Stability of a Szegő-type asymptotics
Peter Müller, Ruth Schulte
TL;DR
This paper proves that the second-order Szegő-type asymptotics for the trace of a spatially truncated Fermi projection remains universal under a compactly supported perturbation $V$ of the Laplacian, i.e., the coefficients $N_{0}(E)$, $\Sigma_{0}(E)$ and the functional $I(h)$ match the Laplacian case for a wide class of test functions $h\in\mathbb{H}_{d}$. The authors combine the traditional Wiener–Hopf/Szegő approach with refined Schatten-norm estimates and interpolation to control the perturbation, showing $\mathrm{tr}\{h(1_{\Lambda_L}1_{<E}(H)1_{\Lambda_L})\} = N_{0}(E) h(1)|\Lambda|L^{d} + \Sigma_{0}(E) I(h) |\partial\Lambda| L^{d-1} \ln L + o(L^{d-1}\ln L)$. The results extend previous work from the von Neumann entropy to a broad class of entanglement-measure functions, including Rényi entropies $h_{\alpha}$, and establish the robustness of the enhanced area-law-type subleading term under short-range perturbations. This has implications for entanglement properties of non-interacting fermions in the continuum with localized perturbations. The analysis provides a framework that potentially generalizes to other self-adjoint perturbations while preserving the universal Szegő coefficients.
Abstract
We consider a multi-dimensional continuum Schrödinger operator $H$ which is given by a perturbation of the negative Laplacian by a compactly supported bounded potential. We show that, for a fairly large class of test functions, the second-order Szegő-type asymptotics for the spatially truncated Fermi projection of $H$ is independent of the potential and, thus, identical to the known asymptotics of the Laplacian.