Spectral bounds for indefinite singular Sturm-Liouville operators with uniformly locally integrable potentials
Consider the maximal operator $A$ associated with the indefinite singular Sturm-Liouville expression $l = \frac{1}{r}(-\frac{d}{dx}p\frac{d}{dx} + q)$ on the whole real line with real coefficients $\frac{1}{p}, q, r \in L^1_{\mathrm{loc}}({\mathbb{R}})$ where $p$ is positive and the weight function $r$ changes its sign. Certain additional conditions are imposed on the coefficients (in particular $q$ is assumed to be uniformly locally integrable) which ensure that the spectrum of $A$ is real except of some isolated eigenvalues of finite algebraic multiplicity lying symmetric to the real axis. The main focus of the paper is on estimates for such a non-real eigenvalue $\lambda \notin {\mathbb R}$. A number of conditions is presented allowing different estimates for $|\lambda|$ and for $|Im \lambda|$ in terms of $p$ of the negative part $q_-$ of the potential $q$ and (in a more implicite way) of the weight $r$. It turns out that in the special case $p=1$, $r=sgn$ the paper ends up with estimates which were already obtained in an earlier paper [Proc. Am. Math. Soc. 146, No. 9, 3935--3942 (2018; Zbl 1396.34052)] by the same authors.
scientific article; zbMATH DE number 7046603
scientific article