Asymptotics of eigenfunctions and eigenvalues of a singularly perturbed relativistic analogue of the Schrödinger equation with an arbitrary potential

The paper is devoted to the study of finite truncations of the radial part of the relativistic analog of the Schrödinger equation (quasipotential equation) which are obtained by means of truncating the operator coshleft( frac {ihslash}{mc}frac d{dr}right) to a finite part of its Taylor series. The constant c is assumed to be large, so that the truncated equation is a singular perturbation of boundary layer type of the limit standard radial Schrödinger equation. The asymptotics of the eigenvalues are constructed by means of a standard technique in terms of the solutions of the unperturbed equation and boundary layer functions; the eigenvalues have regular perturbation series. The paper contains some rather surprising assertions (for example, domains of differential operators are segments of the real line, differential operators in L_2 are continuous, etc.), so that the results of Section 4 on the behavior of eigenvalues as mtoinfty (where m is the order of truncation) are suspect, but formal asymptotic series for a finite truncation are apparently correct.