Improving the accuracy of WKB eigenvalues

Abstract

A simple method is proposed for improving the accuracy of WKB eigenvalues by using the WKB eigenfunctions as trial functions in a variational-principle expression for the eigenvalues. The first-order eigenvalues obtained from this estimate are shown to differ from the exact values by terms of order ε6 (where ε is an appropriately defined small parameter that specifies the accuracy of the WKB approximation). For comparison, the fifth-order WKB eigenvalues also differ from the exact values by terms of order ε6. Higher-order variational eigenvalues are also defined, and the third-order estimate is shown to differ from the exact by terms of order ε10. The accuracy of these variational and WKB estimates are illustrated with a numerical example.

The string of variable density: Further results

“…To make contact with the results of Ref. [6] we consider a simple approximation to E (2) n , which is obtained restricting the sums over j and l in eq. (46) to j = l = 0 2 ; in this case we have…”

“…The two expressions are not equivalent since the results of ref. [6] are obtained assuming that the density is a slowly varying function as stated in their eqns. (11) and (12); we have not made this hypothesis on the density in our calculation.…”

“…The series for A (2) 3 may be approximated by a partial sum over the first N terms in the sum: for instance when we restrict the sum to the first N = 5000 terms we have 6 Notice that…”

Squeezed states and Helmholtz spectra

“…The integration of the 1-st order "angular equation" then turns out one of the most efficient methods to determine the energy spectra [1,5]. In this note we show that the 'classical algorithm' works even better for the Helmholtz problem [6,7,8]; in addition, it exhibits a new aspect of the Helmholtz spectra.…”

Cross-polarization of electromagnetic waves reflected by anisotropic gradient planar structures

“…The Wentzel-Kramers-Brillouin (WKB) method [9] allows us to find a solution for a medium with gradient parameters [10]. Useful approximate solutions obtained using the WKB method were found in [11], and the accuracy of the WKB solution was estimated in [12,13]. In the vicinity of turning points, the method has a limitation, since the solutions of Wenzel, Kramers, and Brillouin diverge in the vicinity of turning points.…”