Monodromy Of Differential Equations Research Articles

Equivariant functions and vector-valued modular forms

For any discrete group Γ and any two-dimensional complex representation ρ of Γ, we introduce the notion of ρ-equivariant functions, and we show that they are parametrized by vector-valued modular forms. We also provide examples arising from the monodromy of differential equations.

LOCAL MONODROMY OF p-ADIC DIFFERENTIAL EQUATIONS: AN OVERVIEW

This primarily expository article collects together some facts from the literature about the monodromy of differential equations on a p-adic (rigid analytic) annulus, though often with simpler proofs. These include Matsuda's classification of quasi-unipotent ∇-modules, the Christol–Mebkhout construction of the ramification filtration, and the Christol–Dwork Frobenius antecedent theorem. We also briefly discuss the p-adic local monodromy theorem without proof.

Analytical calculation of nonadiabatic transition probabilities from the monodromy of differential equations

The nonadiabatic transition probabilities in the two-level systems are calculated analytically by using the monodromy matrix determining the global feature of the underlying differential equation. We study the time-dependent 2x2 Hamiltonian with the tanh-type plus sech-type energy difference and with constant off-diagonal elements as an example to show the efficiency of the monodromy approach. The application of this method to multi-level systems is also discussed.