Trace formulas for Wiener--Hopf operators with applications to entropies of free fermionic equilibrium states

We consider non-smooth functions of (truncated) Wiener--Hopf type operators on the Hilbert space $L^2(\mathbb R^d)$. Our main results are uniform estimates for trace norms ($d\ge 1$) and quasiclassical asymptotic formulas for traces of the resulting operators ($d=1$). Here, we follow Harold Widom's seminal ideas, who proved such formulas for smooth functions decades ago. The extension to non-smooth functions and the uniformity of the estimates in various (physical) parameters rest on recent advances by one of the authors (AVS). We use our results to obtain the large-scale behaviour of the local entropy and the spatially bipartite entanglement entropy (EE) of thermal equilibrium states of non-interacting fermions in position space $\mathbb R^d$ ($d\ge 1$) at positive temperature, $T>0$. In particular, our definition of the thermal EE leads to estimates that are simultaneously sharp for small $T$ and large scaling parameter $\alpha>0$ provided that the product $T\alpha$ remains bounded from below. Here $\alpha$ is the reciprocal quasiclassical parameter. For $d=1$ we obtain for the thermal EE an asymptotic formula which is consistent with the large-scale behaviour of the ground-state EE (at $T=0$), previously established by the authors for $d\ge 1$.


Introduction
The present paper is devoted to the study of (bounded, self-adjoint) operators of the form W α := W α (a; Λ) := χ Λ Op α (a)χ Λ , α > 0, (1.1) on L 2 (R d ), d ≥ 1, where χ Λ is the indicator function of a set Λ ⊂ R d . The parameter 1/α can be interpreted as a quasiclassical parameter that tends to zero in our asymptotic results. The notation Op α (a) stands for the α-pseudo-differential operator with symbol a = a(ξ), which acts on Schwartz functions u on R d as Integrals without indication of the integration domain always mean integration over R d with the value of d which is clear from the context. More general symbols, depending on both variables x and ξ, or operators with matrix-valued symbols can be also treated, but we limit our attention only to ξ-dependent symbols. We call the operator (1.1) a Wiener-Hopf operator. A more precise term would be truncated Wiener-Hopf operator, but we always omit "truncated" for brevity. Our focus is on the operator difference with some suitably chosen functions f . We are interested in the asymptotic properties of the trace tr D α (a, Λ; f ) as α → ∞. If f (0) = 0, Λ is bounded and a decays sufficiently fast at infinity, then it is trivial to observe that the second operator on the right-hand side of (1.2) is trace-class and where |Λ| is the d-dimensional Lebesgue measure of Λ. If |Λ| = ∞, then neither of the terms on the right-hand side of (1.2) is trace class (except in trivial cases), but their difference is trace class, under the conditions adopted in this paper. We must emphasise that it is essential to us to consider Λ in (1.2) of infinite measure.
Asymptotic properties of D α (a, Λ; f ) have been extensively studied in the literature, with the majority of results obtained in the 1980's. All results obtained at that time pertained to the case of smooth functions f (or more precisely, smooth on the range of the symbol a) and bounded Λ. Under these assumptions, the case of a smooth symbol a was understood particularly well: the full asymptotic expansion of tr D α (a, Λ; f ) in powers of α −1 was derived by A. Budylin-V. Buslaev [4] and H. Widom [30]. The paper [30] also provides a brief historical account of this problem. Out of all relevant bibliography we mention just one other paper by H. Widom, [29], whose ideas we exploit in some of our proofs.
Another important and challenging problem is to study the asymptotics of the trace of D α (a, Λ; f ) for discontinuous symbols, in particular, for symbols of the form a = χ Ω with a bounded region Ω ⊂ R d . This problem was studied by H. Landau-H. Widom [13], H. Widom [27] (for d = 1) and by A.V. Sobolev [21,23] (for arbitrary d ≥ 1). It was found that (1.4) tr D α (a, Λ; f ) = W 1 α d−1 log(α) + o(α d−1 log(α)) , α → ∞, for a bounded domain Λ ⊂ R d with an explicitly given coefficient W 1 = W 1 (∂Λ, ∂Ω, f ). The discontinuity of the symbol a can be interpreted as the presence of one of the two Fisher-Hartwig singularities investigated in detail for truncated Toeplitz matrices, that is, for the discrete counterpart of Wiener-Hopf operators, see [6]. In recent years, new demands for the asymptotics of traces of Wiener-Hopf operators emerged, which have been triggered by applications to (quantum) statistical mechanics. Our interest originates from the large-scale behaviour of the spatially bipartite entanglement entropy (EE, also called mutual information) of free fermions in thermal equilibrium. Here one faces several mathematical challenges at the same time.
(1) Non-smooth functions f . One needs to consider the operator (1.2) with functions f that lack smoothness at finitely many points, or, which is the same in view of additivity, at one point. The functions of interest are the γ-Rényi entropy functions η γ , γ > 0, that are defined in (10.1) and (10.2). (2) Unbounded Λ. One needs to consider the operator (1.2) with unbounded domains Λ, in contrast to most of the previously known results. (3) Uniform estimates. In quantum-mechanical applications, apart from the scaling parameter, it is natural to control the dependence of the symbol a on other parameters such as the temperature T ≥ 0. Thus it is necessary to provide estimates and asymptotic remainder estimates that are uniform in the symbol a in some broad sense. For example, in the study of the entanglement entropy the symbol a in the operator (1.2) is given by the Fermi symbol a T,µ , see the definition (1.5), and one needs to control the T -dependence of the estimates. This requires substantial extra work since the results of [29,30] are not directly applicable.
