Energy Eigenvalues of quartic oscillators in d≤3 dimensions

Abstract

The authors present a simple approximate analytical expression for the energy eigenvalues E^(d) of the pure quartic oscillation in d<or=3 dimensions. The formula reproduces with high accuracy the results from accurate numerical computations reported in the literature. The eigenvalue E_n,l^(d) for given n is seen to decrease as l increases contrary to the prediction of the Quigg-Rosner formula (for d=3) which is incorrect in this respect. They also give a generalisation (from the pure quartic to the quartic anharmonic oscillator) of our formula, good for any d. This formula involves four parameters (one less in the pure quartic case) of which all but one are obtainable by consideration of the WKB limit; the values of all the parameters are independent of the dimension d.