A Semiclassical Approach to Level Crossing in Supersymmetric Quantum Mechanics

Much use has been made of the techniques of supersymmetric quantum mechanics (SUSY QM) for studying bound-state problems characterized by a superpotential $\phi(x)$. Under the analytic continuation $\phi(x) \to i\phi(x)$, a pair of superpartner bound-state problems is transformed into a two-state level-crossing problem in the continuum. The description of matter-enhanced neutrino flavor oscillations involves a level-crossing problem. We treat this with the techniques of supersymmetric quantum mechanics. For the benefit of those not familiar with neutrino oscillations and their description, enough details are given to make the rest of the paper understandable. Many other level-crossing problems in physics are of exactly the same form. Particular attention is given to the fact that different semiclassical techniques yield different results. The best result is obtained with a uniform approximation that explicitly recognizes the supersymmetric nature of the system.