On the resolvent of singular q-Sturm-Liouville operators

Authors : Bilender PAŞAOĞLU, Hüseyin TUNA

Pages : 702-718

Doi:10.31801/cfsuasmas.866753

View : 47 | Download : 10

Publication Date : 2021-12-31

Article Type : Research Paper

Abstract :In this paper, we investigate the resolvent operator of the singular q-Sturm-Liouville problem defined as − insert ignore into journalissuearticles values( 1 / q ); D q ⁻ ¹ [D q y insert ignore into journalissuearticles values( x );] + [r insert ignore into journalissuearticles values( x ); - λ ]y insert ignore into journalissuearticles values( x );=0 −insert ignore into journalissuearticles values(1/q);Dq⁻¹Dqyinsert ignore into journalissuearticles values(x);+rinsert ignore into journalissuearticles values(x);yinsert ignore into journalissuearticles values(x);=λyinsert ignore into journalissuearticles values(x); , with the boundary condition y insert ignore into journalissuearticles values( 0 , λ ); c o s β + D q ⁻ ¹ y insert ignore into journalissuearticles values( 0 , λ ); s i n β = 0 yinsert ignore into journalissuearticles values(0,λ);cosβ+Dq⁻¹yinsert ignore into journalissuearticles values(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 insert ignore into journalissuearticles values( 0 , ∞ ); r∈Lq,loc¹insert ignore into journalissuearticles values(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

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