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This article shows that for a large class of discrete periodic Schrödinger operators, most wavefunctions resemble Bloch states. More precisely, we prove quantum ergodicity for a family of periodic Schrödinger operators H on periodic graphs. This means that most eigenfunctions of H on large finite periodic graphs are equidistributed in some sense, hence delocalized. Our results cover the adjacency matrix on Z^d, the triangular lattice, the honeycomb lattice, Cartesian products, and periodic Schrödinger operators on Z^d. The theorem applies more generally to any periodic Schrödinger operator satisfying an assumption on the Floquet eigenvalues.
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Ambarzumian-type Problems for Discrete Schrödinger Operators
Abstract We discuss the problem of unique determination of the finite free discrete Schrödinger operator from its spectrum, also known as the Ambarzumian problem, with various boundary conditions, namely any real constant boundary condition at zero and Floquet boundary conditions of any angle. Then we prove the following Ambarzumian-type mixed inverse spectral problem: diagonal entries except the first and second ones and a set of two consecutive eigenvalues uniquely determine the finite free discrete Schrödinger operator.
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- PAR ID:
- 10320370
- Date Published:
- Journal Name:
- Complex Analysis and Operator Theory
- Volume:
- 15
- Issue:
- 8
- ISSN:
- 1661-8254
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1007/s11785-021-01169-5
