Quasilinearization method applied to multidimensional quantum tunneling.

Abstract

We apply the quasilinearization method of Bellman and Kalaba [Quasilinearization and Nonlinear Boundary-Value Problems (Elsevier, New York, 1965)] to find approximate solutions for the multidimensional quantum tunneling for separable as well as nonseparable wave equations. By introducing the idea of the complex ``semiclassical trajectory'' which is valid for the motion over and under the barrier, and which, in the proper limit, reduces to the real classical trajectory in the allowed region, we obtain an eigenvalue equation for the characteristic wave numbers. This eigenvalue equation is similar to the corresponding equation obtained from the WKB approximation and yields complex eigenvalues with negative imaginary parts. When the barrier changes very rapidly as a function of the radial distance, we can replace the concept of the semiclassical trajectory, which may not be applicable in this case, by the concept of a complex ``quantum trajectory.'' The trajectory defined either way depends on a constant of integration, and by minimizing the action with respect to this constant we can obtain the minimum escape path. The case of two-dimensional tunneling is discussed as an example of this method.