A general approach to the study of operator differences of the form P f (P AP )P − P f (A)P with a self-adjoint operator A, an orthogonal projection P and a non-smooth function f , was put forward in [24]. One application of the results in [24] is the extension of (1.4) to non-smooth functions f under the assumption that either Λ or its complement is bounded, thereby tackling challenges (1) and (2) above.
The special case a(ξ) = χ Ω (ξ) for bounded Ω, Λ ⊂ R d was considered even earlier in [14]. In the quantum-mechanical context, formula (1.4), if used with a = χ Ω and the function f = η γ , gives the large-scale asymptotics of the entanglement entropy at zero temperature with Fermi sea Ω, see also [8] for some other motivation.
In the present paper we work exclusively with smooth symbols a with a fast decay at infinity. The function f is allowed to lack smoothness at one point, see Condition 2.1. A typical example of such a function is f (t) = |t| γ , γ > 0. The region Λ is such that either Λ or R d \ Λ is bounded, see Condition 3.1 for details.
The goal of this paper is two-fold, and it correspondingly splits in two parts. Part 1: Sections 2-7. First we establish some explicit estimates for the (quasi-) norms of the operator (1.2) in the Schatten-von Neumann classes S q , q ∈ (0, 1]. Later on we need only trace class norms, but the more general S q -estimates are obtained at "no extra cost", and are provided for the sake of completeness. Here we rely on the results of [24] where this problem was studied in the abstract setting. We quote these results in Proposition 2.2. Indeed, the very fact that D α (a, Λ; f ) ∈ S q is an almost direct consequence of Proposition 2.2, but this alone is insufficient for us since we need sharp explicit estimates, uniform in a. Thus we identify a class of symbols a that we call multi-scale symbols, and establish explicit estimates for D α (a, Λ; f ) Sq , which are uniform in some suitable sense, see Remark 3.3. They do turn out to be sharp in α and T when used for the symbol (1.5), which serves as our leading example. The main estimate is contained in Theorem 3.5. This takes care of issue (3) above.
Our next result is the asymptotic formula for tr D α (a, f ; Λ) as α → ∞, for spatial dimension d = 1, see Section 4. Here we assume again that a is a multi-scale symbol, and the main objective is to have the explicit control of the remainder, see Theorems 4.4 and 4.7. As mentioned earlier, we follow the seminal ideas of H. Widom, who proved such asymptotic results for smooth functions f already in the 1980's, see [28,29,30,31]. The proofs of the main asymptotic results of Section 4 are presented in Sections 5 and 6. To accomplish this we use rather a standard methodology of quasiclassical analysis: first we prove the required asymptotics for smooth functions f , and then using the bounds from Theorem 3.5 we extend them to non-smooth ones. The starting point is the Helffer-Sjöstrand formula (see Appendix A) which rewrites the trace of D α (a, Λ; f ) for smooth f in terms of D α (a, Λ; r z ) with the resolvent function r z (λ) : Part 2: Sections 8-10. Here we apply the results obtained in Part 1 to the symbol which is nothing but the Fermi symbol of free Fermions. Here the real-valued function h = h(ξ) is the classical one-particle Hamiltonian of the free Fermi gas, h(ξ) → ∞ as |ξ| → ∞, the parameter T > 0 is the (absolute) temperature, and µ ∈ R is the chemical potential. We always assume that µ is fixed and T ∈ (0, T 0 ] for some T 0 > 0. We are interested in the behaviour of D α (T ) = D α (a T,µ , Λ; f ) when α → ∞ and T ↓ 0 simultaneously. The symbol a T,µ fits in the formalism of multi-scale symbols, laid out in Section 3, and as a result we derive from Theorem 3.5 a sharp estimate for the trace norm of D α (T ) with explicit dependence on T and α, under the condition αT ≥ 1, see Theorem 8.3. For d = 1 the sharpness of this estimate is confirmed by the asymptotic formulas (8.24), (8.27) for the trace of D α (T ), that are derived from Theorem 4.7. The extension of this large-scale asymptotics to dimensions d ≥ 2 is the content of a separate paper [26]. In Section 10 we specialise further to the function f = η γ , γ > 0, which brings us to the main application of our results, that is, to the large scale asymptotic formulas for the entanglement entropy (EE) H γ (T, µ; αΛ) of free fermions in thermal equilibrium associated with the bipartition R d = Λ ∪ Λ c with a bounded Λ ⊂ R d , at temperature T > 0.
As pointed out earlier, by now the EE is well-understood at zero temperature (see [8,14]), which corresponds to the case when the Fermi symbol a is given by the indicator function χ Ω of the Fermi sea Ω ⊂ R d . In this case the EE exhibits a logarithmically enhanced area-law scaling of the form (1.4). The case T > 0 is somewhat trickier: the entropy of the total system on R d , that is, tr η γ (W α (a T,µ ; R d )) = tr Op α (a T,µ ), is infinite, and hence it is not clear in advance even how to define the EE in a meaningful way. Intuitively, the EE measures the difference between the sum of the entropies of the states localised to Λ and Λ c and the entropy of the total system. Therefore, a physically and mathematically reasonable definition of the EE is given in (10.4) below. By that we not only ensure the finiteness of the EE, but are also able to obtain sharp (in α and T ) upper bounds in any spatial dimension d ≥ 1. In Theorem 10.1 we show that |H γ (T, µ; αΛ)| ≤ Cα d−1 (| log(T )| + 1) 1 , if α ≥ 1, αT ≥ 1. This bound tallies well with the asymptotics (1.4), and thus supports the intuitive expectation that the scaling behaviour of the EE at T > 0 should resemble more and more the zero temperature behaviour, as T ↓ 0. For d = 1 this expectation is further justified by the asymptotic formulas (10.9) and (10.11), derived from (8.24), see Section 9 for the low T -behaviour of the asymptotic coefficient. As a by-product, this leads to the two-term asymptotic expansion of the local thermal entropy of the free Fermi gas, which extends the hitherto known leading Weyl asymptotics (see [12,2]).
The paper [15] presents results on the EE for the one-dimensional and the multidimensional case without the underlying mathematical details. In combination with [26] and [16] the present paper provides then a full proof of these announcements.

