On the resolvent of singular q-Sturm-Liouville operators

Year 2021, Volume: 70 Issue: 2, 702 - 718, 31.12.2021

Bilender Paşaoğlu , Hüseyin Tuna

https://doi.org/10.31801/cfsuasmas.866753

Cited By: 1

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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