Bounds on density of states and spectral gap in CFT$_{2}$

We improve the recently discovered bounds on the $O(1)$ correction to the Cardy formula for the density of states in 2 dimensional conformal field theory at high energy. We prove a conjectured upper bound on the asymptotic gap between two consecutive Virasoro primaries for a central charge greater than $1,$ demonstrating it to be $1.$ Furthermore, a systematic method is provided to establish a limit on how tight the bound on the $O(1)$ correction to the Cardy formula can be made using bandlimited functions. The techniques and the functions used here are of generic importance whenever the Tauberian theorems are used to estimate some physical quantities.

S δ associated with a particular energy window of width δ around a peak value ∆, which is allowed to go to infinity, and found where ρ(∆) is the density of states, given by a sum of Dirac delta functions peaked at the positions of the operator dimensions. It is shown in [18] that for O(1) energy width, the O(1) correction s(δ, ∆) is bounded from above and below: The purpose of the current note is to improve the bound and provide a systematic way to estimate how tight the bounds can be made using bandlimited functions. We also prove the conjectured upper bound on the asymptotic gap between Virasoro primaries, which turns out to be 1. This gap is optimal since for the Monster CFT, the gap is precisely 1.
Our results can be summarized by figure [1], where the green line and dots denote the lower (upper) bound on the upper (lower) bound. The orange lines denote the improved achievable bounds. The brown dots stand for the lower (upper) bound on the upper (lower) bound obtained from implementing the positive definiteness condition on the Fourier transform of ±(φ ± − Θ) via Matlab. The bound on bounds represented by the green line is thus weaker than that represented by the brown dots. In short, the brown shaded region is not achievable by any bandlimited function.
In particular, we show that the upper bound on s(δ, ∆) is given by  Figure 1: Exp[s ± ] as a function of δ, the half-width of the energy window. The blue line is the bound obtained in [18]. The orange line denotes the improved bound that we report here. The green line is the analytical lower (upper) bound on the upper (lower) bound, while the brown dots stand for the lower (upper) bound on the upper (lower) bound obtained from enforcing the positive definiteness condition on the Fourier transform of ±(φ ± − Θ) via Matlab. The bound on bounds represented by the green line is thus weaker than that represented by the brown dots. The brown shaded region is not achievable by any bandlimited function.
The lower bound s − (δ, ∆) is given by (1.6) The eq. (1.5) is an improvement of the lower bound for δ > 0.94, as evident from figure [2]. The rest of the paper details the derivation of the above. In section 2, we derive the improvement on the bound on the O(1) correction to the Cardy formula. Section 3 describes a systematic way to estimate how tight the bound can be made. We derive the optimal gap on the asymptotic spectra in section 4 and conclude with a brief discussion in section 5.

Derivation of the improvement
The basic ingredients for estimating the asymptotic growth of the density of states are two functions φ ± such that the following holds: (2.1) We refer the readers to section 4 of [18] for details of the procedure leading to a bound when ∆ goes to infinity. The basic result can be summarized as: where ρ 0 (∆) reproduces the contribution from the vacuum at high temperature and is given by The above is in fact the leading result for the density of states at high energy. Furthermore, c ± is defined as The eq. (2.2) holds if the Fourier transform of φ ± has a support on an interval which lies entirely within [−2π, 2π]. With this constraint in mind, we consider the following functions: In order to ensure that the indicator function on the interval [∆ − δ, ∆ + δ] is bounded above by φ + , we need to have The number in the eq. (2.7) is obtained by requiring that φ + (∆ ± δ) > 1. The functions φ ± have Fourier transforms with bounded supports [−Λ ± , Λ ± ] , respectively. Thus, in order for this support to lie within [−2π, 2π], we also require that Λ ± < 2π. The bound is then obtained by minimizing (or maximizing) for a given δ by varying Λ ± subject to the constraint given by the eq. (2.7), as well as Λ ± < 2π. From the eq. (2.2), one can conclude [18] that Since for a fixed δ, c + is a monotonically decreasing function of Λ + , we deduce that c + should be minimized by 15 π ∼ 0.49 , δ > 3 √ 15 11 π ∼ 1.12. (2.11) .
The lower bound can be further improved for δ > 1 by considering the following function whose Fourier transform has a support over This yields c − = 0.5, which is an improvement over the above; see figure 3.
Serendipity -connection to the sphere packing problem: The function in the eq. (2.12) also appears in the context of one dimensional sphere packing problem [19]. In fact, there is an uncanny similarity between the functions required in the two problems, especially if we look at the requirements on the function producing the lower bound 3 . In the sphere packing problem, one has a Fourier transform pair f,f satisfying In our case, we have x ↔ ∆ and k ↔ t and we require thatf (k) has bounded support. In both scenarios, the goal is to maximizef (0). In the case of sphere packing, we also normalize f (0) to one. For more details on the relevance of sphere packing to CFT, we refer the reader to the recent article [20].
It turns out that only in one dimension [19], where the sphere packing problem is trivial, the relevant function as given in the eq. (2.12) has bounded support in the Fourier domain and is positive 4 . This seems to suggest that if we want to further improve our bound, we need a bandlimited function whose Fourier transform becomes negative within the band.
Before moving on to the discussion of the bound on bounds, we pause to remark that the following class of functions parameterized by α can not be used to improve the bound from above: Within this class of functions, α = 4 gives the tightest bound as found in [18].