Estimates
2.1. The Schatten-von Neumann ideals of compact operators. This paper relies on the results obtained in [24] for general quasi-normed ideals of compact operators. Here we limit our attention to the case of Schatten-von Neumann operator ideals S q , q > 0. Detailed information on these ideals can be found e.g. in [3,8,18,20]. We shall point out only some basic facts. For a compact operator A on a separable Hilbert space H denote by s n (A), n = 1, 2, . . . its singular values, that is, the eigenvalues of the operator |A| := √ A * A. We denote the identity operator on H by 1. The Schatten-von Neumann ideal S q , q > 0 consists of all compact operators A, for which If q ≥ 1, then the above functional defines a norm; if 0 < q < 1, then it is a so-called quasi-norm. There is nevertheless a convenient analogue of the triangle inequality, which is called the q-triangle inequality: Sq , A 1 , A 2 ∈ S q , 0 < g ≤ 1, and the Hölder inequality, [19] and also [3]. In what follows we focus on the case q ∈ (0, 1].

2.2.
Non-smooth functions. We study non-smooth functions, satisfying the following condition: with some γ > 0, and is supported on the interval (t 0 − R, t 0 + R) with some R > 0. The case R = ∞ means no restriction on the support of the function f .
Below we denote by χ R the indicator function of the interval (−R, R), R > 0. For a function f satisfying the above condition the following bound holds for t = t 0 : If n ≥ 1, then the above condition implies that with κ := min{1, γ} the function f is κ-Hölder continuous -we denote this set by C 0,κ (R). In particular, one can show that for any t 1 , t 2 ∈ R, The following Proposition was proved in [24]. For simplicity we state it only for bounded self-adjoint operators.
Proposition 2.2. Suppose that f satisfies Condition 2.1 with some γ > 0, n ≥ 2 and some t 0 ∈ R, R ∈ (0, ∞). Let q be a number such that (n − σ) −1 < q ≤ 1 with some number σ ∈ (0, 1], σ < γ. Let A be a bounded self-adjoint operator and let P be an orthogonal projection such that P A(1 − P ) ∈ S σq . Then , with a positive constant C independent of the operators A, P , the function f , and the parameters R, t 0 .
Since the operator A is bounded, one does not have to assume that f is compactly supported. The function f can be always replaced by another function suitably localised to a bounded interval of size 2 A around the origin. This observation allows us to obtain a bound of the correct degree of homogeneity. We state this fact as a corollary of Proposition 2.2.
, with a positive constant C independent of the operators A, P , the function f and the parameter λ.
As far as the λ-behaviour is concerned, the above estimate is sharp, since for f (t) = |t| γ , γ > 0, both sides have the same homogeneity in λ. We include such estimates where an operator (or later, a symbol) is scaled by λ in this paper for completeness although the main application will appear only in [16].
We point out one special case of the non-homogeneous function η defined as which nevertheless leads to a homogeneous estimate: Corollary 2.4. Let q ∈ (0, 1], and let A be a bounded self-adjoint operator and let P be an orthogonal projection such that A ≤ 1 and P A(1 − P ) ∈ S σq for some σ ∈ (0, 1). Then for any λ > 0, (2.9) η(λP AP ) − P η(λA)P Sq ≤ C σ λ P A(1 − P ) σ Sσq , with a positive constant C σ independent of the operators A, P and the parameter λ.
The function η satisfies (2.3) with an arbitrary γ ∈ (σ, 1), and arbitrarily large n, on any bounded interval centred at t 0 = 0. Now Proposition 2.2 leads to the claimed estimate.

Estimates for multidimensional Wiener-Hopf operators
3.1. Definitions. Now we derive from Proposition 2.2 some estimates for Wiener-Hopf operators on L 2 (R d ). In this paper, under Wiener-Hopf operators we understand operators of the form (1.1), with a set Λ ⊂ R d and symbol a = a(ξ). Throughout the paper we assume that a ∈ L ∞ (R d ) so that the operator Op α (a) is bounded with Op α (a) = a L ∞ . Later we will assume that a satisfies some smoothness conditions. Our focus is on the operator difference (1.2) with suitable functions f . The right-hand side of (1.2) is well defined for a large class of functions f . We are mostly interested in functions f satisfying Condition 2.1. Our immediate objective is to obtain for the operator (1.2) estimates in the Schatten-von Neumann classes S q , q ∈ (0, 1]. These will be derived from appropriate S q -bounds for the operator Bounds of this type were proved in [22]. To state them properly we need to specify precise conditions on the set Λ and the symbol a. We call a domain (an open, connected set) Lipschitz, if it can be described locally as a set above the graph of a Lipschitz function, see [22] for details. We call Λ a Lipschitz region if Λ is a union of finitely many Lipschitz domains such that their closures are pair-wise disjoint. (1) If d = 1, then Λ is a finite union of open intervals (bounded or unbounded) such that their closures are pair-wise disjoint. (2) If d ≥ 2, then Λ is a Lipschitz region, and either Λ or R d \ Λ is bounded.
We rely on the bounds for the operator (3.1) obtained in [22]. They were derived for symbols a satisfying the following support condition: with some constant τ > 0 and µ ∈ R d . For methodological purposes we also introduce a smooth function ϕ which is often assumed to satisfy this condition: 3) support of the function ϕ is contained in B(z, ℓ), with some constant ℓ > 0 and z ∈ R d .
The bounds from [22] also allow one to control the scaling properties through the norms (3.4) N (m) (a; τ ) := max 0≤r≤m sup ξ∈R d τ r |∇ r ξ a(ξ)|, m = 1, 2, . . . , and similarlily defined norms N (n) (ϕ; ℓ). Now we can quote the result from [22]. The constants in the estimates below are independent of the symbol a, the function ϕ and the parameters α, τ, ℓ, as well as the points µ, z. If ατ ℓ ≥ α 0 > 0, then for any q ∈ (0, 1] In the next subsection we extend Proposition 3.2 to more general symbols a.

3.2.
Multi-scale symbols, a. We consider C ∞ -symbols a = a(ξ) for which there exist positive continuous functions v = v(ξ) and τ = τ (ξ) and constants C k , k = 0, 1, 2, . . . such that It is natural to call τ the scale (function) and v the amplitude (function). We refer to symbols a satisfying (3.6) as multi-scale symbols. In fact, in what follows, only some finite smoothness of the symbol a is sufficient, but in most cases we impose the C ∞smoothness in order to avoid cumbersome formulations. It is convenient to introduce the notation Apart from the continuity we often need some extra conditions on the scale and the amplitude. First we assume that τ is globally Lipschitz, that is, with some ν > 0. By adjusting the constants C k in (3.6) we may assume that ν < 1. It is straightforward to check that Under this assumption on the scale τ , the amplitude v is assumed to satisfy the bounds (3.10) with some positive constants C 1 , C 2 independent of ξ and η. The condition ν < 1 guarantees that one can construct a covering of R d by open balls centred at some points ξ j , j = 1, 2, . . . of radius τ j := τ (ξ j ), which satisfies the finite intersection property, that is, the number of intersecting balls is bounded from above by a constant depending only on the parameter ν, see [10, Chapter 1, Theorem 1.4.10]. We denote B j := B(ξ j , τ j ). Moreover, there exists a partition of unity φ j ∈ C ∞ 0 (R d ) subordinate to the above covering such that It is useful to think of v and τ as (functional) parameters. They, in turn, can depend on other parameters, e.g. numerical parameters like α. In our leading example of the Fermi symbol (1.5), the function τ is naturally chosen to be dependent on the temperature T > 0, see (8.20).

