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Eigenvalue repulsion estimates and some applications for the one-dimensional Anderson model. (English) Zbl 1221.82059

Summary: 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.

MSC:

82B44 Disordered systems (random Ising models, random Schrödinger operators, etc.) in equilibrium statistical mechanics
81Q10 Selfadjoint operator theory in quantum theory, including spectral analysis
82B20 Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics
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