WKB approximation for quasi-bound states and trapped modes

Abstract

We develop a largely automatic semi-analytic method for calculating weakly damped quasi-bound states and trapped modes supported by a local minimum of an effective potential. The real part of the frequency is obtained from an arbitrarily high-order local WKB expansion and Padé resummation, while the exponentially small imaginary part is estimated using the Gamow approximation for tunnelling through one or two potential barriers. Because the local quantization requires only derivatives of the potential at its minimum, the method applies readily to non-rational effective potentials and to different compact-object geometries. We provide a Mathematica notebook implementing the procedure. Comparisons with continued-fraction and direct numerical results for massive scalar fields around Schwarzschild and Kerr black holes and for axial trapped modes of a uniform-density star show that the method yields accurate real frequencies and useful estimates of decay rates outside the superradiant regime.