Quasi-classical asymptotics for functions of Wiener–Hopf operators: smooth versus non-smooth symbols

Sobolev, Alexander V.

doi:10.1007/s00039-017-0408-9

Cited by 15 publications

(16 citation statements)

References 25 publications

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“…The double asymptotics of the EE and of its coefficient B d are not simple to analyze and hard to guess by heuristic arguments. For low temperatures, that is small T > 0, this analysis has been performed in [17] for d = 1 and in [24] for d ≥ 2 yielding a result consistent with that for T = 0 in [12,15].…”

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confidence: 63%

Rényi entropies of the free Fermi gas in multi-dimensional space at high temperature

Leschke¹,

Sobolev²,

Spitzer³

2022

Preprint

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We study the local and (bipartite) entanglement Rényi entropies of the free Fermi gas in multi-dimensional Euclidean space in thermal equilibrium. We prove the positivity of the entanglement entropies at any temperature for Rényi index γ ≤ 1. Furthermore, for general γ > 0 we establish the asymptotics of the entropies at high temperature and at large scaling parameter for two different regimes -for fixed chemical potential and also for fixed particle density. In particular, we thereby provide the last remaining building block for a complete proof of our low-and high-temperature results presented (for γ = 1) in J. Phys. A: Math. Theor. 49, 30LT04 (2016) (Corrigendum: 50, 129501 (2017)) but being supported there only by the basic proof ideas.

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confidence: 63%

Rényi entropies of the free Fermi gas in multi-dimensional space at high temperature

Leschke¹,

Sobolev²,

Spitzer³

2022

Preprint

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“…The obtained asymptotic formula described the behaviour of the trace of (1.1) as the two parameters, α and T , independently tended to their respective limits: α → ∞ and T → 0. On the other hand, the results of [15] did not cover the case α → ∞, T = const. One aim of the current paper is to bridge this gap.…”

Section: Introductionmentioning

confidence: 86%

“…In this case the asymptotics for the operator (1.1) have a form different from (1.2), and their proof requires different methods. (4) In [15] the transition between the smooth and discontinuous symbol was studied: the smooth symbol a was supposed to depend on an extra "smoothing" parameter T > 0 so that a = a T converged to an indicator function as T → 0. The obtained asymptotic formula described the behaviour of the trace of (1.1) as the two parameters, α and T , independently tended to their respective limits: α → ∞ and T → 0.…”

Section: Introductionmentioning

confidence: 99%

On Szegő Formulas for Truncated Wiener–Hopf Operators

Sobolev

2019

Integr. Equ. Oper. Theory

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We consider functions of multi-dimensional versions of truncated Wiener-Hopf operators with smooth symbols, and study the scaling asymptotics of their traces. The obtained results extend the asymptotic formulas obtained by H. Widom in the 1980's to non-smooth functions, and non-smooth truncation domains.The obtained asymptotic formulas are used to analyse the scaling limit of the spatially bipartite entanglement entropy of thermal equilibrium states of non-interacting fermions at positive temperature.2010 Mathematics Subject Classification. Primary 47G30, 35S05; Secondary 45M05, 47B10, 47B35. Key words and phrases. Non-smooth functions of Wiener-Hopf operators, asymptotic trace formulas, entanglement entropy.The non-smooth generalizations are partly motivated by new applications of the Szegő asymptotics in Statistical Physics, connected with the entanglement entropy for free fermions (EE), see [2], [3], [5], [6] and references therein. In particular, the asymptotic trace formula for smooth symbols a (i.e. the one in Theorem 2.3) is used to describe the EE at a positive temperature (see [6]) , whereas the zero temperature case requires the use of discontinuous symbols (see [5]). We briefly comment on these applications after the main Theorem 2.3.The paper is organized as follows. In Sect. 2 we provide some preliminary information and state the main result, followed by a short discussion of the applications to the EE. It is not so trivial to see that the main asymptotic coefficient B d (a, ∂Λ; f ) is finite, if the function f is non-smooth. This point and other useful properties of B d (a, ∂Λ; f ) are clarified in Sect. 3. In Sect. 4 we collect some known and some new bounds for trace norms of Wiener-Hopf operators. Among other bounds, Sect. 4 contains the crucial trace-norm estimate for the operator (1.

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“…The multidimensional generalization of this result, even with more general discontinuous symbols a was obtained in [23], [24]. Further extension to non-smooth functions h was done in [18], [26], [27]. The formula (1.1) is a continuous analogue of the second-order Szegő limit theorem, see [30], so we loosely refer to (1.1) and (1.2) as Szegő formulas, or formulas of Szegő type.…”

Section: Introductionmentioning

confidence: 87%

Formulas of Szegő Type for the Periodic Schrödinger Operator

Pfirsch

Sobolev

2018

Commun. Math. Phys.

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We prove asymptotic formulas of Szegő type for the periodic Schrödinger operator H = − d 2 dx 2 + V in dimension one. Admitting fairly general functions h with h(0) = 0, we study the trace of the operator h(χ (−α,α) χ (−∞,µ) (H)χ (−α,α) ) and link its subleading behaviour as α → ∞ to the position of the spectral parameter µ relative to the spectrum of H. Date: December 7, 2016. 2010 Mathematics Subject Classification. Primary 47G30, 35S05; Secondary 45M05, 47B10, 47B35, 81Q10. Key words and phrases. Periodic Schrödinger operators, asymptotic trace formulas, non-smooth functions of Wiener-Hopf operators, entanglement entropy.where the integral kernel of Π µ is defined in (4.2). Proof. With the notation of Lemma 4.2 we may write

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Quasi-classical asymptotics for functions of Wiener–Hopf operators: smooth versus non-smooth symbols

Cited by 15 publications

References 25 publications

Rényi entropies of the free Fermi gas in multi-dimensional space at high temperature

Rényi entropies of the free Fermi gas in multi-dimensional space at high temperature

On Szegő Formulas for Truncated Wiener–Hopf Operators

Formulas of Szegő Type for the Periodic Schrödinger Operator

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