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Zeitschrift für Analysis und ihre Anwendungen

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Volume 31, Issue 4, 2012, pp. 473–505
DOI: 10.4171/ZAA/1470

Published online: 2012-10-10

Spectral Gaps for Self-Adjoint Second Order Operators

Denis Borisov[1] and Ivan Veselić[2]

(1) Ufa Scientific Center of RAS, Russian Federation
(2) Technische Universität Chemnitz, Germany

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. We discuss specific examples of operators to which the bounds can be applied.

Keywords: Spectral gap, lower estimate, second order elliptic operator

Borisov Denis, Veselić Ivan: Spectral Gaps for Self-Adjoint Second Order Operators. Z. Anal. Anwend. 31 (2012), 473-505. doi: 10.4171/ZAA/1470