On the solution of some linear integral equation with partial integrals

The study of spectral properties, in particular bound states, of multiparticle operators of quantum mechanics, solid state physics and technic is closely related to the problem of solving integral equations with partial integrals (partial integral equations) for functions of three variables. In this article, we study the question of the existence and uniqueness of the solution of a linear partial integral equation for functions of three variables with degenerate kernels and many parameters in a complex Hilbert space. It is proved that under natural conditions, equation (1) has a unique solution, which is expressed through the data and their integrals of the considered equation. The general view of the solution is found. It is proved that the nonhomogeneous equation has solution if the free function is orthogonal to every solutions of homogeneous equation. When solving the partial integral equation under study, the Fredholm method was developed for solving a linear integral equation of the second kind with a parameter.

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