Remark 3.3.
Our aim is to derive various trace-norm estimates (resp. asymptotics) with explicit or implicit constants that are independent of the functions τ , v, a, but may depend on the constants in (3.6) and the domain Λ. If the functions τ , v are required to satisfy (3.8) and (3.10), then the constants in the trace-norm estimates (resp. asymptotics) may also depend on the constants ν and C 1 , C 2 in (3.10). In all these cases we say that the estimates (resp. asymptotics) are uniform in τ, v and a.
In the example of the symbol (1.5), the above uniformity allows us to control explicitly the dependence of the obtained bounds on the temperature.
In what follows we always assume that The constants in the obtained estimates will be independent of the parameters α, τ inf , ℓ, satisfying the assumption with some α 0 > 0, but may depend on α 0 .
Lemma 3.4. Suppose that the domain Λ satisfies Condition 3.1, and let the functions τ and v be as described above. Let n be as defined in (3.5). Suppose that the symbol a satisfies (3.6), the function ϕ satisfies (3.3), and that (3.14) holds. Then for any q ∈ (0, 1] we have Suppose that (3.13) is satisfied. Then The bound is uniform in τ, v and a in the sense specified in Remark 3.3.
Proof. Without loss of generality assume that N (n) (ϕ; ℓ) = 1. Let m be as defined in (3.5). Denote v j := v(ξ j ), τ j := τ (ξ j ) and B j := B(ξ j , τ j ), j = 1, 2, . . . . Due to (3.6) and (3.9), (3.10), the localised symbol a j = aφ j is supported in the ball B(ξ j , τ j ), and the bound holds: (3.4). Since αℓτ j ≥ α 0 , by Proposition 3.2, we have for any q ∈ (0, 1] that In view of (3.9) and (3.10), and hence the sum on the right-hand side of (3.17) is bounded by At the last step we used the finite intersection property of the covering {B j }. This leads to (3.15). The bound (3.16) immediately follows from (3.15) upon using a finite covering of Λ or R d \ Λ by unit balls and an associated smooth partition of unity.

Lemma 3.4 leads to the following result.
Theorem 3.5. Suppose that f satisfies Condition 2.1 with some n ≥ 2 and γ > 0, and the domain Λ satisfies Condition 3.1. Let a be a real-valued symbol. Let the functions a and τ , v be as in Lemma 3.4, and let (3.13) be satisfied. Then for any σ ∈ (0, 1], σ < γ, and q ∈ ((n − σ) −1 , 1] we have , with a constant independent of t 0 . Furthermore, if t 0 = 0 and a L ∞ ≤ 1, then for any λ > 0, The above bounds are uniform in τ, v and a in the sense specified in Remark 3.3. Furthermore, the constants in (3.18) and (3.19) are independent of α, R, λ, but may depend on α 0 in (3.13).
We also state separately the estimate for the function (2.8): Theorem 3.6. Let the function η be as defined in (2.8). Suppose that the real-valued symbol a is as in Lemma 3.4 with a L ∞ ≤ 1 and that (3.13) is satisfied. Then for any λ > 0 and any q ∈ (0, 1], σ ∈ (0, 1) one has . The bound is uniform in τ, v and a in the sense specified in Remark 3.3. Furthermore, the constant in (3.20) is independent of α, but may depend on α 0 in (3.13).
The proof is similar to that of (3.19), but instead of (2.7) one uses (2.9).

4.
Asymptotic results for the one-dimensional case 4.1. Results for smooth functions. Now we focus on the asymptotic behaviour of the trace of D α (a, Λ; f ) as α → ∞ for dimension d = 1. In line with the general theme of the paper we put the emphasis on non-smooth functions f . Our starting point, however, is the asymptotic formula for smooth f . This type of asymptotics was studied in [29] and later in [17], and we use one result from [29] without proof. Conditions on the smoothness and decay of the symbol a imposed in [29] are quite mild, but we assume stronger restrictions that enable us to utilize the bounds derived in Section 3. More precisely, we impose the following condition. To state the result we first define asymptotic coefficients. For any function g : C → C and any s 1 , s 2 ∈ C denote This integral is finite for functions g ∈ C 0,κ (C), κ ∈ (0, 1]. Note also that (4.2) U(s 1 , s 1 ; g) = 0, U(s 1 , s 2 ; g) = U(s 2 , s 1 ; g), ∀s 1 , s 2 ∈ C.
Note also that the integral equals zero if g(t) = 1 or g(t) = t. Now we define the asymptotic coefficient Note that B is invariant under the change a(ξ) → a(τ ξ) with an arbitrary τ > 0. If g is such that g ′′ ∈ L ∞ (C), then the principal value integral can be replaced by the double integral, and the following bound holds: This estimate was first pointed out in [29, (17)]. As shown in [25], under Condition 4.1, one has (1) Let g be analytic on a neighbourhood of the closed convex hull of the range of the function a. Then the operator D 1 (a; R ± ; g) is trace class and (4.5) tr D 1 (a; R ± ; g) = B(a; g).
(2) If the symbol a is real-valued, then formula (4.5) holds under the condition g ∈ C 4 0 (R). Formula (4.5) was obtained in [29] under weaker conditions on the symbol a. Moreover, for real-valued symbols a the smoothness conditions on g in [29] are less restrictive than in the above proposition. Note also that for real-valued a the paper [17] allows further relaxation on the functions a and g but we omit the details.
By rescaling x → αx one immediately concludes that the left-hand side of (4.5) coincides with tr D α (a; R ± ; g). It is worth pointing out that, formally speaking, the estimates in Section 3 do not ensure that the trace on the left-hand side of (4.5) is finite, since neither R ± itself nor its complement is bounded. However, those estimates in combination with Proposition 6.1 below do guarantee that D 1 (a; R ± ; g) is trace-class.
We apply Proposition 4.2 to the case of a real-valued symbol a and the function g : R → C defined as Now our immediate objective is to derive from (4.5) a similar asymptotic formula for the operator D α (a; Λ; g) with a set Λ satisfying Condition 3.1(1). For d = 1, instead of Λ we use the notation I. According to Condition 3.1(1), with some x 0 ∈ R, and its closure is also disjoint from the other intervals. Below we use the following notation for the number of endpoints of I, namely By writing K = K(I) and ω = ω(I) we emphasize the dependence on the set I. We observe that For arbitrary symbols a, b we introduce the notation Theorem 4.3. Let I and ω be as described in (4.6) and (4.7). Assume that Suppose that a ∈ C m (R), m ≥ 3, is real-valued. Then for any α > 0 we have with a constant C m independent of a, z, δ(z, a), and α, and the intervals I k , k = 0, 1, . . . , K + 1.
Clearly, by scaling we may replace the 1 on the right-hand side in condition (4.11) by any (strictly) positive real number.
In the case of one bounded interval, the convergence of the left-hand side of (4.12) to zero as α → ∞ was proved in [29,Theorem 2], see also [17,Theorem 9]. Note that for an infinitely smooth a the right-hand side of (4.12) decays as α −∞ , α → ∞. For one bounded interval, this effect was pointed out in [4, formula (1.5)]. These conclusions of [4,17,29] are not sufficient for us, as our aim is to have a more explicit control of the remainder as a function of the symbol a as in Theorem 4.3. In particular, when considering symbols a = a T,µ defined in (1.5), Theorem 4.3 allows us to obtain estimates that depend explicitly on the temperature T , and possibly the chemical potential µ. The proof of Theorem 4.3 draws on the ideas of [29], and it is postponed until Section 6.
We extend the above bound to arbitrary functions of finite smoothness satisfying some decay conditions. Precisely, for g ∈ C n (R), n ∈ N 0 and a constant r > 0 we define Theorem 4.4. Let I and ω be as in the previous theorem. Suppose that a ∈ C m (R), m ≥ 3, is real-valued and satisfies the bound (3.6) with some continuous positive functions τ, v. Suppose further that f ∈ C n 0 (R) with n ≥ m + 6. Then for any r ≥ v L ∞ and any α > 0 we have This bound is uniform in τ, v and a in the sense specified in Remark 3.3. The constant C m,n in (4.14) is independent of the parameters α, r and the function f .

