Abstract
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.
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Exner, P., Fraas, M. On the Dense Point and Absolutely Continuous Spectrum for Hamiltonians with Concentric δ Shells. Lett Math Phys 82, 25–37 (2007). https://doi.org/10.1007/s11005-007-0191-x
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DOI: https://doi.org/10.1007/s11005-007-0191-x