Motion of Energy Levels and Energy Dependent Constants of Motion

Abstract

From the eigenvalue equation ( H 0 +λ V )|ψ n (λ)>= E n (λ)|ψ n (λ)> one can derive an autonomous system of first order differential equations for the eigenvalues E n (λ) andthe matrix elements V m n (λ)= where λ is the independent variable. We investigate the case where the Hamiltonian H is given by a finite dimensional symmetric matrix and derive the energy dependent constants of motion. Furthermore we describe the connection with stationary state perturbation theory. Several open questions are also discussed.