Approximations of spectra of Schrödinger operators with complex potentials on ℝd

Bögli, Sabine; Siegl, Petr; Tretter, Christiane (2017). Approximations of spectra of Schrödinger operators with complex potentials on ℝd. Communications in partial differential equations, 42(7), pp. 1001-1041. Taylor & Francis 10.1080/03605302.2017.1330342

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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.

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