On a $q$-analogue of the Sturm–Liouville operator with discontinuity conditions

In this paper, a $q$-analogue of the Sturm–Liouville problem with discontinuity condition on a finite interval is studied. It is proved that the $q$-Sturm–Liouville problem with discontinuity conditions is self-adjoint in $L_q^2(0,\pi)$. The completeness theorem and the sampling theorem are proved.