Derivatives with respect to parameters of the eigenvectors of a Hamiltonian and their application

Abstract

A method is described for finding the first- and higher-order derivatives of the eigenvectors of a Hamiltonian with respect to its parameters. This method is useful even when the explicit dependence of the eigenvectors on the parameters is not known. The method is based on a transfer of the differentiation from the eigenvector to the Hamiltonian and on a separate analysis of the derivatives of the projections of the eigenvector onto the corresponding subspace and onto the orthogonal complement of this subspace. Conditions governing the position of the eigenvector being differentiated in its degenerate subspace are analyzed. This method can be used in certain fundamental problems, and it can be related to steady-state Rayleigh-Schrö-dinger perturbation theory.