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Approximations of spectra of Schrödinger operators with complex potentials on ℝ^d

Boegli, Sabine; Siegl, Petr; Tretter, Christiane

Approximations of spectra of Schrödinger operators with complex potentials on ℝ^d Thumbnail


Authors

Petr Siegl

Christiane Tretter



Abstract

We study spectral approximations of Schrödinger operators T = −Δ+Q with complex potentials on Ω = ℝd, or exterior domains Ω⊂ℝd, by domain truncation. Our weak assumptions cover wide classes of potentials Q for which T has discrete spectrum, of approximating domains Ωn, and of boundary conditions on ∂Ωn such as mixed Dirichlet/Robin type. In particular, Re Q need not be bounded from below and Q may be singular. We prove generalized norm resolvent convergence and spectral exactness, i.e. approximation of all eigenvalues of T by those of the truncated operators Tn without spectral pollution. Moreover, we estimate the eigenvalue convergence rate and prove convergence of pseudospectra. Numerical computations for several examples, such as complex harmonic and cubic oscillators for d = 1,2,3, illustrate our results.

Citation

Boegli, S., Siegl, P., & Tretter, C. (2017). Approximations of spectra of Schrödinger operators with complex potentials on ℝ^d. Communications in Partial Differential Equations, 42(7), 1001-1041. https://doi.org/10.1080/03605302.2017.1330342

Journal Article Type Article
Acceptance Date Jun 14, 2017
Online Publication Date Jul 28, 2017
Publication Date Jul 30, 2017
Deposit Date Dec 11, 2019
Publicly Available Date Dec 17, 2019
Journal Communications in Partial Differential Equations
Print ISSN 0360-5302
Electronic ISSN 1532-4133
Publisher Taylor and Francis Group
Peer Reviewed Peer Reviewed
Volume 42
Issue 7
Pages 1001-1041
DOI https://doi.org/10.1080/03605302.2017.1330342

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