Exact results for models of multichannel quantum nonadiabatic transitions

Abstract

We consider nonadiabatic transitions in explicitly time-dependent systems with Hamiltonians of the form $\hat{H}(t) = \hat{A} +\hat{B} t + \hat{C}/t$, where $t$ is time and $\hat{A}$, $\hat{B}$, $\hat{C}$ are Hermitian $N\times N$ matrices. We show that in any model of this type, scattering matrix elements satisfy nontrivial exact constraints that follow from the absence of the Stokes phenomenon for solutions with specific conditions at $t \rightarrow -\infty$. This allows one to continue such solutions analytically to $t \rightarrow +\infty$, and connect their asymptotic behavior at $t \rightarrow -\infty$ and $t \rightarrow +\infty$. This property becomes particularly useful when a model shows additional discrete symmetries. In particular, we derive a number of simple exact constraints and explicit expressions for scattering probabilities in such systems.