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 ? 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) = ω2r2 2 + λrk, if k > 0. For spiked oscillators with k < -1, the basis of the pseudoharmonic oscillator eigenfunctions with two parameters ? and γ is more suitable, and optimization of the latter appears crucial for a precise determination of the spectrum.

22 pages,8 figures

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