4.2.
Results for non-smooth functions. Now we assume that f satisfies Condition 2.1 with some γ > 0. In this case, if γ > 0 is small, it is not immediately clear why and under which conditions on the symbol a the coefficient B(a; f ) is finite. This issue was investigated in [25]. We quote the appropriate bound, adjusted for the use in the forthcoming calculations. We use the integral V σ,ρ (v, τ ) defined in (3.7) and the notation κ := min{1, γ}.
with a constant C σ independent of f , uniformly in the functions τ, v, and the symbol a in the sense specified in Remark 3.3.
We note another useful result from [25]. It describes the contribution of "close" points ξ 1 and ξ 2 in the coefficient (4.3). Suppose that τ inf := inf ξ∈R τ (ξ) > 0, then we define This quantity is estimated in the following proposition.
Proposition 4.6. Suppose that f satisfies Condition 2.1 with n = 2 and γ > 0. Let a ∈ C ∞ (R) satisfy Condition 4.1. Suppose also that τ inf > 0. Then for any δ ∈ [0, κ), the following bound holds: uniformly in the functions τ, v, and the symbol a in the sense specified in Remark 3.3.
The bound (4.15) plays a central role in the proof of the following theorems. From now on we assume that τ inf > 0 and that τ inf and α satisfy (3.13). The convergence in the next theorems is uniform in the functions τ, v, and the symbol a in the sense specified in Remark 3.3, but no uniformity is claimed in the parameter α 0 in (3.13).
Theorem 4.7. Let I and ω be as described in (4.6) and (4.7). Suppose that f satisfies Condition 2.1 with some γ > 0, n = 2 and some t 0 ∈ R. Let a ∈ C ∞ (R) be a realvalued symbol satisfying Condition 4.1, and let ατ inf ≥ α 0 . Suppose that v L ∞ ≤ 1 and V σ,1 (v, τ ) < ∞ for some σ ∈ (0, 1], σ < γ, and and the convergence is uniform in v, τ and a. In order to avoid possible confusion we recall that v, τ are thought of as functional parameters of the problem, and they may depend on the numerical parameter α. Thus the equality (4.18) is a genuine, non-vacuous assumption.
For the next theorem recall that the function η is defined in (2.8).
We point out that the smoothness conditions on the function f in Theorem 4.8 are much more restrictive than those in Theorem 4.7. This difference will be briefly explained after the proof of Theorem 4.8.
The main difficulty lies in the proof of Theorem 4.3, whereas the remaining theorems are derived from it via relatively standard methods. In the next section we concentrate on this derivation. The proof of Theorem 4.3 is deferred until Section 6.
for any r ≥ v L ∞ . Let us now estimate M (m) (a z , a −1 z ). Lemma 5.1. Suppose that a ∈ C m (R) satisfies (3.6) with some m ≥ 1. Then Moreover, for any r ≥ v L ∞ , and all y with |y| < x r , we have with a constant C m independent of r.
Proof. By definition (4.9), In view of the bound (3.6) the first summand in the above formula is bounded by To estimate the second term on the right-hand side of (5.4) we use the Leibniz formula and (3.6) to obtain Thus the second summand in (5.4) does not exceed This leads to the claimed bound (5.2). For r ≥ v L ∞ and |y| < x r , the bound (5.3) immediately follows from (5.2).
Proof of Theorem 4.4. By (5.3), the integral on the right-hand side of (5.1) is estimated by Since n ≥ m + 6, this integral is finite and it is bounded by where we have used the definition (4.13). Therefore (5.1) yields the bound Step 1: proof of formula (4.19) for f ∈ C 2 (R). Without loss of generality we may assume that a L ∞ ≤ 1/2 and that the function f is supported on the interval [−1, 1]. By the Weierstrass Theorem, for any ε > 0 one can find a real polynomial f ε such that the function g ε := f − f ε satisfies the bound Clearly, In order to estimate D α (g ε ) we extend the function g ε to the interval [−2, 2] as a C 2 0function in such a way that g ε C 2 ≤ Cε with some universal constant C > 0. Observe now that such g ε satisfies Condition 2.1 with t 0 = −3, R = 5, n = 2 and arbitrary γ > 0. Furthermore, g ε 2 < Cε. To be definite we take γ = 2. Since the condition (3.13) is satisfied, we may use Theorem 3.5 with q = 1 and arbitrary σ ∈ (0, 1), so that Moreover, by (4.4), In order to handle the trace of D α (f ε ), extend the polynomial f ε as a C ∞ 0 -function on the interval [−2, 2]. Thus by Theorem 4.
with arbitrary m ≥ 3. In view of the condition (4.18) and by virtue of (5.6) and (5.7), we have Here the lim sup is taken as α → ∞, ατ inf ≥ α 0 , and it is uniform in v, τ and a. Since ε > 0 is arbitrary, this leads to (4.19) for arbitrary C 2 -functions f .
Proof of Theorem 4.8. Instead of f we introduce for λ > 0 the function It is clear that f n = f (λ) n for all n. As in the previous proof, without loss of generality we may assume that a L ∞ ≤ 1/2, so that the function f (λ) may be assumed to be supported on the interval [−1, 1]. Note that Thus the asymptotic formula (4.20) is equivalent to the following relation: tr D α (a, I; f (λ) ) − ωB(a; f (λ) ) = 0.
The further proof now follows essentially Step 2 of the proof of Theorem 4.7. As before, for brevity we use the notation D α (f ) := D α (a, I; f ), B(f ) := B(a; f ).
Since the function η satisfies Condition 2.1 for any γ ∈ (σ, 1), the proclaimed asymptotic formula is a direct consequence of the formula (4.19) for the operator D α (a, I; η).
Observe that the proof of Theorem 4.8 has only one step, in contrast to that of Theorem 4.7. Namely, in the former we do not prove that the sought asymptotics holds for arbitrary f ∈ C 2 (R) since this would require approximating f (λ) with polynomials whose dependence on λ would have to be explicitly controlled. We do not go into these difficulties.

