Differences Between Robin and Neumann Eigenvalues

Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega {\subset } {\mathbb {R}}^2$$\end{document}Ω⊂R2 be a bounded planar domain, with piecewise smooth boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}∂Ω. For \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma >0$$\end{document}σ>0, we consider the Robin boundary value problem \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} -\Delta f =\lambda f, \qquad \frac{\partial f}{\partial n} + \sigma f = 0 \text{ on } \partial \Omega \end{aligned}$$\end{document}-Δf=λf,∂f∂n+σf=0on∂Ωwhere \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \frac{\partial f}{\partial n} $$\end{document}∂f∂n is the derivative in the direction of the outward pointing normal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\partial \Omega $$\end{document}∂Ω. Let \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\lambda ^\sigma _0\le \lambda ^\sigma _1\le \ldots $$\end{document}0<λ0σ≤λ1σ≤… be the corresponding eigenvalues. The purpose of this paper is to study the Robin–Neumann gaps \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} d_n(\sigma ):=\lambda _n^\sigma -\lambda _n^0 . \end{aligned}$$\end{document}dn(σ):=λnσ-λn0.For a wide class of planar domains we show that there is a limiting mean value, equal to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2{\text {length}}(\partial \Omega )/{\text {area}}(\Omega )\cdot \sigma $$\end{document}2length(∂Ω)/area(Ω)·σ and in the smooth case, give an upper bound of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$d_n(\sigma )\le C(\Omega ) n^{1/3}\sigma $$\end{document}dn(σ)≤C(Ω)n1/3σ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.

Abstract: Let ⊂R 2 be a bounded planar domain, with piecewise smooth boundary ∂ . For σ > 0, we consider the Robin boundary value problem where ∂ f ∂n is the derivative in the direction of the outward pointing normal to ∂ . Let 0 < λ σ 0 ≤ λ σ 1 ≤ . . . be the corresponding eigenvalues. The purpose of this paper is to study the Robin-Neumann gaps d n (σ ) := λ σ n − λ 0 n .
For a wide class of planar domains we show that there is a limiting mean value, equal to 2 length(∂ )/ area( ) · σ and in the smooth case, give an upper bound of d n (σ ) ≤ C( )n 1/3 σ and a uniform lower bound. For ergodic billiards we show that along a density-one subsequence, the gaps converge to the mean value. We obtain further properties for rectangles, where we have a uniform upper bound, and for disks, where we improve the general upper bound.

Statement of Results
Let ⊂R 2 be a bounded planar domain, with piecewise smooth boundary ∂ . For σ ≥ 0, we consider the Robin boundary value problem where ∂ f ∂n is the derivative in the direction of the outward pointing normal to ∂ . The case σ = 0 is the Neumann boundary condition, and we use σ = ∞ as a shorthand for the Dirichlet boundary condition f | ∂ = 0.
Robin boundary conditions are used in heat conductance theory to interpolate between a perfectly insulating boundary, described by Neumann boundary conditions σ = 0, and a temperature fixing boundary, described by Dirichlet boundary conditions corresponding to σ = +∞. To date, most studies concentrated on the first few Robin eigenvalues, with applications in shape optimization and related isoperimetric inequalities and asymptotics of the first eigenvalues (see [5]). Our goal is very different, aiming to study the difference between high-lying Robin and Neumann eigenvalues. There are very few studies addressing this in the literature, except for [4,32] which aim at different goals.
We will take the Robin condition for a fixed and positive σ > 0, when all eigenvalues are positive, one excuse being that a negative Robin parameter gives non-physical boundary conditions for the heat equation, with heat flowing from cold to hot; see however [17] for a model where negative σ is of interest, in particular σ → −∞ [10,15,21,22]. Let 0 < λ σ 0 ≤ λ σ 1 ≤ . . . be the corresponding eigenvalues. The Robin spectrum always lies between the Neumann and Dirichlet spectra (Dirichlet-Neumann bracketing) [5] : We define the Robin-Neumann difference (RN gaps) as d n (σ ) := λ σ n − λ 0 n and study several of their properties. See Sect. 2 for some numerical experiments. This seems to be a novel subject, and the only related study that we are aware of is the very recent work of Rivière and Royer [28], which addresses the RN gaps for quantum star graphs.

