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The Widom-Sobolev formula for discontinuous matrix-valued symbols

Leon Bollmann, Peter Müller

TL;DR

The paper generalizes the Widom–Sobolev Szegő-type formula to truncated Wiener–Hopf operators with discontinuous matrix-valued symbols in higher dimensions, establishing a logarithmically enhanced area term with explicit coefficients. It develops a matrix-valued symbol calculus, reduces matrix traces to scalar traces, and proves commutator estimates to handle noncommuting symbols, first for polynomials and then for general analytic, smooth, and Hölder test functions. The leading volume term is given by $L^d\mathfrak{W}_0$ and the boundary/logarithmic correction by $L^{d-1}\log L\,\mathfrak{W}_1$ involving the auxiliary symbol $\mathfrak{A}$ and the boundary functional $\mathfrak{U}$; this explicit structure extends the scalar theory to matrix-valued symbols and supports applications to entanglement entropies in non-interacting Fermi gases. These results unify and extend previous one-dimensional matrix-valued work, enable applications to Dirac-type operators, and provide a robust framework for studying Szegő-type asymptotics with discontinuities in multiple variables.

Abstract

We prove the Widom-Sobolev formula for the asymptotic behaviour of truncated Wiener-Hopf operators with discontinuous matrix-valued symbols for three different classes of test functions. The symbols may depend on both position and momentum except when closing the asymptotics for twice differentiable test functions with Hölder singularities. The cut-off domains are allowed to have piecewise differentiable boundaries. In contrast to the case where the symbol is smooth in one variable, the resulting coefficient in the enhanced area law we obtain here remains as explicit for matrix-valued symbols as it is for scalar-valued symbols.

The Widom-Sobolev formula for discontinuous matrix-valued symbols

TL;DR

The paper generalizes the Widom–Sobolev Szegő-type formula to truncated Wiener–Hopf operators with discontinuous matrix-valued symbols in higher dimensions, establishing a logarithmically enhanced area term with explicit coefficients. It develops a matrix-valued symbol calculus, reduces matrix traces to scalar traces, and proves commutator estimates to handle noncommuting symbols, first for polynomials and then for general analytic, smooth, and Hölder test functions. The leading volume term is given by and the boundary/logarithmic correction by involving the auxiliary symbol and the boundary functional ; this explicit structure extends the scalar theory to matrix-valued symbols and supports applications to entanglement entropies in non-interacting Fermi gases. These results unify and extend previous one-dimensional matrix-valued work, enable applications to Dirac-type operators, and provide a robust framework for studying Szegő-type asymptotics with discontinuities in multiple variables.

Abstract

We prove the Widom-Sobolev formula for the asymptotic behaviour of truncated Wiener-Hopf operators with discontinuous matrix-valued symbols for three different classes of test functions. The symbols may depend on both position and momentum except when closing the asymptotics for twice differentiable test functions with Hölder singularities. The cut-off domains are allowed to have piecewise differentiable boundaries. In contrast to the case where the symbol is smooth in one variable, the resulting coefficient in the enhanced area law we obtain here remains as explicit for matrix-valued symbols as it is for scalar-valued symbols.
Paper Structure (13 sections, 28 theorems, 81 equations)

This paper contains 13 sections, 28 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. Admissible domains
  4. Complex-valued symbols
  5. Matrix-valued symbols
  6. Asymptotic formula for polynomials
  7. Commutation results
  8. Asymptotic formula
  9. Extension to more general Wiener--Hopf operators
  10. Closing the asymptotics
  11. Analytic functions
  12. Smooth functions
  13. More general functions

Key Result

Lemma 3.1

Let $A\in C^{\infty}_{b}(\mathbb{R}^{d} \times \mathbb{R}^{d} \times \mathbb{R}^{d}, \mathbb{C}^{n\times n})$ be a matrix-valued amplitude. Then, for every $L\geqslant 1$ we have where $m:=\lfloor \frac{d}{2}\rfloor+1$, and the constant $C$ is independent of $L$. Here, $\lfloor u\rfloor$ stands for the largest integer not exceeding $u\in\mathbb{R}$. Clearly, this carries over to symbols $A\in C^{

Theorems & Definitions (38)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Lemma 3.1
  • Definition 3.2
  • Definition 3.3
  • Lemma 3.4
  • Remark 3.5
  • ...and 28 more