通过非交换Picard-Vessiot理论研究通用微分方程的解

Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials

In the framework of differential Galois theory we treat the classical spectral problem $Ψ"-u(x)Ψ=λΨ$ and its finite-gap potentials as exactly solvable in quadratures by Picard--Vessiot without involving special functions; the ideology goes back to the 1919 works by J. Drach. We show that duality between spectral and quadrature approaches is realized through the Weierstrass permutation theorem for a logarithmic Abelian integral. From this standpoint we inspect known facts and obtain new ones: an important formula for the $Ψ$-function and $Θ$-function extensions of Picard--Vessiot fields. In par…

Spectral Picard-Vessiot fields for Algebro-geometric Schrödinger operators

This work is a galoisian study of the spectral problem $LΨ=λΨ$, for algebro-geometric second order differential operators $L$, with coefficients in a differential field, whose field of constants $C$ is algebraically closed and of characteristic zero. Our approach regards the spectral parameter $λ$ an algebraic variable over $C$, forcing the consideration of a new field of coefficients for $L-λ$, whose field of constants is the field $C(Γ)$ of the spectral curve $Γ$. Since $C(Γ)$ is no longer algebraically closed, the need arises of a new algebraic structure, generated by the solutions of the s…

The Picard Group of a Noncommutative Algebraic Torus

We compute the Picard group $ Pic(A_q) $ of the noncommutative algebraic 2-torus $A_q$, describe its action on the space $ R(A_q) $ of isomorphism classes of rk 1 projective modules and classify the algebras Morita equivalent to $ A_q $. Our computations are based on a quantum version of the Calogero-Moser correspondence relating projective $A_q$-modules to irreducible representations of the double affine Hecke algebras (DAHA) $ H_{t, q^{-1/2}}(S_n) $ at $ t = 1 $. We show that, under this correspondence, the action of $ Pic(A_q) $ on $ R(A_q) $ agrees with the action of $ SL_2(Z) $ on $ H_{t,…

Differential equations of order 1 in fields of zero characteristic

I begin from a particular field of generalised Puiseux series and investigate a class of nonlinear differential equations in the field. It is appeared that the main part of differential equation determines solvability and positions of resonance i.e. the appearance of a free constant in solutions. Secondly, for any singular differential equation in a differential field of the form $Q(y)\dot y=P(y)$, $P,Q$ polynomials. Then the greatest Picard-Vessiot extension exists. It is shown that the arising set of solutions is obtained from algebraic equations labelled by constants. Sufficient conditions …

Liouvillian solutions for second order linear differential equations with Laurent polynomial coefficient

This paper is devoted to a complete parametric study of Liouvillian solutions of the general trace-free second order differential equation with a Laurent polynomial coefficient. This family of equations, for fixed orders at $0$ and $\infty$ of the Laurent polynomial, is seen as an affine algebraic variety. We proof that the set of Picard-Vessiot integrable differential equations in the family is an enumerable union of algebraic subvarieties. We compute explicitly the algebraic equations of its components. We give some applications to well known subfamilies as the doubly confluent and biconflue…

Laurent序列、扩展Rota代数与动力系统的范畴离散化

Laurent Sequences, Extended Rota Algebras and Categorical Discretization of Dynamical Systems

提出一种保可积性的离散化方法，通过范畴论统一处理连续与离散动力系统，基于广义Rota有限算子微积分和扩展Rota代数。