Self-Similar Interpolation in Quantum Mechanics

TL;DR

This paper introduces a self-similar approximation method for accurately deriving analytical solutions to quantum eigenvalue problems across various models, ensuring correct asymptotic behavior and broad applicability.

Contribution

It develops a novel self-similar renormalization approach combined with boundary conditions to produce accurate, analytical interpolations for quantum mechanical problems.

Findings

Effective interpolation formulae for anharmonic oscillators

Accurate eigenvalues and wave functions for nonlinear Schrödinger equation

Applicability demonstrated on diverse quantum models

Abstract

An approach is developed for constructing simple analytical formulae accurately approximating solutions to eigenvalue problems of quantum mechanics. This approach is based on self-similar approximation theory. In order to derive interpolation formulae valid in the whole range of parameters of considered physical quantities, the self-similar renormalization procedure is complimented here by boundary conditions which define control functions guaranteeing correct asymptotic behaviour 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 Schr\"odinger equation is considered, for which both eigenvalues and wave functions are constructed.