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

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