TY - JOUR

T1 - Eigenvalue repulsion estimates and some applications for the one-dimensional Anderson model

AU - Rivkind, Alexander

AU - Krivolapov, Yevgeny

AU - Fishman, Shmuel

AU - Soffer, Avy

PY - 2011/7/29

Y1 - 2011/7/29

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=79960566393&partnerID=8YFLogxK

U2 - 10.1088/1751-8113/44/30/305206

DO - 10.1088/1751-8113/44/30/305206

M3 - Article

AN - SCOPUS:79960566393

VL - 44

JO - Journal of Physics A: Mathematical and Theoretical

JF - Journal of Physics A: Mathematical and Theoretical

SN - 1751-8113

IS - 30

M1 - 305206

ER -