Pseudospectra of the Schroedinger operator with a discontinuous complex potential

Abstract

We study spectral properties of the Schroedinger operator with an imaginary sign potential on the real line. By constructing the resolvent kernel, we show that the pseudospectra of this operator are highly non-trivial, because of a blow-up of the resolvent at infinity. Furthermore, we derive estimates on the location of eigenvalues of the operator perturbed by complex potentials. The overall analysis demonstrates striking differences with respect to the weak-coupling behaviour of the Laplacian.

32 pages, 4 figures; version accepted for publication in J. Spectr. Theory (amendments following the referee's recommendations, new figure by Mark Embree)

References in corpus (5)

• Mixed-state evolution in the presence of gain and loss
• Pseudospectra in non-Hermitian quantum mechanics
• Lieb-Thirring inequalities for Schrödinger operators with complex-valued potentials
• Differential operators admitting various rates of spectral projection growth
• Spectral projections of the complex cubic oscillator