On the Pseudospectrum of the Harmonic Oscillator with Imaginary Cubic Potential

Abstract

We study the Schrödinger operator with a potential given by the sum of the potentials for harmonic oscillator and imaginary cubic oscillator and we focus on its pseudospectral properties. A summary of known results about the operator and its spectrum is provided and the importance of examining its pseudospectrum as well is emphasized. This is achieved by employing scaling techniques and treating the operator using semiclassical methods. The existence of pseudoeigenvalues very far from the spectrum is proven, and as a consequence, the spectrum of the operator is unstable with respect to small perturbations and the operator cannot be similar to a self-adjoint operator via a bounded and boundedly invertible transformation. It is shown that its eigenfunctions form a complete set in the Hilbert space of square-integrable functions; however, they do not form a Schauder basis.

A Bound on the Pseudospectrum for a Class of Non-normal Schrödinger Operators

“…A similar result for c = 1 was shown by Novák in [Nov14] and is easily extended to arbitrary c > 0. Figure 2 shows a numerical computation of the pseudospectrum of H 1 .…”

“…This shows that the large eigenvalues of H 0 are highly unstable under small perturbations. A similar result for c = 1 was shown by Novák in [Nov14] and is easily extended to arbitrary c > 0. Figure 2 shows a numerical computation of the pseudospectrum of H 1 .…”

“…More precisely, we will prove a bound on the pseudospectrum of the operator H = −∆+V , where Re V (x) ≥ c|x| 2 −d for some c, d > 0 on L 2 (R n ), which complements the results of [SK12,Nov14].…”

“…Moreover, H c is closed and has compact resolvent [CGM80,Mez01] so the spectrum is also discrete. On the other hand, Novak recently obtained the following result Theorem 1.1 ( [Nov14]). The operator H c has the following properties:…”

Pseudomodes for Schrödinger operators with complex potentials

“…On the other hand, the by now well-known examples of potentials for which (1.2) holds in vast complex regions Ω are just purely imaginary monomials V (x) := ix n and their perturbations, see e.g. [5,4,19,20,12,16]. Hence, the state of the art of the current research in construction of the "large-energy" pseudomodes for (non-semiclassical) Schrödinger operators is by far incomplete and the objective of this paper is to fill up the gap.…”