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.
Categories: cond-mat.mes-hall quant-ph