Perturbations of embedded eigenvalues for the planar bilaplacian

Abstract

Operators on unbounded domains may acquire eigenvalues that are embedded in the essential spectrum. Determining the fate of these embedded eigenvalues under small perturbations of the underlying operator is a challenging task, and the persistence properties of such eigenvalues are linked intimately to the multiplicity of the essential spectrum. In this paper, we consider the planar bilaplacian with potential and show that the set of potentials for which an embedded eigenvalue persists is locally an infinite-dimensional manifold with infinite codimension in an appropriate space of potentials

Persistence of embedded eigenvalues

“…By (13) it follows that δ(H + W − λ)f, f = 0. Note that δ(H + W − λ)f, g defines a sesquilinear form on H s , for which the Schwarz inequality holds.…”

“…See [13] for another approach to the problem where the continuous spectrum of the operators involved has infinite multiplicity. In [13], the structure of the set of local perturbations which do not remove an embedded simple eigenvalue is determined for a specific example. where s > s 1 and the inclusion is continuous.…”

Perturbations of embedded eigenvalues for self-adjoint ODE systems

“…On the other hand, we in this case are interested in those potentials for which the embedded eigenvalue exists. Some examples of similar approach to such problems can be found in [1,2,4,7,8,9]. More specifically, our goal is to determine the structure Date: June 8, 2021.…”

“…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].…”

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

“…[10,13,18,13]. The current paper builds on methods developed for solving perturbation problems for the bilaplacian [11,12], but instead of the bilaplacian, we study in this paper a perturbation problem for a magnetic Schrödinger operator on a cylindrical domain, where the small perturbation enters as an addition to the magnetic potential. Other perturbation problems for embedded eigenvalues of magnetic Schrödinger operators have been studied in e.g.…”