A lower bound.
Recall that a domain is "star-shaped with respect to a point x ∈ " if the segment between x and every other point of lies inside the domain; so convex means star-shaped with respect to any point; "star-shaped" just means that there is some x so that it is star-shaped with respect to x. Theorem 1.3. Let ⊂R 2 be a bounded star-shaped planar domain with smooth boundary. Then the Robin-Neumann differences are uniformly bounded below: For all σ > 0, ∃C = C( , σ ) > 0 so that Note that for quantum star graphs, this lower bound fails [28].

A general upper bound.
We give a quantitative upper bound: Theorem 1.4. Assume that has a smooth boundary. Then ∃C = C( ) > 0 so that for all σ > 0, While quite poor, it is the best individual bound that we have in general. Below, we will indicate how to improve it in special cases. Question 1.5. Are there planar domains where the differences d n (σ ) are unbounded?
We believe that this happens in several cases, e.g. the disk, but at present can only show this for the hemisphere [30], which is not a planar domain.

Ergodic billiards.
To a piecewise smooth planar domain one associates a billiard dynamics. When this dynamics is ergodic, as for the stadium billiard (see Fig. 2 If the billiard dynamics is uniformly hyperbolic, we expect that more is true, that all the gaps converge to the mean. A key ingredient in the proofs of the above results is that they can be connected to L 2 restriction estimates for eigenfunctions on the boundary via a variational formula for the gaps (Lemma 3.1) where u n,τ is any L 2 ( )-normalized eigenfunction associated with λ τ n .
We use Theorem 1.7 to draw a consequence for the level spacing distribution of the Robin eigenvalues on a rectangle: Let x 0 ≤ x 1 ≤ x 2 ≤ . . . be a sequence of levels, and δ n = x n+1 − x n be the nearest neighbour gaps. We assume that x N = N + o(N ) so that the average gap is unity. The level spacing distribution P(s) of the sequence is then defined as (assuming that the limit exists). It is well known that the level spacing distribution for the Neumann (or Dirichlet) eigenvalues on the square is a delta-function at the origin, due to large arithmetic multiplicities in the spectrum. Once we put a Robin boundary condition, we can show [31] that the multiplicities disappear for σ > 0 sufficiently small, except for systematic doubling due to symmetry. Nonetheless, even after desymmetrizing (removing the systematic multiplicities) we show that the level spacing does not change: Theorem 1.8. The level spacing distribution for the desymmetrized Robin spectrum on the square is a delta-function at the origin.
1.7. The disk. As we will explain, upper bounds for the gaps d n can be obtained from upper bounds for the remainder term in Weyl's law for the Robin/Neumann problem. While this method will usually fall short of Theorem 1.4, for the disk it gives a better bound. In that case, Kuznetsov and Fedosov [16] (see also Colin de Verdiére [8]) gave an improved remainder term in Weyl's law for Dirichlet boundary conditions, by relating the problem to counting (shifted) lattice points in a certain cusped domain. With some work, the argument can also be adapted to the Robin case (see Sect. 10.2 and "Appendix A"), which recovers Theorem 1.4 in this special case. The remainder term for the lattice count was improved by Guo, Wang and Wang [13], from which we obtain: Theorem 1.9. For the unit disk, for any fixed σ > 0, we have d n (σ ) = O(n 1/3−δ ), δ = 1/990.

Numerics
We present some numerical experiments on the fluctuation of the RN gaps. In all cases, we took the Robin constant to be σ = 1. Displayed are the run sequence plots of the RN gaps. The solid (green) curve is the cumulative mean. The solid (red) horizontal line is the limiting mean value 2 length(∂ )/ area( ) obtained in Theorem 1.1.
In Fig. 1 we present numerics for two domains where the Neumann and Dirichlet problems are solvable, by means of separation of variables, the square and the disk. These were generated using Mathematica [35]. For the square, we are reduced to finding Robin eigenvalues on an interval as (numerical) solutions to a secular equation, see Sect. 8, and have used Mathematica to find these.
The disk admits separation of variables, and as is well known the Dirichlet eigenvalues on the unit disk are the squares of the positive zeros of the Bessel functions J n (x). The positive Neumann eigenvalues are squares of the positive zeros of the derivatives J n (x), and the Robin eigenvalues are the squares of the positive zeros of x J n (x) + σ J n (x). We generated these using Mathematica, see Fig. 1B. For the remaining cases we used the finite elements package FreeFem [11,20]. In Fig. 2 we display two ergodic examples, the quarter-stadium billiard and a uniformly hyperbolic, Sinai-type dispersing billiard which was investigated numerically by Barnett [2].
It is also of interest to understand rational polygons, that is simple plane polygons all of whose vertex angles are rational multiples of π ( Fig. 3), when we expect an analogue of Theorem 1.6 to hold, compare [23].
The case of dynamics with a mixed phase space, such as the mushroom billiard investigated by Bunimovich [6] (see also the survey [26]) also deserves study, see Fig. 4.

