Nonlinear evolution equations associated with the Schrödinger spectral problem

Evolution equations associated with the Schrödinger equation are derived for an arbitrary time-dependent potential. It is shown that the eigenvalues evolve according to the Hellmann-Feynman theorem, while the eigenfunction evolution can be determined either by solving a system of coupled differential equations or by a contour integration in the complex k-domain. A possible application to solving a class of Schrödinger spectral problems is also discussed.