Proof of Theorem 4.3: the case of a single interval
We recall the notation (1.1) for the Wiener-Hopf operator: W α (a; I) = χ I Op α (a) χ I with the notation Λ replaced by a subset I ⊂ R. A central role in our argument plays the operator with C m -symbols a = a(ξ) and b = b(ξ). At the first step of the proof we assume that the set I is just a bounded interval (x 0 , y 0 ) with y 0 − x 0 ≥ 1.

Preliminary bounds. For any z ∈ R denote
x 0 ). Most of the estimates for the introduced operators will follow from the next proposition, which is a consequence of [22, Theorem 2.7]. Proposition 6.1. Let a be a symbol such that ∂ m ξ a ∈ L 1 (R) with some m ≥ 3. Let z, t be numbers such that z − t = ℓ > 0. Then for any α > 0, we have Proof. The operator χ , for any m ≥ 3. This is the required estimate.
Remark. Theorem 2.7 in [22] contains two misprints: the number n should be defined by the formula n := ⌈2q −1 ⌉ + 1, and the main estimate of the Theorem should have the factor r 2q −1 −m instead of r q −1 −m . Now we proceed to estimating trace norms of the operators H α , Z α introduced above.
Recall that M (m) (a, b) is defined in (4.9). Lemma 6.2. Let I = (x 0 , y 0 ) with y 0 − x 0 ≥ 1. Then for m ≥ 3 and any α > 0 we have Proof. We denote A := Op α (a), B := Op α (b). Clearly, the operator Z := Z α (a, b; I) splits into the sum Then, by Proposition 6.1, a, b). Adding it up with the same bound for the operator Z (2) completes the proof of (6.2) for Z(a, b; I).
Proof of (6.3): Let z 0 := (x 0 + y 0 )/2, so that the trace norm of the operator x 0 can be estimated by x 0 S 1 . Now Proposition 6.1 leads to the bound (6.3) for the first term on the left-hand side of (6.3). In the same way one proves the same bound for the second term on the left-hand side. Lemma 6.3. Let the conditions of Lemma 6.2 be satisfied, and let g ∈ C m (R) be another symbol. Then x 0 and estimate For the second term on the right-hand side of (6.5) let z 0 := (x 0 + y 0 )/2. Then x 0 S 1 . Now Proposition 6.1 leads to inequality (6.4) for the first term on the left-hand side of (6.4). The remaining inequality is derived in the same way. 6.2. Estimates for D α (a, I; r z ): one-dimensional case. We apply definition (6.1) to the symbols a z := a − z and a −1 z . Now we assume that a is a real-valued symbol and δ(z, a) > 0, see (4.10) for definition. Thus we obtain (replace Λ by I) Clearly, both operators W α (a z ; I) and W α (a −1 z ; I) are invertible on L 2 (I) and W α (a z ; I)| −1 As a consequence, Let us analyse the part of the right-hand side which contains H a .
Lemma 6.5. For z ∈ C let g be the function defined as g(λ) := r z (λ) := (λ − z) −1 for λ ∈ R. Also, with the symbol a as above, let a z := a − z. Then for any α > 0, Proof. We use the notation H α , H α , H α from Lemma 6.4, and . By Lemma 6.3 and the bound (6.6), for k = 1, 2. Together with Lemma 6.4 this gives Now formula (6.7) leads to the proclaimed estimate.
Proof of Theorem 4.3 for the case I = (x 0 , y 0 ). Lemma 6.5 shows that for the function r z defined as r z (λ) := (λ − z) −1 the trace of D α (a, I; r z ) coincides with the sum x 0 ; r z ) up to the remainder specified in the lemma. As we have observed earlier, due to the translation and reflection invariance, each of the intervals R (+) y 0 in the above trace sum can be replaced by (0, ∞). When calculating the traces in (6.8), by making the change of variables x → αx we can take α = 1. Now Theorem 4.3 follows from Proposition 4.2(1).

Proof of Theorem 4.3: the case of multiple intervals
In this section we consider general sets I of the form (4.6), and assume that (4.11) is satisfied. Throughout this section we assume that a ∈ C m (R) with some m ≥ 3 and that a is real-valued. The parameter α is allowed to take any positive value and the constants in all estimates obtained are independent of the function a or the parameters z with δ(z, a) > 0 and α. Our strategy is to reduce the case of general I's either to the case of one bounded interval, considered in the previous section, or to the case of the half-line, covered by Proposition 4.2. More precisely, our objective is to prove the following result: The proof consists of several steps: Lemma 7.2. Under the above conditions and with H α (a, b; I) defined in (6.1), Proof of Theorem 7.1. As in the previous proof we use the notation H α , H . In view of (6.7), The first term on the right-hand side satisfies (7.1) by (7.2) and (6.6), (7.4). The second term satisfies (7.1) by Lemma 7.3 and due to the bound W Proof of Theorem 4.3. By (7.1), it remains to use the results for individual operators D α (a, I k ; r z ). For k = 1, 2, . . . , K, that is, when I k is a bounded interval, we use the bound (4.12) proved previously. If k = 0 or k = K + 1, that is, when I k is a half-line, we use the identity (4.5). This leads to the bound (4.12), and completes the proof of Theorem 4.3.