Robin-Neumann bracketing and positivity of the RN gaps.
We recall the min-max characterization of the Robin eigenvalues where H 1 ( ) is the Sobolev space. This shows that λ σ n ≥ λ 0 n if σ > 0. Likewise, there is a min-max characterization of the Dirichlet eigenvalues with H 1 ( ) replaced by the This shows that λ σ n ≤ λ ∞ n . In fact, we have strict inequality, This is proved (in greater generality) in [29] using a unique continuation principle.

A variational formula for the gaps.
where u n,τ is any L 2 ( )-normalized eigenfunction associated with λ τ n .
Proof. According to [1, Lemma 2.11] (who attribute it as folklore), for any bounded Lipschitz domain ⊂R d , and n ≥ 1, the function σ → λ σ n is strictly increasing for σ ∈ [0, ∞), is differentiable almost everywhere in (0, ∞), is piecewise analytic, and the non-smooth points are locally finite (i.e. finite in each bounded interval). It is absolutely continuous, and in particular its derivative dλ σ n /dσ (which exists almost everywhere) is locally integrable, and for any 0 ≤ α < β, Moreover, there is a variational formula valid at any point where the derivative exists: where u n,σ is any normalized eigenfunction associated with λ σ n . Therefore We can ignore the finitely many points τ where (3.2) fails, as the derivative is integrable.

Indeed, for the case of smooth boundary, [3, Proposition 2.4] 2 give an upper bound on the boundary integrals of eigenfunctions
As a consequence of the variational formula (3.1), we deduce and in particular for planar domains, using Weyl's law, we obtain for n ≥ 1 d n (σ ) n 1/3 · σ. 1 Here and in the sequel we use A B as an alternative to A = O(B). 2 Their Proposition 2.4 is stated only for the Neumann case, but as is pointed out in Remark 2.7, the proof applies to Robin case as well, uniformly in σ ≥ 0; and they attribute it to Tataru [34,Theorem 3]. Note that their convention for the normal derivative is different than ours.

The Mean Value
In this section we give a proof of Theorem 1.1, that Using Lemma 3.1 gives The local Weyl law [14] (valid for any piecewise smooth ) shows that for any fixed σ , It remains to prove a uniform upper bound for W N (σ ).
Proof. What we use is an upper bound on the heat kernel on the boundary. Let K σ (x, y; t) be the heat kernel for the Robin problem. Then [14, Lemma 12.1], where C, δ > 0 depend only on the domain . Moreover, on the regular part of the boundary, Thus we find a uniform upper bound We note that the mean value result is valid in any dimension d ≥ 2 for piecewise smooth domains ⊂R d as in [14], in the form Indeed [14] prove the local Weyl law in that context, and Lemma 4.1 is also valid in any dimension.

A Uniform Lower Bound for the Gaps
To obtain the lower bound of Theorem 1.3 for the gaps, we use the variational formula (3.1) to relate the derivative dλ σ n /dσ to the boundary integrals ∂ u 2 n,σ ds, where u n,σ is any eigenfunction with eigenvalue λ σ n , and for that will require a lower bound on these boundary integrals.

A lower bound for the boundary integral.
The goal here is to prove a uniform lower bound for the boundary data of Robin eigenfunctions on a star-shaped, smooth planar domain .
Theorem 5.1. Let ⊂R 2 be a star-shaped bounded planar domain with smooth boundary. Let f be an L 2 ( ) normalized Robin eigenfunction associated with the n-th eigenvalue λ σ n . Then there are constants C > 0, A, B ≥ 0 depending on so that for all n ≥ 1, For σ = 0 (Neumann problem), this is related to the L 2 restriction bound of Barnett-Hassell-Tacy [3, Proposition 6.1].

