Nonintegral Maslov indices

The phase loss of a wave reflected by a smooth potential generally varies continuously from π in the long-wave limit to π/2 in the limit of short waves. Incorporating the corresponding nonintegral multiples of π/2 as nonintegral Maslov indices in the formulation of the WKB approximation leads to a substantial improvement of accuracy when the conditions for applicability of the WKB method are violated only near the classical turning points. We demonstrate the efficacy of using nonintegral Maslov indices for a Woods-Saxon potential and a repulsive 1/${\mathit{x}}^2$ potential. The nonintegral Maslov index for a given 1/${\mathit{x}}^2$ potential yields far more accurate wave functions than the conventional Langer modification of the potential in conjunction with phase loss π/2. The energy spectrum of the radial harmonic oscillator (including the centrifugal potential), which is reproduced exactly by the standard WKB method with the Langer modification, is also reproduced exactly without the Langer modification when the nonintegral Maslov index is used. We suggest a method for approximately calculating the nonintegral Maslov index near the long-wave limit from the decaying WKB wave function in the classically forbidden region. © 1996 The American Physical Society.