8.
Estimates for D α (a, Λ; f ) with Fermi symbol a = a T,µ : multi-dimensional case As explained in the Introduction, the asymptotic analysis in this paper was partly motivated by the study of the entanglement entropy of free fermions. Thus in this section we apply the results obtained above to the special choice of the symbol a featuring in definition (1.2). We choose the symbol a to be the Fermi symbol a T,µ defined in (1.5). The choice of the (non-smooth) function f remains arbitrary for the time-being. Further on, in Section 10, we specialise to the Rényi entropy function f = η γ , γ > 0.
The physical context of the various quantities is as follows. We assume that the energy of a single particle in position space R d consists only of kinetic energy in the absense of external forces and is determined by a Hamiltonian h = h(ξ) and that, for simplicity, particles do not have a spin-degree of freedom. The free Fermi gas is then a collection of infinitely many such particles obeying the (Pauli-)Fermi-Dirac statistics. An equilibrium state of this free Fermi gas is uniquely determined by specifying the temperature T > 0, the chemical potential µ ∈ R, and the Fermi symbol (1.5). We will assume that µ is fixed and T ∈ (0, T 0 ], and, in particular, T is allowed to become small, that is, T ↓ 0. Our aim is to find estimates with explicit dependence on T and α.
In what follows it will be convenient to use the following notation. For any two non-negative functions x and y depending on all or some of the variables/parameters α, T, ξ, we write x ≍ y if there exist two constants C, c independent of α, T, ξ such that cy ≤ x ≤ Cy.
Let us record some useful inequalities for the symbol a = a T,µ from (1.5).
with constants C n depending on T 0 , µ, and the constants in (8.2).
The proof is elementary and thus omitted. A straightforward calculation leads to the bounds Our objective is to obtain the following estimate.
Suppose that the function f satisfies Condition 2.1 with some n ≥ 2 and γ > 0, Suppose also that the region Λ and the function h satisfy Conditions 3.1 and 8.1 respectively. Let αT ≥ α 0 > 0, 0 < T ≤ T 0 for some α 0 and T 0 . Then for any σ ∈ (0, γ), σ ≤ 1, we have with a constant C independent of T, R, t 0 , α, and the function f , but depending on α 0 , T 0 , µ.
Until the end of this section we always assume that the region Λ and the function h satisfy Conditions 3.1 and 8.1 respectively.
Because of (8.1) the set Ω is bounded, so that Ω ⊂ B(0, R 0 ) with some R 0 > 0. Assume first that d ≥ 2. Due to condition (8.3), the set S is locally a C ∞ -surface, which is called the Fermi surface. More precisely, for any ξ 0 ∈ S there is a radius r > 0 such that |∂ ξ d h(ξ)| ≥ c for all ξ ∈ B(ξ 0 , 2r) with a suitable choice of coordinates ξ = (ξ, ξ d ), and hence there exists a function Ψ ∈ C ∞ (R d−1 ) such that For definiteness we assume that B(ξ 0 , 2r) ⊂ B(0, R 0 ). We may also assume that This can be achieved by replacing ξ d and Ψ(ξ) with −ξ d and −Ψ(ξ) and by taking a smaller r, if necessary. Without loss of generality we may assume that ∇Ψ L ∞ ≤ M with some constant M > 0. By choosing a sufficiently small r > 0, due to the condition |∂ ξ d h| ≥ c one can also guarantee that On the other hand, |ξ − η| ≥ (1 + M 2 ) −1/2 |ξ d − Ψ(ξ)|, for any ξ ∈ B(ξ 0 , 2r) and any η ∈ S ∩ B(ξ 0 , 2r). Consequently, Since the set Ω is in fact a C ∞ -region, we can cover its boundary S with finitely many open balls {D j (r)} of radius r centred at some ξ j ∈ S, such that in each D j (2r) one can find an appropriate function Ψ = Ψ j that satisfies the properties (8.7)-(8.9) after an appropriate choice of coordinates in every ball D j (2r). From now on for brevity we denote D j = D j (r). LetD ⊂ R d be a region such thatD ∩ S = ∅, and If d = 1, then we modify the definitions of {D j } andD in an obvious way. For example, each D j is now an interval such that with an appropriate choice of the coordinate ξ the open set D j ∩ Ω is simply D j ∩ {ξ ∈ R : ξ > 0}. Thus the covering (8.11) holds for d = 1 as well.
The idea of the proof of Theorem 8.3 is to observe that the symbol (1.5) satisfies (3.6) on each element of the covering (8.11) with some functions τ and v defined individually on each of the domains D j andD. After that Theorem 3.5 produces Theorem 8.3.
Let us first describe the construction of the scaling function τ and amplitude function v on D j andD. We do this for the case d ≥ 2, as for d = 1 only obvious modifications are required.
Let Ψ = Ψ (j) ∈ C ∞ (R d−1 ) be a function describing the surface S inside D j , see (8.7). Recall that we always assume that ∇Ψ L ∞ ≤ C. We introduce the functions ℓ (j) and w (j) defined on R d as Due to (8.4), (8.5) and (8.9), the constant c 1 can be chosen to guarantee that Since D j ⊂ B(0, R 0 ), we get from Lemma 8.2 that for ξ ∈ D j |∇ n a(ξ)| ≤ C n T −n a(ξ)(1 − a(ξ)) ≤C n T −n w (j) (ξ), C n = C n (R 0 ).
The case of a homogeneous function h deserves special attention since in this case one can explicitly control the dependence on the chemical potential µ. We illustrate this with the example of the function h(ξ) = |ξ| 2 . The parameter µ can be "scaled out" with the help of the following formula: , Λ; f ). Thus Theorem 8.3 leads to the following result.
The final result in this section is specific to dimension one.
The above formulas hold for arbitrary T satisfying the condition αT ≥ α 0 . If we assume additionally that T ↓ 0, then the asymptotics (8.24) can be written in a more explicit form, thanks to the asymptotic formula for B(a T,µ ; f ), T ↓ 0, obtained in Theorem 9.1, which, incidentally, confirms the sharpness of the estimate (8.25). Recall that according to Condition 8.1, for d = 1 the set Ω is represented as Proof. The claimed asymptotics follows immediately from Theorems 8.6 and 9.1.

9.
Asymptotics of B(a T,µ ; f ) as T ↓ 0 Here we study the behaviour of B(a T,µ ; f ) with the Fermi symbol a T,µ defined in (1.5) as T ↓ 0. The number N below is as in the representation (8.26). Let τ and v be as defined in (8.20), so that τ inf = θT . We study separately the integral B (1) defined in (4.16) and Using (4.17) and (8.22), we conclude that for all T ∈ (0, T 0 ], To study B (2) we intend to replace a with the indicator function χ Ω . To this end we note the following properties of the function f , and as a result, of the integral (4.1). The bound (2.5) says that the function f is Hölder: γ}.
Assume in addition that with some d 0 ∈ (0, d 1 ]. Let ϕ ∈ C(R) ∩ L ∞ (R) be a function. Then, as T ↓ 0, where O(1) depends only on d 0 and d 1 .
Proof. Proof of (9.9). Without loss of generality assume that ϕ L ∞ = 1. It is immediate to see that so that (9.9) reduces to showing that Hence it suffices to prove (9.9) for one integral only, that is, that for a bounded interval J = (s 0 , t 0 ) with |s 0 − t 0 | ≥ d 0 . Without loss of generality assume that J = (0, 1). Split the sought integral into the sum X 1 + X 2 + X 3 + X 4 , with Direct calculations show that X 3 + X 4 ≤ C uniformly in T ∈ (0, T 0 ]. The integral X 1 differs from ds (t − s) 2 at most by a constant independent of T . An elementary calculation shows that . Thus X 1 satisfies the same formula. In the same way one proves the appropriate formula for X 2 . This leads to (9.10), and hence to (9.9).
The bound (9.8) is proved in a similar way by estimating integrals of the same type as in the first part of the proof. We omit the details.