The
Neumann case σ = 0. We first show the corresponding statement for Neumann eigenfunctions (which are Robin case with σ = 0), which is much simpler. Let f be a Neumann eigenfunction, that is ( We may assume that λ > 0, the result being obvious for λ = 0 when f is a constant function. After translation, we may assume that the domain is star-shaped with respect to the origin.
We start with a Rellich identity ([27, For every function f on Using (5.2) in dimension d = 2 for a normalized eigenfunction, so that L f = 0 and f 2 = 1, and recalling that for Neumann eigenfunctions ∂ f ∂n = 0 on ∂ , gives The term x ∂ x ∂n + y ∂ y ∂n is the inner product n(x) · x between the outward unit normal n(x) = ( ∂ x ∂n , ∂ y ∂n ) at the point x ∈ ∂ and the radius vector x = (x, y) joining x and the origin. Since the domain is star-shaped w.r.t. the origin, we have on the boundary ∂ so that we can drop 3 the term with ||∇ f || 2 and get an inequality Replacing (n(x) · x) ≤ 2C on ∂ gives Theorem 5.1 for σ = 0:

The Robin case.
Using the Rellich identity (5.2) in dimension d = 2 for a normalized eigenfunction, so that L f = 0 and Now n(x) · x ≥ 0 on the boundary ∂ since is star-shaped with respect to the origin, and λ > 0, so we may drop the term with ||∇ f || 2 and get an inequality Due to the boundary condition, we may replace the normal derivative ∂ f ∂n by −σ f , and obtain, after using 0 ≤ n(x) · x ≤ 2C = 2C (we may take 2C to be the diameter of ), that To proceed further, we need: Assume that ∂ is smooth. There are numbers P, Q ≥ 0, not both zero, depending only on ∂ , so that for any normalized σ -Robin eigenfunction f , Proof. Decompose the vector field A = x ∂ ∂ x + y ∂ ∂ y into its normal and tangential components along the boundary: where p, q are functions on the boundary . For example, for the circle x 2 + y 2 = ρ 2 , we have A = ρ ∂ ∂n and the normal derivative is just the radial derivative ∂ ∂n = ∂ ∂r , so that p ≡ ρ, and q ≡ 0.

Ergodic Billiards
In this section we give a proof of Theorem 1.6. By Chebyshev's inequality, it suffices to show: Proposition 6.1. Let ⊂R 2 be a bounded, piecewise smooth domain. Assume that the billiard map for is ergodic. Then for every σ > 0, Proof. We again use the variational formula (3.1) We have Hassell and Zelditch [14, eq 7.1] (see also Burq [7]) show that if the billiard map is ergodic then for each σ ≥ 0, Therefore, by Cauchy-Schwarz, S N (τ ) tends to zero for all τ ≥ 0, by (6.2); by Lemma 4.1 we know that S N (τ ) ≤ C is uniformly bounded for all τ ≤ σ , so that by the Dominated Convergence Theorem we deduce that the limit of the integrals tends to zero, hence that We note that Theorem 1.6 is valid in any dimension d ≥ 2 for piecewise smooth domains ⊂R d with ergodic billiard map as in [14], with the mean value interpreted as

Variable Robin Function
In this section, we indicate extensions of our general results to the case of variable boundary conditions. 7.1. Variable boundary conditions. The general Robin boundary condition is obtained by taking a function on the boundary σ : ∂ → R which we assume is always nonnegative: σ (x) ≥ 0 for all x ∈ ∂ . Thus we look for solutions of which is interpreted in weak form as saying that for all v ∈ H 1 ( ). We will assume that σ is continuous. Then we obtain positive Robin eigenvalues except that in the Neumann case σ ≡ 0 we also have zero as an eigenvalue. Robin to Neumann bracketing is still valid here, in the following form: if σ 1 , σ 2 ∈ C(∂ ) are two continuous functions with 0 ≤ σ 1 ≤ σ 2 and such that there is some point x 0 ∈ ∂ such that there is strict inequality σ 1 (x 0 ) < σ 2 (x 0 ) (by continuity this therefore holds on a neighborhood of x 0 ), then we have a strict inequality [29] λ σ 1 n < λ σ 2 n , ∀n ≥ 1. For instance, the universal lower bound for star-shaped domains (Theorem 1.3) follows because d n (σ ) ≥ d n (σ min ) ≥ C(σ min ) > 0, etcetera.
The existence of mean values (Theorem 1.1) and the almost sure convergence of the gaps to the mean value in the ergodic case (Theorem 1.6) require an adjustment of the variational formula (Lemma 3.1) which is provided in Sect. 7.3. Once that is in place, the result is Given the mean value formula (7.2), Theorem 1.6 (almost sure convergence of the RN gaps to the mean in the ergodic case) also follows.
Proof. The proof is verbatim that of [1, Lemma 2.11] where σ ≡ 1. As is explained there, each eigenvalue depends locally analytically on α, with at most a locally finite set of splitting points. We just repeat the computation of the derivative at any α which is not a splitting point for λ n (α): We use the weak formulation of the boundary condition, as saying that for all v ∈ H 1 ( ), In particular, applying (7.5) with v = u n,β gives ∇u n,α · ∇u n,β + ∂ ασ u n,α u n,β ds = λ n (α) u n,α u n,β .

