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.
I. INTRODUCTION AND RESULT
A classical result of Szegő describes the asymptotic growth of the determinant of a truncated Toeplitz matrix as the truncation parameter tends to infinity.24,25 Jump discontinuities in the spectral function (or symbol) of the Toeplitz matrix are of crucial importance for the growth of the subleading term in such an asymptotic expansion.1,7
Recent years have witnessed considerable interest in Szegő-type asymptotics for spectral projections of Schrödinger operators.2,3,6,12,14–17,19–21,27 The first mathematical proof of such an asymptotics was established by Leschke, Sobolev, and Spitzer12 and relies on extensive work of Sobolev,22,23 making rigorous a long-standing conjecture of Widom.26 The authors of Ref. 12 consider the simplest and most prominent Schrödinger operator, the self-adjoint (negative) Laplacian H0 ≔ −Δ. It acts on a dense domain in the Hilbert space of complex-valued square-integrable functions over d-dimensional Euclidean space . More precisely, given a (Fermi) energy E > 0, they consider the spectral projection 1<E(H0) of H0 associated with the interval . Besides physical motivation, this spectral projection provides a prototypical example for a symbol with a single discontinuity. The spectral projection 1<E(H0) gives rise to a Wiener–Hopf operator by truncation with the multiplication operator from the left and from the right. The truncation corresponds to multiplication with the indicator function of the spatial subset , L > 0, which is the scaled version of some “nice” bounded subset . We state the details for Λ in Assumption 1.1(i). The final ingredient is a “test function” , which is piecewise continuous and vanishes at zero, h(0) = 0. Furthermore, it is required that h grows at most algebraically near the endpoints of the interval, that is, there exists a “Hölder exponent” α > 0 such that
Here, we employed the Bachmann–Landau notation for asymptotic equalities: We will use the big-O and little-o symbols throughout this paper. Under these hypotheses, the second-order asymptotic formula for the trace
as L → ∞ is proved in Ref. 12. Here, |Λ| denotes the (Lebesgue) volume of Λ, and |∂Λ| denotes the surface area of the boundary ∂Λ of Λ. The leading-order coefficient in (1.2) is determined by the integrated density of states,
of H0. Here, Γ denotes Euler’s gamma function. The coefficient of the subleading term factorizes into a product of
We follow the usual terminology and refer to (1.2) as a (second-order) Szegő-type asymptotics. The occurrence of the logarithmic factor ln L multiplying the surface area Ld−1 in the subleading term of (1.2) is attributed to the discontinuity of the symbol 1<E together with the dynamical delocalization of the Schrödinger time evolution generated by the Laplacian.15 In the context of entanglement entropies for non-interacting Fermi gases [see (1.8)], the occurrence of this additional ln L-factor is also coined an enhanced area law.12
A natural question concerns the fate of the asymptotics (1.2) when H0 = −Δ is replaced by general self-adjoint Schrödinger operators H ≔ −Δ + V with a (suitable) electric potential V. Unfortunately, there exists no general approach, which allows us to derive a two-term asymptotics like (1.2) for H. Mathematical proofs are restricted to special examples or classes of examples. The exact determination of the coefficient in the subleading term beyond bounds poses a particularly challenging task.
First, we describe two situations in which—in contrast to (1.2)—the subleading term does not exhibit a logarithmic enhancement. Such a behavior is generally referred to as an area law and caused an enormous attraction in the physics literature over several decades until now (see, e.g., Refs. 5 and 10 and references therein). An area law is typically expected if H has a mobility gap in its spectrum and if the Fermi energy falls inside the mobility gap. It was proved for discrete random Schrödinger operators, a Fermi energy lying in the region of complete localization, and for test functions obeying some smoothness assumption, including the ones for entanglement entropies in (1.7).6,17,18 An area law also shows up in the Szegő asymptotics for rather general test functions when H equals the Landau Hamiltonian in two dimensions with a perpendicular constant magnetic field and if the Fermi energy E coincides with one of the Landau levels14 (see also Ref. 2). Very recently, Ref. 19 established the stability of this area law under suitable magnetic and electric perturbations and for the test functions (1.7) corresponding to entanglement entropies.
Now, we return to situations where the subleading term exhibits a logarithmic enhancement as in (1.2). One-dimensional Schrödinger operators H = −Δ + V with an arbitrarily often differentiable periodic potential V were studied in Ref. 21. If the Fermi energy lies in the interior of a Bloch band, Pfirsch and Sobolev21 established an enhanced area law for the same class of test functions as considered in (1.2). Surprisingly, the coefficient of the subleading term of the Szegő asymptotics remains the same as for H0 = −Δ in (1.2). However, as the integrated density of states of the periodic Schrödinger operator H differs from that of H0, in general, this affects the leading-order term of the asymptotics. In contrast to the area law in the two-dimensional Landau model,14 a logarithmic enhancement occurs when this model is considered in three space dimensions due to the free motion in the direction parallel to the magnetic field.20 Another situation was studied by the present authors in Ref. 16. There, H0 is perturbed by a compactly supported and bounded potential. The enhanced area law for H0 is then proven to persist for the test function h = h1 from (1.7), which corresponds to the von Neumann entropy. Yet, the bounds in Ref. 16 are not good enough as to allow for a conclusion concerning the coefficient.
