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**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.

Reviewer: Henri Schurz (Minneapolis)

### 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 |