On Embedded Eigenvalues of Perturbed Periodic Schrödinger Operators

Peter Kuchment () and Boris Vainberg ()
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Peter Kuchment: Wichita State University, Department of Mathematics and Statistics
Boris Vainberg: University of North Carolina at Charlotte, Department of Mathematics

Chapter 5 in Spectral and Scattering Theory, 1998, pp 67-75 from Springer

Abstract: Abstract The problem of non-existence of eigenvalues imbedded into the continuous spectrum is considered for Schrödinger operators with periodic potentials perturbed by a sufficiently fast decaying “impurity” potentials. Absence of embedded eigenvalues is shown in dimensions two and three if the periodic potential satisfies some additional condition on the corresponding Fermi surface. It is conjectured that generic periodic potentials satisfy this condition. It is stated that separable periodic potentials satisfy it, and hence in dimensions two and three a Schrödinger operator with a separable periodic potential perturbed by a sufficiently fast decaying “impurity” potential has no embedded eigenvalues. The proofs are only sketched. The complete proofs will be provided elsewhere.

Keywords: Entire Function; Continuous Spectrum; Periodic Potential; Real Zero; Hill Operator (search for similar items in EconPapers)
Date: 1998
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-1-4899-1552-8_5

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DOI: 10.1007/978-1-4899-1552-8_5

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