A criterion for the unique solvability of the spectral Poincare problem for a class of multidimensional hyperbolic equations

Two-dimensional spectral problems for hyperbolic equations are well studied, and their multidimensional analogs, as far as the author knows, have been little studied. This is due to the fact that in the case of three or more independent variables there are difficulties of a fundamental nature, since the very attractive and convenient method of singular integral equations used for two-dimensional problems cannot be used here due to the absence of any complete theory of multidimensional singular integral equations. The theory of multidimensional spherical functions, on the contrary, has been adequately and fully studied. These functions have an important application in mathematical and theoretical physics, and in the theory of multidimensional singular equations. In the cylindrical domain of Euclidean space for a class of multidimensional hyperbolic equations, the Poincare spectral problem is considered. The solution is sought as an expansion in multidimensional spherical functions. The existence and uniqueness theorems are proved. The conditions for the unique solvability of the problem, which significantly depend on the height of the cylinder, are obtained.