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

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 Ω enables determination of bound-state 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 , if k > 0. For spiked oscillators with k < −1, the basis of the pseudoharmonic oscillator eigenfunctions with two parameters Ω and γ is more suitable, and optimization of the latter appears crucial for a precise determination of the spectrum.