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Resonances for Schrödinger operators on infinite cylinders and other products

T. J. Christiansen

Vol. 16 (2023), No. 7, 1497–1546

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.

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

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