: Wave propagation in complex media. IMA Vol. Math. Appl. 96, 143-154 (1998; Zbl 0905.65107)]. This work is highly recommended to those who study random Schrödinger operators by serious mathematical methods.

MSC:

60H25 Random operators and equations (aspects of stochastic analysis)
34L40 Particular ordinary differential operators (Dirac, one-dimensional Schrödinger, etc.)
34L25 Scattering theory, inverse scattering involving ordinary differential operators
35P05 General topics in linear spectral theory for PDEs
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis

Citations:

Zbl 0905.65107
PDFBibTeX XMLCite