Accumulation of complex eigenvalues of an indefinite Sturm--Liouville operator with a shifted Coulomb potential

TL;DR

This paper studies how complex eigenvalues of a specific indefinite Sturm--Liouville operator with a shifted Coulomb potential accumulate near zero, revealing their asymptotic distribution along certain complex curves.

Contribution

It establishes the asymptotic accumulation of eigenvalues along specific curves and links their behavior to the eigenvalues of related self-adjoint operators.

Findings

Eigenvalues accumulate near zero along specific complex curves

Asymptotic behavior of eigenvalues is characterized

Relation to two-term asymptotics of self-adjoint operator eigenvalues

Abstract

For a particular family of long-range potentials $V$, we prove that the eigenvalues of the indefinite Sturm--Liouville operator $A = \mathrm{sign}(x)(-\Delta + V(x))$ accumulate to zero asymptotically along specific curves in the complex plane. Additionally, we relate the asymptotics of complex eigenvalues to the two-term asymptotics of the eigenvalues of associated self-adjoint operators.