Approximations of spectra of Schrödinger operators with complex potentials on ℝd

Sabine Bögli, Petr Siegl, Christiane Tretter*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

9 Citations (Scopus)
18 Downloads (Pure)

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.

Original languageEnglish
Pages (from-to)1001-1041
Number of pages41
JournalCommunications in Partial Differential Equations
Volume42
Issue number7
DOIs
Publication statusPublished - 28 Jul 2017
Externally publishedYes

Keywords

  • Complex potential
  • domain truncation
  • eigenvalue approximation
  • harmonic oscillator
  • Laplace operator
  • non-selfadjoint Schrödinger operator
  • pseudospectra
  • resolvent convergence
  • spectral exactness
  • spectral pollution
  • spurious eigenvalue

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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