Trace formula for a perturbed operator of Jacobi type

There is a broad class of problems of mathematical physics that lead to the solution of second-order differential equations of some special form. In particular, systems of solutions of such equations are given by classical polynomials (Jacobi, Laguerre, and Hermite polynomials). Such equations are naturally related to second-order differential operators in appropriate Hilbert spaces and the corresponding spectral problems. We consider a Jacobi operator and its perturbation by the operator of multiplication by a function. We derive a trace formula for the perturbed operator and a closed-form expression for the first correction.