Quantum curves

One says that a pair (P,Q) of ordinary differential operators specify a quantum curve if [P,Q]=const. If a pair of difference operators (K,L) obey the relation KL=const LK we say that they specify a discrete quantum curve. This terminology is prompted by well known results about commuting differential and difference operators, relating pairs of such operators with pairs of meromorphic functions on algebraic curves obeying some conditions. The goal of this paper is to study the moduli spaces of quantum curves. We will show how to quantize a pair of commuting differential or difference operators (i.e. to construct the corresponding quantum curve or discrete quantum curve). The KP-hierarchy acts on the moduli space of quantum curves; we prove that similarly the discrete KP-hierarchy acts on the moduli space of discrete quantum curves.Comment: New results, some correction

Computation of formal fundamental solutions

AbstractA new proof for the existence of formal fundamental solutions of meromorphic systems of linear ordinary differential equations is given. The proof is presented in a totally constructive form. Moreover, we give a corresponding proof of existence for highest level formal solutions

Formal Solutions of Singularly Perturbed Linear Differential Systems

In this article, we discuss formal invariants of singularly-perturbed linear differential systems in neighborhood of turning points and give algorithms which allow their computation. The algorithms proposed are implemented in the computer algebra system Maple