Bound on bounds
In this section, we provide a systematic algorithm to estimate how tight the bounds can be made using bandlimited functions φ ± . This provides us with a quantitative estimate of the limitation of the procedure which produces these bounds on the O(1) correction to the Cardy formula. If one drops the requirement that the function be bandlimited, one might hope to do better. For the rest of this section, we will restrict ourselves to bandlimited functions only.
We recall that the functions φ ± are chosen in such a way that they satisfy This inequality gives a trivial bound on c ± : In what follows, we make this inequality tighter. In this context, the following characterization of the Fourier transform of a positive function in terms of a positive definite function turns out to be extremely useful. Before delving into the proof, let us define the notion of positive definiteness of a function. Unless otherwise specified, here we will be dealing with functions from the real line to the complex plane. A function f (t) is said to be positive definite if for every positive integer n and for every set of distinct points t 1 , . . . , t n chosen from the real line, the n × n matrix A defined by is positive definite. A function g(∆) is said to be positive if g(∆) > 0 for every ∆. One can show that the Fourier transform of a positive function is positive definite 5 . Now, let us explore how this characterization can improve the eq. (3.2). Without loss of generality, we set ∆ = 0 henceforth, and define At this point we use the fact that φ ± is a bandlimited function, i.e., it has a bounded support [−Λ ± , Λ ± ] , and that Λ ± < 2π. This requirement stems from the procedure followed in [18]. Thus we arrive at the following: The eq. (3.2) states thatg(0)/2δ > 0. In order to improve this, we construct 2 × 2 matrices with t 2 > 2π: For a fixed δ, we consider the first positive peak ofg ± outside t > 2π. If this occurs at t = t(δ), we choose t 2 = t(δ). Subsequently, the positive definiteness of the matrix G ± boils down to the inequality where t(δ) is the first positive peak of g ± outside t > 2π. For example, we can show that (see the green lines in Fig. 4):

Positive function ⇔ Positive definite Function: Fourier transform
We will show that the Fourier transform of an even and positive function is a positive definite function. Consider a function g(∆) and let us define the Fourier transform as (3.11) Now, we construct the matrix In order to show that G is a positive definite matrix, i.e., ij v i v j G ij > 0 for v i ∈ R such that i v 2 i = 0, we think of an auxiliary 2 dimensional space with n vectors v (i) , (for clarity, we remark that i labels the vector itself, not its component) such that we have v (i) ≡ (|v i | cos(∆t i ), |v i | sin(∆t i )) . (3.13) if t 1 , . . . , t n are distinct. Here, V is given by This completes the proof that the Fourier transform of an even positive function is a positive definite function. First of all, it is easy to see that c ± , and hence the inequality, is insensitive to the midpoint of the interval, i.e., ∆, so we set it to 0 and this makes the functions φ ± and Θ even. In particular, we will be applying this theorem to φ . We make one more remark before exploring the consequences of this. The above result is true for any function, not necessarily even. The converse is also true due to Bochner's Theorem, but in what follows, we do not require the converse statement.

Matlab implementation
We implement the above argument using more than two points and making sure that |t i − t j | ≥ 2π. For a fixed δ, we use a random number generator to sample the points t i with the mentioned constraint. We do this multiple times and each time, we test the positive definiteness of the matrix G by providing as an input the value of ±(c ± − 1). The range of ±(c ± − 1) is chosen to be from the first peak t(δ) till some value larger than the achievable bound given in (1.3) and (2.11). This in turn yields a lower bound (or upper bound) for c ± for each trial 6 . Subsequently, we pick out the best possible bound among all the trials. For example, we provide a table [1] showing the outputs from a typical run for improving the bound on the upper bound. The tables [1] and [2] improve the lower (upper) bound for c ± and this is shown in the figure [1], where the brown dots are the stronger bounds over the green lines and disallow a larger region.

