Spectral bounds for singular indefinite Sturm-Liouville operators with $L^1$--potentials

Abstract

The spectrum of the singular indefinite Sturm-Liouville operator $$A=\text{sgn}(\cdot )(-\frac{d^2}{d{x}^2}+q)$$ with a real potential $q\in L^1(\mathbb{R})$ covers the whole real line and, in addition, non-real eigenvalues may appear if the potential $q$ assumes negative values. A quantitative analysis of the non-real eigenvalues is a challenging problem, and so far only partial results in this direction were obtained. In this paper the bound $$|\hat{I}»|\le |q{|}_{L^1}^2$$ on the absolute values of the non-real eigenvalues $\hat{I}»$ of $A$ is obtained. Furthermore, separate bounds on the imaginary parts and absolute values of these eigenvalues are proved in terms of the $L^1$-norm of the negative part of $q$.

to appear in Proc. Amer. Math. Soc