Inverse problems of spectral analysis for the Sturm-Liouville operator with regular boundary conditions

TL;DR

This paper investigates inverse spectral problems for Sturm-Liouville operators with regular boundary conditions, showing that potentials leading to asymptotically multiple spectra are densely distributed among summable functions.

Contribution

It demonstrates the density of potentials producing asymptotically multiple spectra under certain conditions, extending understanding of spectral properties of Sturm-Liouville operators.

Findings

Potentials with asymptotically multiple spectra are dense in the space of summable functions.

The results apply to Sturm-Liouville operators with regular but not strongly regular boundary conditions.

The paper establishes conditions under which spectral multiplicity occurs asymptotically.

Abstract

We consider the Sturm-Liouville operator Lu=u''-q(x)u with regular but not strongly regular boundary conditions. Under some supplementary assumptions we prove that the set of potentials q(x) that ensure an asymptotically multiple spectrum is everywhere dense in the space of summable functions.