High-precision methods are developed to evaluate eigenvalues for all states |m, N) of the x 2m anharmonic oscillators with m from 2 to 6, for all values of the anharmonicity parameter. There are three basic steps: rescaling to introduce a length scale natural to the problem; use of fifth-order JWKB to generate an accurate starting estimate of the rescaled energy; and shifted (resolvent-based) Lanczos algorithm to sharpen the initial JWKB estimate. JWKB itself gives 33-figure accuracy for N greater than 1500 (for m=2) to 3500 (for m=6). With the JWKB starting energy, the shifted Lanczos algorithm converges to 33 figures in 3 iterations or less for all states. These methods are used in a study of the systematics of anharmonic-oscillator spectra and of the physical effects of the rescaling transformation.
1999 Academic Press
The techniques developed are then used in an extensive study of the systematics of anharmonic-oscillator spectra.
The calculations described here are modest in scale 500 to 1000 kilobytes of memory, running times of a fractions of a second on a PC with a 200-MHz chip. They involve conventional floating-point arithmetic only. All results are accurate to