Asymptotics of determinants of 4-th order operators at zero

TL;DR

This paper analyzes the behavior of the Fredholm determinant for fourth order differential operators with compactly supported coefficients, focusing on its asymptotic properties at zero for operators on the line and half-line.

Contribution

It provides a detailed description of the determinant's behavior at zero, revealing the order of poles in generic cases for both line and half-line operators.

Findings

Determinant has a pole of order 4 on the line at zero.

Determinant has a pole of order 1 on the half-line at zero.

The analysis applies to operators with compactly supported coefficients.

Abstract

We consider fourth order ordinary differential operators with compactly supported coefficients on the half-line and on the line. The Fredholm determinant for this operator is an analytic function in the whole complex plane without zero. We describe the determinant at zero. We show that in the generic case it has a pole of order 4 in the case of the line and of order 1 in the case of the half-line.