Eigenvalue estimates for singular left-definite Sturm-Liouville operators

TL;DR

This paper analyzes the spectral properties and eigenvalue estimates of singular left-definite Sturm-Liouville operators by relating them to their right-definite counterparts, providing bounds on eigenvalues and insights into their accumulation behavior.

Contribution

It introduces new eigenvalue estimates for singular left-definite Sturm-Liouville operators by connecting their spectra to right-definite operators and explores eigenvalue accumulation and special cases with symmetric and periodic coefficients.

Findings

Eigenvalues of $JA$ are closely related to those of $A$ with at most a three eigenvalue difference.

Intervals in the spectrum of $A$ correspond to symmetric intervals in the spectrum of $JA$.

Eigenvalue accumulation behavior is characterized for singular left-definite Sturm-Liouville operators.

Abstract

The spectral properties of a singular left-definite Sturm-Liouville operator $JA$ are investigated and described via the properties of the corresponding right-definite selfadjoint counterpart $A$ which is obtained by substituting the indefinite weight function by its absolute value. The spectrum of the $J$-selfadjoint operator $JA$ is real and it follows that an interval $(a,b)\subset\mathbb R^+$ is a gap in the essential spectrum of $A$ if and only if both intervals $(-b,-a)$ and $(a,b)$ are gaps in the essential spectrum of the $J$-selfadjoint operator $JA$. As one of the main results it is shown that the number of eigenvalues of $JA$ in $(-b,-a) \cup (a,b)$ differs at most by three of the number of eigenvalues of $A$ in the gap $(a,b)$; as a byproduct results on the accumulation of eigenvalues of singular left-definite Sturm-Liouville operators are obtained. Furthermore, left-definite problems with symmetric and periodic coefficients are treated, and several examples are included to illustrate the general results.