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

Abstract

Abstract. We study spectral approximations of Schrödinger operators T = −∆ + Q with complex potentials on Ω = R d , or exterior domains Ω ⊂ R d , 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.

Eigenvalues of Magnetohydrodynamic Mean-Field Dynamo Models: Bounds and Reliable Computation

“…Note that exact a priori error estimates for eigenvalue approximations cannot be derived in the nonnormal case. For differential operators with compact resolvents that are truncated to finite sections/subintervals, it is a classical result that asymptotic error estimates depend on the behavior of the eigenfunctions truncated to finite sections/subintervals; see, e.g., [15] or [4,Thm. 5.2].…”

Non-accretive Schrödinger operators and exponential decay of their eigenfunctions

“…The first problem that we tackle in our analysis is finding of a Dirichlet realisation of (1.1) with non-empty resolvent set. This is not a trivial task as we do not restrict the signs of Re V and Im V and so the standard sectorial form techniques of [ Here we can even go beyond operators like (1.2) for which the suitable Dirichlet realisation can be actually found by available methods in [4,5]. We allow much wilder behaviour of V in terms of the possible growth at infinity and oscillations.…”

“…In a different context (absence of eigenvalues), a certain analogy between the magnetic field and Im V was observed in [17]. 3) is an improvement comparing to [4,5] where the power 1 is assumed; in these references (where (1.2) fits already), a big-O instead of the little-o is used. In the present paper, we can therefore treat examples like…”

“…balls, and subject to Dirichlet boundary conditions. The rate of convergence for a given eigenvalue of (1.1) on R d was estimated by the decay rate of the corresponding eigenfunctions (and generalised eigenfunctions in the case of Jordan blocks) at infinity (see [5,Thm. 5.2]).…”

On a spectral problem in magnetohydrodynamics and its relevance for the geodynamo

“…Remarkably, this simple model exhibited quite a number of the typical reversal features, in particular a temporal asymmetry with slow decay and fast recovery. However, since the eigenvalue problems underlying these time‐evolutions are not symmetric, corresponding numerical computations are prone to be unreliable and thus have to be interpreted with care . Moreover, there are questions that can never be answered by numerical computations, for example, are there only finitely many nonreal eigenvalues?…”