论文信息 - On the spectrum of Schrödinger operators with a random potential

On the spectrum of Schrödinger operators with a random potential

We investigate the spectrum of Schrodinger operatorsHω of the type:Hω=−Δ+∑qi(ω)f(x−xi+ζi(ω))(qi(ω) and ζi(ω) independent identically distributed random variables,i∈ℤd). We establish a strong connection between the spectrum ofHω and the spectra of deterministic periodic Schrodinger operators. From this we derive a condition for the existence of “forbidden zones” in the spectrum ofHω. For random one- and three-dimensional Kronig-Penney potentials the spectrum is given explicitly.

Werner Kirsch | Fabio Martinelli | F. Martinelli | W. Kirsch

[1] W. Penney,et al. Quantum Mechanics of Electrons in Crystal Lattices , 1931 .

[2] A. Grossmann,et al. The one particle theory of periodic point interactions , 1980 .

[3] D. C. Mattis,et al. Mathematical physics in one dimension , 1966 .

[4] S. P. Lloyd,et al. Electron Levels in a One-Dimensional Random Lattice , 1960 .

[5] R. Borland. Existence of Energy Gaps in One-Dimensional Liquids , 1961 .

[6] Mark S. C. Reed,et al. Method of Modern Mathematical Physics , 1972 .

[7] J. Fenstad,et al. Singular perturbations and nonstandard analysis , 1979 .

[8] A. Grossmann,et al. Spectral properties of reduced Bloch Hamiltonians , 1977 .

[9] L. Pastur. Spectra of Random Self Adjoint Operators , 1973 .

[10] William G. Faris. Self-Adjoint Operators , 1975 .

[11] L. Pastur. Spectral properties of disordered systems in the one-body approximation , 1980 .

[12] H. Nagai,et al. On an Asymptotic Property of Spectra of a Random Difference Operator , 1975 .

[13] S. Nakao. On the spectral distribution of the Schrödinger operator with random potential , 1977 .

[14] Hervé Kunz,et al. Sur le spectre des opérateurs aux différences finies aléatoires , 1980 .