Multidimensional WKB Approximation for Tunneling Along Curved Escape Paths

Abstract

Asymptotics of the perturbation series for the ground state energy of the coupled anharmonic oscillators for the positive coupling constant is related to the lifetime of the quasistationary states for the negative coupling constant. The latter is determined by means of the multidimensional WKB approximation for tunneling along curved escape paths. General method for obtaining such approximation is described. The cartesian coordinates (x,y) are choosen in such a way that the x-axis has the direction of the probability flux at large distances from the well. The WKB wave function is then obtained by the simultaneous expansion of the wave function in the coordinate y and the parameter γ determining the curvature of the escape path. It is argued, both physically and mathematically, that these two expansions are mutually consistent. Several simplifications in the integrations of equations are pointed out. It is shown that to calculate outgoing probability flux it is not necessary to deal with inadequacy of the WKB approximation at the classical turning point. The WKB formulas for the large-order behavior of the perturbation series are compared with numerical results and an excellent agreement between the two is found.