On the resolvent of singular q-Sturm-Liouville operators

Abstract

In this paper, we investigate the resolvent operator of the singular q-Sturm-Liouville problem defined as −(1/q)Dq⁻¹[Dqy(x)]+[r(x)-λ]y(x)=0−(1/q)Dq⁻¹Dqy(x)+r(x)y(x)=λy(x), with the boundary condition y(0,λ)cosβ+Dq⁻¹y(0,λ)sinβ=0y(0,λ)cosβ+Dq⁻¹y(0,λ)sinβ=0, where λ∈Cλ∈C, $r$ is a real function defined on $[0,∞)$, continuous at zero and r∈Lq,loc¹(0,∞)r∈Lq,loc¹(0,∞). We give an integral representation for the resolvent operator and investigate some properties of this operator. Furthermore, we obtain a formula for the Titchmarsh-Weyl function of the singular $q$-Sturm-Liouville problem.