Self-similar interpolation in quantum mechanics

An approach is developed for constructing simple analytical formulas accurately approximating solutions to eigenvalue problems of quantum mechanics. This approach is based on self-similar approximation theory. In order to derive interpolation formulas valid in the whole range of parameters of considered physical quantities, the self-similar renormalization procedure is complemented here by boundary conditions which define control functions guaranteeing correct asymptotic behavior in the vicinity of boundary points. To emphasize the generality of the approach, it is illustrated by different problems that are typical for quantum mechanics, such as anharmonic oscillators, double-well potentials, and quasiresonance models with quasistationary states. In addition, the nonlinear Schrödinger equation is considered, for which both eigenvalues and wave functions are constructed.