Langer’s method for weakly bound states of the Helmholtz equation with symmetric profiles

• Published in: Journal of mathematical physics 1982-11, Vol.23 (11), p.2122-2126
• Main Author: Johnston, George L.
• Format: Article
• Language: English
• ISSN: 0022-2488; 1089-7658
• DOI: 10.1063/1.525266

Use of the harmonic oscillator equation as the comparison equation in the application of Langer’s method to bound states of the Helmholtz equation, w″+k 2 0 g(z)w(z)=0, with symmetric profiles k 2 0 g(z), produces the WKB eigenvalue condition, which asserts the equality of the phase integral of the original equation between the turning points to (n+1/2)π. In the case of weakly bound states, this condition gives eigenvalue estimates of low accuracy. Use of the Helmholtz equation with the symmetric Epstein profile, G(x)=[Ẽ+U 0(cosh αx)− 2], as the comparison equation provides the basis for a convenient method to obtain eigenvalue estimates of substantially increased accuracy in the case of weakly bound states. In addition to the usual condition of equality of the phase integrals of the original and comparison equations between the turning points, the conditions k 2 0 g(0) =G(0) and k 2 0 g(∞) =G(∞) are imposed. An eigenvalue condition which is a simple generalization of the usual WKB eigenvalue condition is obtained. Its application to selected diverse examples of the Helmholtz equation indicates that it has a broad range of utility.