06-98 V.Grinshpun

On Properties of Impurity Spectrum in the Disordered Exactly Solvable Model (102K, "LaTex" ) Mar 30, 06

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Abstract. The random point interaction Hamiltonian (H) is considered on L^2(R^d), d=2, or d=3. Existence and certain bounds of the non-empty pure point component and exponential decay of the corresponding eigenfunctions with probability 1, within region of impurity spectrum of H, are rigorously established. In order to prove the localization result, the structure of the generalized eigenfunctions of H is explicitly described, and the relation between its spectral properties, and the properties of spectra of finite-difference infinite-order operators on l^2(Z^d), is established. The multiscale analysis scheme is applied to investigate the point spectrum of finite-difference operators. In addition, the generalized spectral theorem, and absolute continuity of the integrated density of states of H at the negative (impurity) part of the spectrum, rigorously proved. Applications of the new approximation scheme include straightforward analysis of absolutely continuous conductivity spectrum, subject to a possible separate publication by the author.

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