Abstract

We consider Schrödinger operators of the form H_R = - ,text {{d}}^2/,text {{d}}x^2 + q + i gamma chi _{[0,R]} for large R>0, where q in L^1(0,infty ) and gamma > 0. Bounds for the maximum magnitude of an eigenvalue and for the number of eigenvalues are proved. These bounds complement existing general bounds applied to this system, for sufficiently large R.

Highlights

  • There has recently been a surge of interest concerning bounds for the magnitude of eigenvalues and the number of eigenvalues of Schrodinger operators with complex potentials

  • Perturbations of the form iγχ[0,R] are referred to as dissipative barriers and arise in spectral approximation, where they can be utilised as part of numerical schemes for the computation of eigenvalues

  • Korotyaev has proved in [19, Theorem 1.6] a bound specific to Schrodinger operators with compactly supported potentials: the number of eigenvalues N of a Schrodinger operator − d2/ dx2 +V on L2(R+) endowed with a Dirichlet boundary condition at 0, with V ∈ L1(R+) and suppV ⊆ [0, Q], satisfies

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Summary

Introduction

There has recently been a surge of interest concerning bounds for the magnitude of eigenvalues and the number of eigenvalues of Schrodinger operators with complex potentials. We consider Schrodinger operators of the form. Endowed with a Dirichlet boundary condition at 0, where γ > 0 and the background potential q ∈ L1(0, ∞) (which may be complex-valued) are regarded as fixed parameters. Perturbations of the form iγχ[0,R] are referred to as dissipative barriers and arise in spectral approximation, where they can be utilised as part of numerical schemes for the computation of eigenvalues [2,22,23,24,31,34]. Our aim is to prove estimates for the magnitude and number of eigenvalues of HR for large R

Existing Bounds for the Magnitude and Number of Eigenvalues
Stepanenko
Summary of Results
Literature
Notations and Conventions
Magnitude Bound
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Preliminaries
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Exponentially Decaying Potentials
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