Tunneling solutions of the Hamilton-Jacobi equation for multidimensional semiclassical theory

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Abstract
A class of complex solutions to the time-independent Hamilton-Jacobi equation in the real-valued configuration space that represent multidimensional nonclassical motions such as dynamical tunneling, namely, energetically allowed but dynamically forbidden transition, as well as the ordinary tunneling are shown. We introduce a quantity called ‘‘parity of motion’’ into each coordinate in configuration space for the Hamilton-Jacobi equation and thereby construct the solutions. Positive parity induces merely ordinary classical motion, while negative parity allows nonclassical motion such as tunneling. These solutions are classified by a given set of parities, each class of which forms a sheet in the entire solution space. New canonical equations of motion are derived with which nonclassical paths are generated in each sheet. Furthermore, it is shown that each sheet is associated with two kinds of action integrals: One, which is real valued, satisfies the principle of least action, thereby generating both ordinary and tunneling trajectories, but is not a solution to the Hamilton-Jacobi equation, while the other action is a solution to the time-independent Hamilton-Jacobi equation, and is complex valued in a tunneling region. Only in Newtonian mechanics, which forms an extreme sheet having all the parities positive, do these two actions happen to coincide with each other. Numerical examples for dynamical tunneling among tori in the Hénon-Heiles system and ordinary potential tunneling in a three-dimensional system are presented.