The purpose of this paper is to show that the second-order Szegő asymptotics (1.2) remains valid with the same coefficients if H0 is replaced by H = −Δ + V with a compactly supported and bounded potential . This improves the result in Ref. 16 in two ways: (i) the statement is strengthened as to cover also universality of the coefficients and (ii) it is extended from the von Neumann entropy to a fairly large class of test functions. The general approach we follow here is different from that in Ref. 16. It combines the traditional way11 for proving (1.2) (see also, e.g., Refs. 12 and 21) with improved estimates from Ref. 16.
Our assumptions on the spatial domain coincide with those in Ref. 16.
We consider a bounded Borel set such that
it is a Lipschitz domain with, if d ⩾ 2, a piecewise C1-boundary and
the origin is an interior point of Λ.
We specify the set of test functions to which our main result applies.
(i) The restriction to symmetric test functions in is technical. It relates to the incommodious fact that the trace of a sequence of operators may converge, whereas convergence in trace norm need not hold. We refer to Remark 2.5(ii) for more details.
(ii) The class of test functions for which (1.2) holds requires less regularity for h near the endpoints of the interval [0,1] as compared to . We expect that Theorem 1.4 extends to this more general class, i.e., to test functions h with arbitrarily small “Hölder exponents” α. Possibly, such an extension requires additional smoothness of the potential V.
(iii) It is only for the sake of simplicity that we confined ourselves to compactly supported and bounded potentials. The theorem should remain valid for potentials with sufficient integrability properties.
We conclude this section with an application of Theorem 1.4 to entanglement entropies for the ground state of a system of non-interacting fermions with single-particle Hamiltonian H. We introduce the one-parameter family of test functions hα: [0, 1] → [0, 1], given by
and refer to them as Rényi entropy functions. The particular case h1 is also called the von Neumann entropy function, for which we use the convention 0 log2 0 ≔ 0 for the binary logarithm. Following Ref. 8, the quantity
α > 0, defines the Rényi-entanglement entropy with respect to a spatial bipartition for the ground state of a quasi-free Fermi gas characterized by the single-particle Hamiltonian H and Fermi energy E. Here, is any bounded Borel set. It is obvious from the definition of the space of test functions that
Therefore, Theorem 1.4 has the following immediate:
Again, it would be desirable to remove the restriction α > d−1 in the corollary.
II. PROOF OF THEOREM 1.4
The following notion will be useful in the proof of Theorem 2.1.
Step (i). is a polynomial.
Steps (ii)–(iv) establish the “closure” of the asymptotics.
Step (ii). is of the form h = s1f with f ∈ C([0, 1]).
Step (iii). is continuous.
Step (iv). .
We follow the argument in Ref. 21 and assume—without restriction—that h is real-valued. Otherwise, we decompose h into its real part Reh and imaginary part Imh.
Since h is piecewise continuous and continuous at 0 and at 1, there exists δ > 0 such that h is continuous in [0, 2δ] ∪ [1 − 2δ, 1]. Let ɛ > 0. We choose continuous functions such that
h, h1, and h2 coincide on [0, δ] ∪ [1 − δ, 1],
h1 ⩽ h ⩽ h2, and
(ii) Likewise, we believe that the trace on the rhs of the first line of (2.19) remains bounded as L → ∞. On the other hand, the trace norm ‖(PL − PL,0)sn(PL,0)‖1 may grow logarithmically in L in some one-dimensional situations, which is correctly captured by the product of Hilbert–Schmidt norms in the second line of (2.19). Thus, it is the second inequality in (2.19), which is responsible for the additional symmetry constraint in , respectively, , as compared to higher dimensions.
The difference QL − QL,0 was previously estimated in Lemma 2.5 of Ref. 16 in suitable von Neumann–Schatten norms ‖·‖s. The next lemma improves that result by accessing smaller values of s and obtaining a weaker growth in L as compared to Lemma 2.5 of Ref. 16.
We summarize the results of this section in the following structural reformulation of Theorem 2.1, which is independent of the concrete forms of the unperturbed operator H0 and of the perturbed operator H:
This paper is dedicated to Abel Klein in honor of his numerous fundamental contributions to mathematical physics. As a mentor and friend of P. M., Abel has been a constant source of inspiration to him.
Conflict of Interest
The authors have no conflicts to disclose.
Peter Müller: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (lead). Ruth Schulte: Conceptualization (equal); Investigation (equal); Writing – original draft (equal); Writing – review & editing (supporting).
Data sharing is not applicable to this article as no new data were created or analyzed in this study.