On a Method of Investigation of the Self-Consistent NonlinearBoundary-Value Problem for Eigen-Valueswith Growing Potentials

One of the most common methods for investigating multiparticle problems in the frameworkof the variational approach is the transition to a nonlinear one-particle problem by introducinga self-consistent field that depends on the states of these particles. The paper considers anonlinear boundary value eigenvalue problem for the Schr¨odinger equation with a growingpotential including a dependence on the wave function and a power dependence on the coordinate = where = 1,2,3.... For n = 2, the boundary value problem for the Schr¨odinger equation(linear problem) has an exact solution. For even powers of , it is shown that solutions of sucha problem can be expressed in terms of solutions corresponding to the linear problem, and for= 2 the solution can be obtained in explicit form. The set of solutions obtained for= 2 ischaracterized by equal distances between neighboring eigenvalues. It is shown that the solutionof the nonlinear problem differs from the solution of the linear problem by the shift of theeigenvalues. In the case of a potential higher than the quadratic one, new growing potentialsof a lesser degree appear. For the case of odd values of, the transition is discussed, from theintegro-differential formulation of the problem to a system of differential equations which can besolved numerically on the basis of the method of successive approximations, which has provedits effectiveness in the study of the polaron model.