Spectral properties of Sturm-Liouville equations with singular energy-dependent potentials. (English) Zbl 1313.34268

Pronska, Nataliya

Methods Funct. Anal. Topol. 19, No. 4, 327-345 (2013).

The author studies spectral properties of the Dirichlet problem (or a problem with more complicated boundary conditions involving the quasi-derivative) for the equation \[ -y''+qy+2\lambda py=\lambda^2 y, \] where \(p\) is a real-valued function from \(L_2(0,1)\), \(q\) is a real-valued distribution from the Sobolev space \(W_2^{-1}(0,1)\), \(\lambda \in \mathbb C\), is the spectral parameter. This spectral problem is linearized in a certain Pontryagin space. The notion of norming constants is introduced for this situation. Sufficient conditions are found for the spectrum to be real and simple.

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

34L05	General spectral theory of ordinary differential operators
34B07	Linear boundary value problems for ordinary differential equations with nonlinear dependence on the spectral parameter
34B24	Sturm-Liouville theory
47E05	General theory of ordinary differential operators

Keywords:

Sturm-Liouville equation; energy-dependent potential; Pontryagin space; norming constants