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

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 $$-\ln \left(\det (1-1_I(A)1_{\mathbb{R}\backslash I}(B)1_I(A))\right)={\int }_I\text{d}x{\int }_{\mathbb{R}\backslash I}\text{d}y\frac{\xi (x)\xi (y)}{(y-x{)}^2},$$ where $1_J(\cdot )$ 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.

version as published

References in corpus (4)

• Notes on the sin2Θ theorem
• Anderson's orthogonality catastrophe in one dimension induced by a magnetic field
• The asymptotics of an eigenfunction-correlation determinant for Dirac-$\delta$ perturbations (Anderson's Orthogonality Catastrophe for Dirac-$\delta$)
• The exponent in the orthogonality catastrophe for Fermi gases