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.
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Kuchment, P., Vainberg, B. (1998). On Embedded Eigenvalues of Perturbed Periodic Schrödinger Operators. In: Ramm, A.G. (eds) Spectral and Scattering Theory. Springer, Boston, MA. https://doi.org/10.1007/978-1-4899-1552-8_5
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DOI: https://doi.org/10.1007/978-1-4899-1552-8_5
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