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Middle Laplace transform and middle convolution for linear Pfaffian systems with irregular singularities

Shunya Adachi

TL;DR

The paper develops a unified framework for transforming linear Pfaffian systems with irregular singularities via the middle Laplace transform $\mathcal{ML}$, inspired by Katz theory, and introduces a generalized middle convolution $mc_{\beta}$. It provides a rigorous categorical interpretation using meromorphic connections, extending the construction from one variable to several variables, and proves invertibility and irreducibility properties for the transforms. The addition of $mc_{\beta}=\mathcal{ML}^{-1}\circ add_{-\beta}\circ\mathcal{ML}$ generalizes Dettweiler–Reiter’s middle convolution to irregular settings and enables recursive construction of systems with irregular singularities, including multivariable confluent hypergeometric-type equations. The framework yields a robust, recursive method to generate and study holonomic systems with irregular singularities and links the local data transformations (spectral types, exponents) to global properties, with potential implications for Stokes phenomena and isomonodromic deformations.

Abstract

We introduce a transformation of linear Pfaffian systems, which we call the middle Laplace transform, as a formulation of the Laplace transform from the perspective of Katz theory. While the definition of the middle Laplace transform is purely algebraic, its categorical interpretation is also provided. We then show the fundamental properties (invertibility, irreducibility) of the middle Laplace transform. As an application of the middle Laplace transform, we define the middle convolution for linear Pfaffian systems with irregular singularities. This gives a generalization of Haraoka's middle convolution, which was defined for linear Pfaffian systems with logarithmic singularities. The fundamental properties (additivity, irreducibility) of the middle convolution follow from the properties of the middle Laplace transform. Some examples related to hypergeometric functions with two variables are also given.

Middle Laplace transform and middle convolution for linear Pfaffian systems with irregular singularities

TL;DR

The paper develops a unified framework for transforming linear Pfaffian systems with irregular singularities via the middle Laplace transform , inspired by Katz theory, and introduces a generalized middle convolution . It provides a rigorous categorical interpretation using meromorphic connections, extending the construction from one variable to several variables, and proves invertibility and irreducibility properties for the transforms. The addition of generalizes Dettweiler–Reiter’s middle convolution to irregular settings and enables recursive construction of systems with irregular singularities, including multivariable confluent hypergeometric-type equations. The framework yields a robust, recursive method to generate and study holonomic systems with irregular singularities and links the local data transformations (spectral types, exponents) to global properties, with potential implications for Stokes phenomena and isomonodromic deformations.

Abstract

We introduce a transformation of linear Pfaffian systems, which we call the middle Laplace transform, as a formulation of the Laplace transform from the perspective of Katz theory. While the definition of the middle Laplace transform is purely algebraic, its categorical interpretation is also provided. We then show the fundamental properties (invertibility, irreducibility) of the middle Laplace transform. As an application of the middle Laplace transform, we define the middle convolution for linear Pfaffian systems with irregular singularities. This gives a generalization of Haraoka's middle convolution, which was defined for linear Pfaffian systems with logarithmic singularities. The fundamental properties (additivity, irreducibility) of the middle convolution follow from the properties of the middle Laplace transform. Some examples related to hypergeometric functions with two variables are also given.

Paper Structure

This paper contains 41 sections, 28 theorems, 444 equations.

Table of Contents

  1. Introduction
  2. Middle Laplace transform in one variable
  3. Definition
  4. Inverse (middle) Laplace transform
  5. Fundamental properties of the middle Laplace transform
  6. Categorical interpretation (one variable case)
  7. Category of meromorphic connections
  8. Subcategories
  9. Category of linear ordinary differential equations
  10. Category of ordinary linear differential equations of Birkhoff-Okubo normal form
  11. Functors
  12. Birkhoff-Okubo extension functor
  13. Laplace transform functor
  14. Middle Laplace transform functor
  15. Inversion formula and irreducibility (one variable case)
  16. ...and 26 more sections

Key Result

Lemma 2.3

The subspace $\mathcal{K}$ is a common invariant subspace of the tuple of matrices $(T,B_1,\ldots,B_{\hat{q}})$.

Theorems & Definitions (83)

  • Definition 2.1
  • Remark 2.2
  • Lemma 2.3
  • proof
  • Definition 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Remark 2.8
  • Definition 2.9
  • ...and 73 more