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

Boegli, Sabine; Siegl, Petr; Tretter, Christiane

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
• Public URL: https://durham-repository.worktribe.com/output/1275517