Spectral gaps for self-adjoint second order operators

Details Spectral gaps for self-adjoint second order operators Spectral gaps for self-adjoint second order operators

2010-10-02

arxiv.org/abs/1010.0314v1

Mathematics

Spectral Theory

Scientific paper

We consider a second order self-adjoint operator in a domain which can be bounded or unbounded. The boundary is partitioned into two parts with Dirichlet boundary condition on one of them, and Neumann condition on the other. We assume that the potential part of this operator is non-negative. We add a localized perturbation assuming that it produces two negative isolated eigenvalues being the two lowest spectral values of the resulting perturbed operator. The main result is a lower bound on the gap between these two eigenvalues. It is given explicitly in terms of the geometric properties of the domain and the coefficients of the perturbed operator. We apply this estimate to several asymptotic regimes studying its dependence on various parameters. Several specific examples are given.

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