Quantum Mechanics and a Completely Integrable Dynamical System

From the eigenvalue equation (H0 + XV)\y/n (A)) = En (A) | y/„ ( /)) one can derive an autono­ mous system of first order differential equations for the eigenvalues En (A) and the matrix elements Vmn(X) = (ym(X) \ V\ i//n(X), where X is the independent variable. If the initial values En (X = 0) and y/n (A = 0) are known the differential equations can be solved. Thus one finds the “motion” of the energy levels En(X). Here we give two applications of this technique. Fur­ thermore we describe the connection with the stationary state perturbation theory. We also derive the equations of motion for the extended case H = H0 + X\ Vt + X2 V2■ Finally we in­ vestigate the case where the Hamiltonian is given by a finite dimensional symmetric matrix and derive the energy dependent constants of motion. Several open questions are also discussed.