Zeta and Fredholm determinants of self-adjoint operators

TL;DR

This paper explores the relationship between zeta-regularized and Fredholm determinants for self-adjoint operators, providing formulas and criteria that connect these concepts through heat trace coefficients and spectral assumptions.

Contribution

It derives a formula linking zeta and Fredholm determinants of self-adjoint operators and offers criteria involving heat trace coefficients for their equivalence.

Findings

Derived a formula relating zeta and Fredholm determinants.

Expressed derivatives of zeta determinants in terms of heat trace coefficients.

Provided criteria ensuring the equivalence of determinants for elliptic operators.

Abstract

Let $L$ be a self-adjoint invertible operator in a Hilbert space such that $L^{-1}$ is $p$-summable. Under a certain discrete dimension spectrum assumption on $L$, we study the relation between the (regularized) Fredholm determinant, $\det_{p}(I+z\cdot L^{-1})$, on the one hand and the zeta regularized determinant, $\det_{\zeta}(L+z)$, on the other. One of the main results is the formula \begin{equation*} \frac{\det_{\zeta} (L + z)}{\det_{\zeta} (L)} = \exp \left( \sum_{j=1}^{p-1} \frac{z^j}{j!} \cdot \frac{d^j}{dz^j} \log \det\nolimits_{\zeta} (L+z) |_{z=0} \right) \cdot \det\nolimits_{p}(I + z \cdot L^{-1} ). \end{equation*} We show that the derivatives $\frac{d^j}{dz^j} \log \det\nolimits_{\zeta} (L+z) |_{z=0}$ can be expressed in terms of (regularized) zeta values and heat trace coefficients of $L$. Furthermore, we give a general criterion in terms of the heat trace coefficients (and which is, e.g., fulfilled for large classes of elliptic operators) which guarantees that the constant term in the asymptotic expansion of the Fredholm determinant, $\log\det_{p} (I+z\cdot L^{-1})$, equals the zeta determinant of $L$.