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

Abstract

We study spectral approximations of Schr\"odinger 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, ${\rm Re} \, Q$ 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.