The optimized Rayleigh-Ritz scheme for determining the quantum-mechanical spectrum

TL;DR

This paper demonstrates an optimized Rayleigh-Ritz scheme that accurately computes quantum spectra for various potentials by optimizing basis parameters, outperforming traditional methods especially for complex anharmonic systems.

Contribution

It introduces a trace-minimization approach to optimize nonlinear parameters in the Rayleigh-Ritz method, enhancing spectral accuracy for diverse quantum systems.

Findings

Accurate bound-state energies for one-dimensional oscillators.

Effective for highly anharmonic multi-well potentials.

Suitable basis choice depends on potential type and parameters.

Abstract

The convergence of the Rayleigh-Ritz method with nonlinear parameters optimized through minimization of the trace of the truncated matrix is demonstrated by a comparison with analytically known eigenstates of various quasi-solvable systems. We show that the basis of the harmonic oscillator eigenfunctions with optimized frequency ? enables determination of boundstate energies of one-dimensional oscillators to an arbitrary accuracy, even in the case of highly anharmonic multi-well potentials. The same is true in the spherically symmetric case of V (r) = {\omega}2r2 2 + {\lambda}rk, if k > 0. For spiked oscillators with k < -1, the basis of the pseudoharmonic oscillator eigenfunctions with two parameters ? and {\gamma} is more suitable, and optimization of the latter appears crucial for a precise determination of the spectrum.