Eigenvalue repulsion estimates and some applications for the one-dimensional Anderson model

Alexander Rivkind, Yevgeny Krivolapov, Shmuel Fishman, Avy Soffer

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4 Scopus citations

Abstract

We show that the spacing between eigenvalues of the discrete 1D Hamiltonian with arbitrary potentials which are bounded and with Dirichlet or Neumann boundary conditions is bounded away from zero. We prove an explicit lower bound, given by C e-bN, where N is the lattice size, and C and b are some finite constants. In particular, the spectra of such Hamiltonians have no degenerate eigenvalues. As applications we show that to a leading order in the coupling, the solution of a nonlinearly perturbed Anderson model in one dimension (on the lattice) remains exponentially localized in probability and average sense for initial conditions given by a unique eigenfunction of the linear problem. We also bound the derivative of the eigenfunctions of the linear Anderson model with respect to a potential change.

Original languageEnglish
Article number305206
JournalJournal of Physics A: Mathematical and Theoretical
Volume44
Issue number30
DOIs
StatePublished - 29 Jul 2011
Externally publishedYes

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