Narrowly avoided crossings

Abstract

In order to create a degeneracy in a quantum mechanical system without symmetries one must vary two parameters in the Hamiltonian. When only one parameter, lambda say, is varied, there is only a finite closest approach Delta E of two eigenvalues, and never a crossing. Often the gaps Delta E in these avoided crossings are much smaller than the mean spacing between the eigenvalues, and it has been conjectured that in this case the gap results from tunnelling through classically forbidden regions of phase space and decreases exponentially as h(cross) to 0: Delta E=Ae-S/h(cross). The author reports the results of numerical calculations on a system with two parameters, epsilon , lambda , which is completely integrable when epsilon =0. It if found that the gaps Delta E obtained by varying lambda decrease exponentially as h(cross) to 0, consistent with the tunnelling conjecture. When epsilon =0, Delta E=0 because the system is completely integrable. As epsilon to 0, the gaps do not vanish because the prefactor A vanishes; instead it is found that S diverges logarithmically. Also, keeping h(cross) fixed, the gaps are of size Delta E=O( epsilon nu ), where nu is usually very close to an integer. Theoretical arguments are presented which explain this result.