- Communications Faculty of Sciences University Ankara Series A1 Mathematics and Statistics
- Vol: 70 Issue: 2
- On the resolvent of singular q-Sturm-Liouville operators
On the resolvent of singular q-Sturm-Liouville operators
Authors : Bilender Paşaoğlu, Hüseyin Tuna
Pages : 702-718
Doi:10.31801/cfsuasmas.866753
View : 9 | Download : 5
Publication Date : 2021-12-31
Article Type : Research
Abstract :In this paper, we investigate the resolvent operator of the singular q-Sturm-Liouville problem defined as − ( 1 / q ) D q ⁻ ¹ [D q y ( 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 ) c o s β + D q ⁻ ¹ y ( 0 , λ ) s i n β = 0 y(0,λ)cosβ+Dq⁻¹y(0,λ)sinβ=0 , where λ ∈ C λ∈C , $r$ is a real function defined on $[0,∞)$, continuous at zero and r ∈ L q , l o c ¹ ( 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.Keywords : q-Sturm-Liouville operator, spectral function, resolvent operator, Titchmarsh-Weyl function