On the semi-classical analysis of Schr "odinger operators with purely\n imaginary electric potentials in a bounded domain

2014/05/23 by Raphaël Henry, Henry, Raphaël · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #FOS: Mathematics #Numerical methods in inverse problems #Spectral Theory (math.SP) #Spectral Theory in Mathematical Physics

paper · pdf · doi:10.48550/arxiv.1405.6183

openalex publication_date 2014/05/23 · openalex created_date 2022/10/01 · openalex updated_date 2026/07/28

Abstract

In this paper, we describe the leftmost eigenvalue of the non-selfadjoint\noperator \A_h = -h²\Δ+iV(x) with Dirichlet boundary conditions\non a smooth bounded domain \Ω\⊂\ℝⁿ ,, as h\→0 ,.\nV is assumed to be a Morse function without critical point at the boundary of\n\Ω ,. More precisely, we compare \inf Re\σ(\A_h) with the\nminimum of the spectrum's real part for some model operator. In the case where\nV has no critical point, the spectrum is determined by the boundary points\nwhere \∇ V is orthogonal, and the model operator involves a\n1-dimensional complex Airy operator in \ℝ⁺ ,. If V is a Morse\nfunction with critical points in \Ω ,, the behavior of the operator near\nthe critical points prevails, and the model operator is a complex harmonic\noscillator.\n This question is related to the decay of associated semigroups. In\nparticular, it allows to recover, in a simplified setting, some stability\nresults by Almog in superconductivity theory.\n

Cited by

• Localization of the eigenfunctions of a Bloch-Torrey operator on the half-plane

Related

• On a Schrödinger operator with a purely imaginary potential in the semiclassical limit
• The spectrum of a Schrödinger operator in a wire-like domain with a purely imaginary degenerate potential in the semiclassical limit
• Spectral analysis of a complex Schr "odinger operator in the\n semiclassical limit
• Semiclassical tunneling for some 1D Schrödinger operators with complex-valued potentials
• Eigenvalue asymptotics for confining magnetic Schrödinger operators with complex potentials