Anharmonic oscillator and the optimized basis expansion

TL;DR

This paper develops optimization schemes for accurately calculating eigenvalues and eigenfunctions of one-dimensional anharmonic oscillators, improving precision through analytical fixing of variational parameters and basis function optimization.

Contribution

It introduces new methods for fixing nonlinear variational parameters and demonstrates their effectiveness in achieving arbitrary accuracy in energy spectrum calculations.

Findings

Optimized parameters enable highly accurate eigenvalue calculations.

Optimal frequency aligns with the minimal sensitivity principle.

Methods improve computational efficiency for anharmonic oscillators.

Abstract

We introduce various optimization schemes for highly accurate calculation of the eigenvalues and the eigenfunctions of the one-dimensional anharmonic oscillators. We present several methods of analytically fixing the nonlinear variational parameter specified by the domain of the trigonometric basis functions. We show that the optimized parameter enables us to determine the energy spectrum to an arbitrary accuracy. Also, using the harmonic oscillator basis functions, we indicate that the resulting optimal frequency agrees with the one obtained by the principle of the minimal sensitivity.