Construction of an asymptotic approximation of eigenfunctions and eigenvalues of a boundary value problem for the singular perturbed relativistic analog of the Schrodinger equation with an arbitrary potential

The singular perturbed differential equation of the relativistic quantum mechanics (the analog of Schrodinger equation in the relativistic configuration space) with small parameters at the higher derivatives and with an arbitrary potential is considered. Boundary value problems on a final segment and on a positive semiray for determinating of eigenfunctions and eigenvalues are formulated for this equation and their asymptotical behavior is investigated at diminuting of the small parameter. Asymptotic methods of the singular perturbed theory were applied for this purpose and small parameter asymptotic approximations of solutions are constructed. The obtained results show that an application of asymptotic methods for the given class of boundary value problems is very effective.