Bound on spectral gap: towards optimality
In this section, we switch gear and explore the asymptotic spectral gap. In [18], it has recently been shown that the asymptotic gap between Virasoro primaries are bounded above by 2 3 π 2 1.1 and it has been conjectured that the optimal gap should be 1. The example of Monster CFT tells us that the gap can not be below than 1, hence 1 should be the optimal number. In this section, we show that the previous bound 2 3 π 2 can be improved and made arbitrarily closer to the optimal value 1. Ideally, to prove this one should find out a function f (which will eventually play the role of φ − in this game, to be precise f (∆ ) = φ − (∆ + ∆ )) such that following holds:  Now what would happen iff (0) = 0 ? One need to go back to the original derivation and reconsider it carefully. Hence instead of the eq. (2.2), we consider a more basic inequality [18]: where Λ − = 2π and Z H (β) is the contribution from the heavy states and defined as Now we make the following choice for φ − : This function f has following properties: Since c − = 0, one can not readily evaluate the integral appearing in (4.5) by saddle point method and deduce exp [β(∆ − δ)] d∆ ρ 0 (∆ )e −β∆ φ − (∆ ) = c − ρ 0 (∆), so we look for subleading corrections to the saddle point approximation. We find that the leading behavior is given by, after setting β = π c 3∆ , where C turns out to be (4.12) We remark that C > 0 for any finite ∆ and it becomes 0 only at infinitely large ∆. The second piece in the eq. (4.5) for large ∆ goes as ρ 0 (∆) 1− 1 2 (1− 1 2 ) . The analysis for this second term is exactly same as done in [18]. For sufficiently large ∆, it can be numerically verified that ρ 0 (∆) 1− 1 2 (1− 1 2 ) is subleading compared to Cρ 0 (∆) as long as > 1 (we also provide an analytical proof later on). Here we have In fact one can analytically show that ρ 0 (∆) 1− 1 2 (1− 1 2 ) is subleading to Cρ 0 (∆) for large ∆. One way to show this is to have an estimate for C. We start with the observation that the integrand is positive in 0, 2 and negative in 2 , ∞ . Furthermore, we have ∞ 0 d∆ f (∆ ) = 0 (4.14) Using the above facts, one can always choose 0 < 1 < 2 and 2 < 2 < ∞ such that This is basically guaranteed by the continuity. We choose 1 such that 0 < 1 < 2 and consider the function F (y) = y 1 dx f (x). Now F (y) is a continuous function. It is positive when y = 2 and negative when y → ∞. Thus by continuity, there exists 2 < 2 < ∞ such that the eq. (4.15) holds. The shaded region in the figure. 5 is the area under the function f restricted to the interval [ 1 , 2 ] so that the eq. (4.15) is satisfied. as long as > 1, we can neglect the second piece i.e. contributions from the heavy states due to its subleading nature. In fact, one can do much better and show that 7 C falls like ∆ −3/4 by noting the following: To summarize, we have proved that for sufficiently large ∆,

Function for Spectral Gap
Therefore we have been able to show that the asymptotic gap between two consecutive operators is bounded above by , where > 1. Now one can choose to be arbitrarily close to 1, which proves that the optimal bound is exactly 1. The analysis can be carried over to the case for Virasoro primaries, as pointed out in [18]. This implies that the asymptotic gap between two consecutive Virasoro primaries is bounded above by 1, thereby proves the conjecture made in [18].

Brief discussion
In this work, we have improved the existing bound on the O(1) correction to the density of states in 2D CFT at high energy and proven the conjectured upper bound on the gap between Virasoro primaries. In particular, we have shown that there always exists a Virasoro primary in the energy window of width greater than 1 at large ∆.
We have provided a systematic way to estimate how tight the bound can be made using bandlimited functions. Since there is still a gap between the achievable bound and the bound on the bound, there is scope for further improvement. Ideally, one would like to close this gap, which might be possible either by sampling more points and leveraging the positive definiteness condition on a bigger matrix, or by choosing some suitable function which would make the achievable bound closer to the bound on the bound. Another possible way to obtain the bound on bound is to use a known 2D CFT partition functions, for example 2D Ising model and explicitly evaluate s(δ, ∆). It would be interesting to see how the bound on bound obtained in this paper compares to the one which can be obtained from the 2D Ising model. For example, one can verify that the bound on bound obtained here is stronger than that could be obtained from 2D Ising model 8 for δ = 1. It would be interesting to further explore this.
The utility of the technique developed here lies beyond the O(1) correction to the Cardy formula. We expect the technique to be useful whenever one wants to leverage the complex Tauberian theorems. As emphasized in [18], the importance of Tauberian theorems lies beyond the discussion of 2D CFT partition functions, especially in investigating Eigenstate Thermalization Hypothesis [21][22][23][24] in 2D CFTs [25][26][27][28][29][30][31][32][33][34][35]. We end with a cautious remark that if we relax the condition of using bandlimited functions, the bound on bounds would not be applicable and it might be possible to obtain nicer achievable bounds on the O(1) correction to the Cardy formula.