Perturbations of embedded eigenvalues for a magnetic Schrödinger operator on a cylinder

Abstract

Abstract. Perturbation problems for operators with embedded eigenvalues are generally challenging since the embedded eigenvalues cannot be separated from the rest of the spectrum. In this paper we study a perturbation problem for embedded eigenvalues for a magnetic Schrödinger operator, when the underlying domain is a cylinder. The magnetic potential is C 2 with an algebraic decay rate as the unbounded variable of the cylinder tends to ±∞. In particular, no analyticity of the magnetic potential is assumed. We also assume that the embedded eigenvalue of the unperturbed problem is not the square of an integer, thus avoiding the thresholds of the continuous spectrum of the unperturbed operator. We show that the set of nearby potentials, for which a simple embedded eigenvalue persists, forms a smooth manifold of finite codimension.

Magnetic Schrödinger operators as the quasi-classical limit of Pauli–Fierz-type models

“…The relevance of Schrödinger operators with magnetic fields, also known as magnetic Schrödinger operators (MSO) in modern Quantum Mechanics is testified by the amount of work in the mathematical physics community on the topic, see [BNDP16,FH06,HILo17,LMS17,Ray17], to mention just a few among the most recent ones. The main reason for this interest is mainly related to the role played by MSO in several key phenomena of solid state and condensed matter physics.…”

Perturbations of embedded eigenvalues for self-adjoint ODE systems

“…In particular, we prove that the set of all perturbations for which the embedded eigenvalue persists form a manifold in the Banach space of perturbations, and we determine the co-dimension of this manifold. Our method has previously been developed for solving perturbation problems for partial differential operators, as for example in [8], [11] and [9]. Other problems concerning persistence of embedded eigenvalues have been studied in [1] and [2].…”

Perturbations of embedded eigenvalues for self-adjoint ODE systems

“…the co-dimension of the manifold of all those perturbations for which the embedded eigenvalue persists. Our method has previously been developed for solving perturbation problems for partial differential operators, as for example in [11,8,9]. Other problems concerning persistence of embedded eigenvalues have been studied in [1] and [2].…”