On the Dense Point and Absolutely Continuous Spectrum for Hamiltonians with Concentric δ Shells
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We consider Schrödinger operators in dimension ν ≥ 2 with a singular interaction supported by an infinite family of concentric spheres, analogous to a system studied by Hempel and coauthors for regular potentials. The essential spectrum covers a half line determined by the appropriate one-dimensional comparison operator; it is dense pure point in the gaps of the latter. If the interaction is nontrivial and radially periodic, there are infinitely many absolutely continuous bands; in contrast to the regular case the lengths of the p.p. segments interlacing with the bands tend asymptotically to a positive constant in the high-energy limit.
Mathematics Subject Classification (2000)35J10 35P99 81Q10
KeywordsSchrödinger operator singular interaction concentric spheres spectral properties
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