An eigenvalue problem for the Schrödinger-Maxwell equations

In this paper we study the eigenvalue problem for the Schrödinger operator coupled with the electromagnetic field E,H. The case in which the electromagnetic field is given has been mainly considered ([1]–[3]). Here we do not assume that the electromagnetic field is assigned, then we have to study a system of equations whose unknowns are the wave function ψ = ψ(x, t) and the gauge potentials A = A(x, t), φ = φ(x, t) related to E,H. We want to investigate the case in which A and φ do not depend on the time t and ψ(x, t) = u(x)e, u real function and ω a real number In this situation we can assume A = 0 and we are reduced to study the existence of real numbers ω and real functions u, φ satisfying the system