Separation of variables in the two-dimensional wave equation with potential

where u = u(t, x) ∈ C(R,R) and V (x) ∈ C(R,R), by using the method of separation of variables (SV). Equations belonging to class (1) are widely used in the modern quantum physics and can be related to other linear and nonlinear equations of mathematical physics (these relations will be discussed below, at the end of the article). In particular, class (1) contains the d’Alembert equation (with V (x) = 0) and the Klein– Gordon–Fock equation (with V (x) = m ≡ const). The separation of variables in twoand three-dimensional Laplace, Helmholtz, d’Alembert, and Klein–Gordon–Fock equations had been carried out in the classical works by Bocher [1], Darboux [2], Eisenhart [3], Stepanov [4], Olevsky [5], and Kalnins and Miller (see [6] and references therein). Nevertheless, a complete solution of the problem of SV in equation (1) is not obtained yet. When speaking about solution of equation (1) with separated variables ω1, ω2, we mean the ansatz