\(q\)-Titchmarsh-Weyl theory: series expansion (Q2880837)

The authors establish a \(q\)-Titchmarsh-Weyl theory for singular \(q\)-Sturm-Liouville problems. The theory for singular Sturm-Liouville problems of the type NEWLINE\[NEWLINE \begin{aligned} -y'' + \nu(x) y = \lambda y,&\quad 0\leq x < +\infty, \\ \cos \alpha y(0) + \sin \alpha y'(0) = 0, &\end{aligned} NEWLINE\]NEWLINE has been established by Weyl. The goal of this paper is to establish a corresponding theory for singular Sturm-Liouville \(q\)-difference operators when the derivative is replaced by Jackson's \(q\)-difference operator \(D_{q}\). The \(q\)-limit-point and \(q\)-limit circle singularities are defined, sufficient conditions which guarantee that the singular point is in a limit-point case are given. The resolvent is constructed in terms of Green's function of the problem. Moreover, the eigenfunction expansion in its series form is derived. Finally, a detailed example involving Jackson \(q\)-Bessel functions is given, which leads to the completeness of a wide class of \(q\)-cylindrical functions.