Entanglement entropy and local entropy
In this section we keep using the Fermi symbol a = a T,µ as in (1.5) and investigate the special case of the function f given by the γ-Rényi entropy function η γ : R → [0, log (2)] defined for all γ > 0 as follows. If γ = 1, then for t ∈ (0, 1), and for γ = 1 (the von Neumann case) it is defined as the limit for t ∈ (0, 1).
From now one we assume that the region Λ and the Hamiltonian h satisfy Conditions 3.1 and 8.1 respectively. The operator D α ( · ) is as defined in (1.2) and the notation Ω is used for the Fermi sea, see Condition 8.1. If Λ is bounded, then the local (thermal) γ-Rényi entropy of the equilibrium state at temperature T > 0 and chemical potential µ ∈ R is defined as S γ (T, µ; Λ) := tr η γ (W 1 (a T,µ ; Λ)) , (10.3) see for example [11]. If one lifts the condition of boundedness, then the above quantity may be infinite, but the γ-Rényi entanglement entropy (EE) with respect to the bipartition R d = Λ ∪ (R d \ Λ), defined as is finite, as the next theorem shows. Note that these definitions also make sense for T = 0, if one adopts the notation a 0,µ := lim T ↓0 a T,µ = χ Ω . A somewhat surprising fact is that for bounded Λ, (10.5) H γ (0, µ; Λ) = 2 S γ (0, µ; Λ).
As explained in [14], this is a consequence of the following two identities: tr η γ (χ Λ P Ω χ Λ ) = tr η γ (P Ω χ Λ P Ω ), where P Ω = Op 1 (χ Ω ), and η γ (P Ω χ Λ P Ω ) = η γ (P Ω χ Λ c P Ω ). The first identity holds since the non-zero spectra of χ Λ P Ω χ Λ and P Ω χ Λ P Ω coincide. The second one follows from the symmetry of η γ , that is, from the equality η γ (t) = η γ (1−t), t ∈ [0, 1]. We are interested in the behaviour of the above quantities when Λ is replaced with αΛ, with a large scaling parameter α. While the case T = 0 was investigated in detail in [14], in the current paper we concentrate on the case T > 0 and the limit T ↓ 0. The next theorem shows that the entropies (10.3) and (10.4) are both finite, and establishes sharp bounds when α and T both vary within certain limits. where s γ (T, µ) := 1 (2π) d η γ (a T,µ (ξ))dξ. The constants in (10.6) and (10.7) are independent of α and T , but may depend on the parameters α 0 , T 0 , µ, the function h and the region Λ.
For d = 1, apart from the bounds, we can also determine the asymptotic behaviour of the local (or thermal) entropy and of the EE.
A proof of the leading large-scale behaviour of the local entropy S γ (T, µ; αΛ) at fixed T > 0 appeared (among other things) first in [12,2] (for γ = 1). The sub-leading correction in dimension d = 1 in (10.10) is new. The extension to dimension d ≥ 2 is subject of [26].
If in Theorem 10.2 we also assume that T ↓ 0, then the asymptotic formulas take a more explicit form. To state this result recall that due to Condition 8.1, the Fermi sea has the form (8.26), with a finite N ∈ N. It is worth pointing out that the coefficient in front of | log(T )| agrees with the asymptotic coefficient found in [14] for the zero temperature case. Indeed, with the notation that we presently use, the main Theorem of [14] states that for a bounded I, that is, with ω = 2K, we have (see (10.5)) H γ (0, µ; αI) = 2 S γ (0, µ; αI) = ωN 1 + γ 6γ log(α) + o(log(α)), α → ∞.
Clearly, the coefficient in this formula is the same as in Corollary 10.3. Therefore, if we identify α inside the logarithm with 1/T we recover the above asymptotic expansion in Corollary 10.3.
Together with the bound (10.14) for the second trace, this yields (10.7).
The asymptotics in (10.10) is obtained in the same way using (10.15) and (1.3).
Proof of Corollary 10.3. The claimed formula immediately follows from (10.9) and the asymptotic relation (9.1) after observing that (cf. [14]) One should note that in the same way one could replace the coefficient B(a T,µ , η γ ) by its asymptotics (9.1) in formula (10.10) as well. However, the specific entropy density s γ (T, µ) in the leading term would also need to be expanded in T ↓ 0, and the precise place of the B( · )-term in the resulting expansion of S γ will depend on the relationship between αT and T . We do not go into these details.
Appendix A. The Helffer-Sjöstrand formula When studying functions of self-adjoint operators we rely on the Helffer-Sjöstrand formula which holds for arbitrary operators A = A * and arbitrary smooth functions f ∈ C n 0 (R), n ≥ 2 (z := x + iy, ∂ ∂z := 1 2 ( ∂ ∂x + i ∂ ∂y )): wheref =f (x, y) is an almost analytic extension of the function f , see [5,Chapter 2]. An almost analytic extension of f ∈ C n (R) is a C 1 (R 2 )-functionf , such that f (x) =f (x, 0) and ∂ ∂zf (x, y) ≤ C|y|. For the sake of brevity we use the representation (A.1) for compactly supported functions only, so that the integral (A.1) is norm-convergent.
Let us describe a convenient almost analytic extension of a function f ∈ C n 0 (R). For an arbitrary r > 0 introduce the function U r (x, y) := 1, |y| < x r , 0, |y| ≥ x r , x r := √ x 2 + r 2 .
Lemma A.1. Let f ∈ C n (R), n ≥ 2. Then for any r > 0 the function f has an almost analytic extensionf =f ( · , · ; r) ∈ C 1 (R 2 ) such thatf (x, y; r) = 0 if |y| > x r . Moreover, the derivative, ∂ ∂zf (x, y; r), satisfies the bound The constant C n does not depend on f or the constant r.
The proof of this lemma is a marginal modification of the proof contained in [5, Chapter 2] and is thus omitted.