Quantum integrability in the multistate Landau-Zener problem

We analyze Hamiltonians linear in the time variable for which the multistate Landau-Zener problem is known to have an exact solution. We show that they either belong to families of mutually commuting Hamiltonians polynomial in time or reduce to the 2 x 2 Landau-Zener problem, which is considered trivially integrable. The former category includes the equal slope, bow-tie, and generalized bow-tie models. For each of these models we explicitly construct the corresponding families of commuting matrices. The equal slope model is a member of an integrable family that consists of the maximum possible number (for a given matrix size) of commuting matrices linear in time. The bow-tie model belongs to a previously unknown, similarly maximal family of quadratic commuting matrices. We thus conjecture that quantum integrability understood as the existence of nontrivial parameter-dependent commuting partners is a necessary condition for the Landau-Zener solvability. Descendants of the 2 x 2 Landau-Zener Hamiltonian are e.g. general SU(2) and SU(1,1) Hamiltonians, time-dependent linear chain, linear, nonlinear, and double oscillators. We explicitly obtain solutions to all these Landau-Zener problems from the 2 x 2 case.