Spectral bounds for singular indefinite Sturm-Liouville operators with L1-potentials

Behrndt, Jussi GND; Schmitz, Philipp; Trunk, Carsten GND

The spectrum of the singular indefinite Sturm-Liouville operator A=sgn(.) (-d^2/dx^2)+q with a real potential q in L^1(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 l lambda | <= |q|_{L^1}^2 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 q and its negative part q_-.

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Behrndt, Jussi / Schmitz, Philipp / Trunk, Carsten: Spectral bounds for singular indefinite Sturm-Liouville operators with L1-potentials. Ilmenau 2017.

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