The one-dimensional case.
Let σ > 0 be the Robin constant. The Robin problem on the unit interval is −u n = k 2 n u n , with the one-dimensional Robin boundary conditions The eigenvalues of the Laplacian on the unit interval are the numbers −k 2 n where the frequencies k n = k n (σ ) are the solutions of the secular equation (k 2 − σ 2 ) sin k = 2kσ cos k, or tan(k) = 2σ k k 2 − σ 2 (8.1) (see Fig. 5) and the corresponding eigenfunctions are u n (x) = k n cos(k n x) + σ sin(k n x). As a special case 5 of Dirichlet-Neumann bracketing (1.1), we know that given σ > 0, for each n ≥ 0 there is a unique solution k n = k n (σ ) of the secular equation (8.1) with k n ∈ (nπ, (n + 1)π ), n ≥ 0.
From (8.1), we have as n → ∞, We can interpret, for being the unit interval, 4 = 2#∂ / length so that we find convergence of the RN gaps to their mean value in this case. From (8.2) we deduce: We have From the one-dimensional result (8.3), we deduce that We now pass from the m,n (σ ) to the ordered eigenvalues {λ σ k : k = 0, 1, . . .}. We know that λ σ k ≥ λ 0 k , and want to show that λ σ k ≤ λ 0 k + C L (σ ). For this it suffices to show that the interval I k := [0, λ 0 k + C L (σ )] contains at least k + 1 Robin eigenvalues, since then it will contain λ σ 0 , . . . , λ σ k and hence we will find λ σ k ≤ λ 0 k + C L (σ ). The interval I k contains the interval [0, λ 0 k ] and so certainly contains the first k + 1 Neumann eigenvalues λ 0 0 , . . . , λ 0 k , which are of the form m,n (0) with (m, n) lying in a set S k . Since m,n (σ ) ≤ m,n (0) + C L (σ ), the interval I k must contain the k + 1 eigenvalues { m,n (σ ) : (m, n) ∈ S k }, and we are done.

