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

Boegli, Sabine and Siegl, Petr and Tretter, Christiane (2017) 'Approximations of spectra of Schrödinger operators with complex potentials on ℝ^d.', Communications in partial differential equations., 42 (7). pp. 1001-1041.

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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1080/03605302.2017.1330342
Publisher statement:This is an Accepted Manuscript of an article published by Taylor & Francis in Communications in Partial Differential Equations on 28th July 2017, available online: http://www.tandfonline.com/10.1080/03605302.2017.1330342
Date accepted:14 June 2017
Date deposited:17 December 2019
Date of first online publication:28 July 2017
Date first made open access:17 December 2019

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