A solvable model of the breakdown of the adiabatic approximation

Let L >= 0 and 0 < <much less than> 1. Consider the following time-dependent family of 1D Schrodinger equations with scaled harmonic oscillator potentials i epsilon partial derivative tu epsilon=-<mml:mfrac>12</mml:mfrac>partial derivative x2</mml:msubsup>u epsilon +V(t,x)<mml:msub>u epsilon, u<INF></INF>(-L - 1, x) = pi -1/4 exp(-x2/2), where V(t, x) = (t + L)2x2/2, t < - L, V(t, x) = 0, - L <less than or equal to> t <= L, and V(t, x) = (t - L)2x2/2, t > L. The initial value problem is explicitly solvable in terms of Bessel functions. Using the explicit solutions, we show that the adiabatic theorem breaks down as -> 0. For the case L = 0, complete results are obtained. The survival probability of the ground state pi -1/4 exp(-x2/2) at microscopic time t = 1/ is 1/<mml:msqrt>2</mml:msqrt>+O(epsilon). For L > 0, the framework for further computations and preliminary results are given.