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Eigenvalue bounds for Schrödinger operators with complex radial potentials

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posted on 2024-10-25, 14:13 authored by Solomon Keedle-IsackSolomon Keedle-Isack

We consider sums of eigenvalues zj of Schrodinger operators −Δ + V on L2(Rd) with complex radial potentials V ∈ Lq(Rd), q < d. We prove quantitative bounds on the distribution of (zj) in terms of the Lq norm of V . A consequence of our bounds is that if (zj) accumulates to a point in (0,∞) then (Im zj) is summable. The key technical tools are resolvent estimates in Schatten spaces, which follow from spectral measure estimates by an epsilon removal argument. We also derive bounds on the number of eigenvalues of Schrodinger operators with complex-valued step potentials in one dimension. An investigation of higher-dimensional analogues leads us to a generalisation of a result on the number of resonances of complex step potentials on non-physical sheets. Finally, we present results on asymptotic formulae for eigenvalues of truncated potentials. Our modifications to the procedure outlined by Dyatlov and Zworski allow us to treat several new examples. We derive explicit formulae for eigenvalues of the truncated complex harmonic oscillator, which shows that Zworksi’s bound on the resonance counting function for compactly-supported potentials can be saturated by eigenvalues. An analogous treatment of the Poschl-Teller potential allows us to show that the local bound of Abramov et al. for d = 1 is sharp for compactly-supported potentials.

Funding

EPSRC Doctoral Training Partnership Student Grant

History

School

  • Science

Department

  • Computer Science

Publisher

Loughborough University

Rights holder

© Solomon Keedle-Isack

Publication date

2024

Notes

A Doctoral Thesis. Submitted in partial fulfilment of the requirements for the award of the degree of Doctor of Philosophy of Loughborough University.

Language

  • en

Supervisor(s)

Jean-Claude Cuenin

Qualification name

  • PhD

Qualification level

  • Doctoral

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