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

Abstract:

The spectrum of the singular indefinite Sturm-Liouville operator \begin{equation*} A=\operatorname {sgn}(\cdot )\bigl (-\tfrac {d^2}{dx^2}+q\bigr ) \end{equation*} 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 have been obtained. In this paper the bound \begin{equation*} \vert \lambda \vert \leq \Vert q\Vert _{L^1}^2 \end{equation*} on the absolute values of the non-real eigenvalues $\lambda$ 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 $q$.