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$$)$$D^{}$${}_q$${}_{⁻}$${}_{¹}$${\mathrm{[D}}^{}$${}_q$$y$$($$x$$)]$$+$$\mathrm{[r}$$($$x$$)$$-\lambda$$\mathrm{]y}$$($$x$$)=0$, with the boundary condition $y(0,\lambda )cos\beta +D_{q⁻¹}y(0,\lambda )sin\beta =0$, where $\lambda \in C$, $r$ is a real function defined on $[0,∞)$, continuous at zero and $r\in L_{q,loc}¹(0,\infty )$. 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.

Keywords

• q-Sturm-Liouville operator
• spectral function
• resolvent operator
• Titchmarsh-Weyl function

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