The exact WKB for dynamical and geometric exponents in generalized and non-linear Landau-Zener transitions

The geometric phases and exponents introduced by Berry have been discussed in wide areas, while the exact WKB is a mathematical method that treats perturbative expansion with respect to the Planck constant as a singular perturbation. This method, initiated by Voros, is a powerful computational tool that can replace conventional approximations with exact results and makes the analytical computation much easier for mathematical software. Using the exact WKB, we analyze the derivation of the dynamical and the geometric exponents in generalized Landau-Zener models, highlighting the differences from the other calculational methods.