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Resonances for Schrödinger operators on infinite cylinders and other products

T. J. Christiansen

Vol. 16 (2023), No. 7, 1497–1546
Abstract

We study the resonances of Schrödinger operators on the infinite product X = d × 𝕊1, where d is odd, 𝕊1 is the unit circle, and the potential V lies in Lc(X). This paper shows that at high energy, resonances of the Schrödinger operator Δ + V on X = d × 𝕊1 which are near the continuous spectrum are approximated by the resonances of Δ + V 0 on X, where the potential V 0 is given by averaging V over the unit circle. These resonances are, in turn, given in terms of the resonances of a Schrödinger operator on d which lie in a bounded set. If the potential is smooth, we obtain improved localization of the resonances, particularly in the case of simple, rank 1 poles of the corresponding scattering resolvent on d. In that case, we obtain the leading order correction for the location of the corresponding high-energy resonances. In addition to direct results about the location of resonances, we show that at high energies away from the resonances, the resolvent of the model operator Δ + V 0 on X approximates that of Δ + V on X. If d = 1, in certain cases this implies the existence of an asymptotic expansion of solutions of the wave equation. Again for the special case of d = 1, we obtain a resonant rigidity type result for the zero potential among all real-valued smooth potentials.

Keywords
Schrödinger operator, resonance, infinite cylindrical end, scattering theory
Mathematical Subject Classification
Primary: 58J50, 81U24
Secondary: 35L05, 35P25
Milestones
Received: 11 December 2020
Revised: 19 October 2021
Accepted: 4 February 2022
Published: 21 September 2023
Authors
T. J. Christiansen
Department of Mathematics
University of Missouri
Columbia, MO
United States

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