On the semi-classical analysis of Schrödinger operators with purely imaginary electric potentials in a bounded domain

Abstract

In this paper, we describe the leftmost eigenvalue of the non-selfadjoint operator ${\mathcal{A}}_h=-h^2\Delta +iV(x)$ with Dirichlet boundary conditions on a smooth bounded domain $\Omega \subset {\mathbb{R}}^n$, as $h\to 0$. $V$ is assumed to be a Morse function without critical point at the boundary of $\Omega$. More precisely, we compare $\inf \Re \sigma ({\mathcal{A}}_h)$ with the minimum of the spectrum's real part for some model operator. In the case where $V$ has no critical point, the spectrum is determined by the boundary points where $\nabla V$ is orthogonal, and the model operator involves a $1$-dimensional complex Airy operator in ${\mathbb{R}}^{+}$. If $V$ is a Morse function with critical points in $\Omega$, the behavior of the operator near the critical points prevails, and the model operator is a complex harmonic oscillator. This question is related to the decay of associated semigroups. In particular, it allows to recover, in a simplified setting, some stability results by Almog in superconductivity theory.

References in corpus (2)

• Non-self-adjoint harmonic oscillator, compact semigroups and pseudospectra
• Global stability of the normal state of superconductors in the presence of a strong electric current

Cited by in corpus (7)

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• Critical time for the observability of Kolmogorov-type equations
• The spectrum of a Schrödinger operator in a wire-like domain with a purely imaginary degenerate potential in the semiclassical limit
• Semigroup expansions for non-selfadjoint Schrödinger operators
• On the spectrum of some Bloch-Torrey vector operators
• Subelliptic Resolvent Estimates for Non-self-adjoint Semiclassical Schrodinger Operators