# On an integral formula for Fredholm determinants related to pairs of spectral projections

## TL;DR

This paper derives an integral formula for Fredholm determinants related to spectral projections of self-adjoint operators with rank-one perturbations, linking them to the spectral shift function and applying it to subspace perturbation problems.

## Contribution

It introduces a novel integral identity connecting Fredholm determinants of spectral projections to the spectral shift function for rank-one perturbations of self-adjoint operators.

## Key findings

- Derived an explicit integral formula for Fredholm determinants involving spectral shift functions.
- Established conditions under which the formula applies to spectral projections of self-adjoint operators.
- Applied the formula to analyze subspace perturbation problems.

## Abstract

We consider Fredholm determinants of the form identity minus product of spectral projections corresponding to isolated parts of the spectrum of a pair of self-adjoint operators. We show an identity relating such determinants to an integral over the spectral shift function in the case of a rank-one perturbation. More precisely, we prove $$ -\ln \left(\det \big(\mathbf{1} -\mathbf{1} _{I}(A) \mathbf{1}_{\mathbb R\backslash I}(B)\mathbf{1}_{I}(A)\big) \right) = \int_I \text{d} x \int_{\mathbb R\backslash I} \text{d} y\, \frac{\xi(x)\xi(y)}{(y-x)^2}, $$ where $\mathbf{1}_J (\cdot)$ denotes the spectral projection of a self-adjoint operator on a set $J\in \text{Borel}(\mathbb R)$. The operators $A$ and $B$ are self-adjoint, bounded from below and differ by a rank-one perturbation and $\xi$ denotes the corresponding spectral shift function. The set $I$ is a union of intervals on the real line such that its boundary lies in the resolvent set of $A$ and $B$ and such that the spectral shift function vanishes there i.e. $I$ contains isolated parts of the spectrum of $A$ and $B$. We apply this formula to the subspace perturbation problem.