Sturm-Liouville operators with singularities and generalized Nevanlinna functions
The authors consider the equation \[ -y''(x)+(q_0/x^2+q_1/x)y(x)=\lambda y(x),\quad x\in(0,\infty),\tag{E} \] or its generalization where the potential contains the additional term \(\sum_{n=0}^\infty q_{n+2}x^n\). They recall some results from [the authors, Math. Nachr. 281, No.~10, 1418--1475 (2008; Zbl 1165.34011)] as well as the definition of generalized Nevanlinna functions introduced there. It is shown that the Titchmarsh-Weyl function as introduced in the above mentioned paper is a generalized Nevanlinna function in the limit point case (this is done for the case of the more general potential). The Fourier transformation with respect to a Frobenius solution in the limit point case and to a linear combination of Frobenius solutions in the limit circle case and the corresponding spectral functions are defined and studied. They work out the spectral theory for (E) for all cases of \(q_0,q_1\) satisfying the assumptions \(-1/4\leq q_0<\infty\), \(q_1\in\mathbb{R}\), \(q_0^2+q_1^2\neq0\), and the Friedrichs boundary condition at \(x=0\), and generate explicit formulas for the eigenfunction expansions.
scientific article; zbMATH DE number 5795252
scientific article