Approximations of spectra of Schrödinger operators with complex potentials on ${\mathbb{R}}^d$

Abstract

We study spectral approximations of Schrödinger operators $T=-\Delta +Q$ with complex potentials on $\Omega ={\mathbb{R}}^d$, or exterior domains $\Omega \subset {\mathbb{R}}^d$, by domain truncation. Our weak assumptions cover wide classes of potentials $Q$ for which $T$ has discrete spectrum, of approximating domains ${\Omega }_n$, and of boundary conditions on $\partial {\Omega }_n$ such as mixed Dirichlet/Robin type. In particular, $\mathrm{Re}Q$ 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 $T_n$ 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.

32 pages, 4 figures