The similarity problem for indefinite Sturm-Liouville operators with periodic coefficients

TL;DR

This paper studies when certain indefinite Sturm-Liouville operators with periodic coefficients are similar to self-adjoint operators, revealing new classes with singular critical points and extending regularity conditions.

Contribution

It identifies conditions under which these operators have singular critical points and extends existing regularity criteria to periodic coefficient cases.

Findings

0 is a singular critical point for certain operators

New class of operators with singular critical point 0 identified

Extended Beals and Parfenov regularity conditions to periodic coefficients

Abstract

We investigate the problem of similarity to a self-adjoint operator for $J$-positive Sturm-Liouville operators $L=\frac{1}{\omega}(-\frac{d^2}{dx^2}+q)$ with $2\pi$-periodic coefficients $q$ and $\omega$. It is shown that if 0 is a critical point of the operator $L$, then it is a singular critical point. This gives us a new class of $J$-positive differential operators with the singular critical point 0. Also, we extend the Beals and Parfenov regularity conditions for the critical point $\infty$ to the case of operators with periodic coefficients.