The reducibility Of An Airy Operator

TL;DR

This paper proves the non-vanishing of a specific determinant related to Airy operators and demonstrates the irreducibility of a class of Setoyanagi operators over the complex rational function field.

Contribution

It introduces and analyzes a new determinant associated with Airy operators and establishes the irreducibility of Setoyanagi operators, advancing understanding in differential operator theory.

Findings

The determinant $ abla(d,eta)$ is non-zero.

Setoyanagi operator $S_{p,q}$ is irreducible over $C(x)[ d]$.

New connections between determinants and operator irreducibility.

Abstract

We show that the determinant $\nabla(d,\alpha),$ which seems to be not considered in the past, is not zero. As an application of this result we prove that the Setoyanagi operator $S_{p,q}=\partial^{2}-(ax^{p}+bx^{q}) $ is irreducible over $\mathbb{C}(x) [ \partial] $.