Energy-shell bases (Q2773149)

The paper introduces energy-shell bases, which are usable in quantum dynamics with a Hamiltonian \(H(X)\) depending upon a parameter \(X\), which in turn vary on time. The bases are formed by vectors \(\xi_n(X)=f(H(X)-\overline{E}_n(X))\alpha_n\), where \(\alpha_n\) is a fixed basis; \(\overline{E}_n(X)\) is an analytic approximation to an eigenvalue \(E_n(X)\); and \(f(E)\) is an analytic function mainly supported in a neighbourhood of \(0\) with a diameter smaller than typical energy level separation. The energy-shell bases have advantages in a study of time-dependant Schrödinger equation because they vary more slowly as functions of \(X\) than the eigenfunctions of \(H(X)\).