Application of Boundedness of the RN Gaps to Level Spacings
In this section, we show that the level spacing distribution of the Robin eigenvalues for the desymmetrized square is a delta function at the origin, as is the case with Neumann or Dirichlet boundary conditions. Recall the definition of the level spacing distribution: We are given a sequence of levels Recall that the Robin spectrum has systematic double multiplicities m,n (σ ) = n,m (σ ) (see (8.4) with L = 1), which forces half the gaps to vanish for a trivial reason. To avoid this issue, one takes only the levels m,n (σ ) with m ≤ n, which we call the desymmetrized Robin spectrum. In other words, if we denote by λ σ 0 ≤ λ σ 1 ≤ . . . the ordered (desymmetrized) Robin eigenvalues, then the cumulant of the level spacing distribution satisfies: For all y > 0, Proof. The Neumann spectrum for the square consists of the numbers m 2 + n 2 (up to a multiple), with m, n ≥ 0. There is a systematic double multiplicity, manifested by the symmetry (m, n) → (n, m). We remove it by requiring m ≤ n. Denote the integers which are sums of two squares by We define index clusters N i as the set of all indices of desymmetrized Neumann eigenvalues which coincide with s i : For instance, s 0 = 0 = 0 2 + 0 2 has multiplicity one, and gives the index set N 1 = {1}; s 1 = 1 = 0 2 + 1 2 has multiplicity 1 (after desymmetrization) and gives N 2 = {2}; s 3 = 2 = 1 2 + 1 2 giving N 3 = {3}, . . . s 14 = 25 = 0 2 + 5 2 = 3 2 + 4 2 , N 14 = {14, 15}, etcetera. Then these are sets of consecutive integers which form a partition of the natural numbers {1, 2, 3, . . .}, and if i < j then the largest integer in N i is smaller than the smallest integer in N j . Denote by λ σ n the ordered desymmetrized Robin eigenvalues: λ σ 0 ≤ λ σ 1 ≤ . . ., so for σ = 0 these are just the integers s i repeated with multiplicity #N i . For each σ ≥ 0, we define clusters C i (σ ) as the set of all desymmetrized Robin eigenvalues λ σ n with n ∈ N i : Now use the boundedness of the RN gaps (Theorem 1.7): 0 ≤ λ σ n − λ 0 n ≤ C(σ ), to deduce that the clusters have bounded diameter: If #N i = 1 then diam C i (σ ) = 0, so we may assume that #N i ≥ 2 and write For the first N eigenvalues, the number I of clusters containing them is the number of the s i involved, which is at most the number of s i ≤ λ σ N ≈ N . A classical result of Landau [18] states that the number of integers ≤ N which are sums of two squares is about N / √ log N , in particular 6 is o(N ). Hence We count the number of nearest neighbour 7 gaps δ σ n = λ σ n+1 − λ σ n of size bigger than y. Of these, there are at most I such that λ σ n+1 and λ σ n belong to different clusters, and since I = o(N ) their contribution is negligible. For the remaining ones, we group them by cluster to which they belong: . 6 This is much easier to show using a sieve. 7 For simplicity we replace 1 2 area( ) 4π by 1, that is we don't bother normalizing so as to have mean gap unity; the result is independent of this normalization.
We have The sum of nearest neighbour gaps in each cluster is Thus we find Since I = o(N ), and C, y are fixed, we conclude that Thus the cumulant of the level spacing distribution satisfies: For all y > 0, so that P(s) is a delta function at the origin.
Note that the claim is not that all gaps λ σ n+1 − λ σ n tend to zero. On the contrary, it is possible to produce thin sequences {n} so that λ σ n+1 − λ σ n tend to infinity. Looking at the proof of Theorem 9.1, these correspond to the rare cases when λ σ n and λ σ n+1 belong to neighboring "clusters" which are far apart from each other.

Upper bounds for d n via Weyl's law.
In this section we prove Theorem 1.9. We first show how to obtain upper bounds for the gaps d n from upper bounds in Weyl's law for the Robin/Neumann problem. The result is that Lemma 10.1. Let be a bounded planar domain. Assume that there is some θ ∈ (0, 1/2) so that Proof. We first note that (10.1) gives (10.2) and likewise for the Robin counting function, as will be explained below. Now compare the counting functions N σ (λ σ n ) and N 0 (λ 0 n ) for the Robin and Neumann spectrum using (10.1) and (10.2): and Subtracting the two gives and therefore To show (10.2), denote λ = λ 0 n , and pick ε ∈ (0, 1) sufficiently small so that in the interval [λ − ε 2 , λ + ε 2 ] there are no eigenvalues other than λ, which is repeated with multiplicity K ≥ 1. Then On the other hand, by Weyl's law (with A = area( )/4π , B = length(∂ )/4π ) Now use |N (λ 0 n ) − n| ≤ K λ θ n θ which gives (10.2).
Below we implement this strategy for the disk to obtain Theorem 1.9.  [16] and Colin De Verdière [8].
where δ = 1/990. Noting that we obtain from Proposition 10.2 that Applying with eigenvalues κ 2 n,k , where κ n,k is the k-th positive zero of x J n (x) + σ J n (x). In particular, for the Neumann case (σ = 0), we get zeros of the derivative J n (x), denoted by j n,k ; since zero is a Neumann eigenvalue we use the standard convention that x = 0 is counted as the first zero of J 0 (x). Let and let F : S → R be the degree 1 homogeneous function satisfying F ≡ 1 on the graph of g. Obviously, on the other hand, as will be shown in Lemma 10.3 below, the numbers κ n,k are well approximated by F n, k + max (0, −n) − 3 4 . This will give the desired connection between Weyl's law on the disk and the lattice count problem in dilations of D. The proof of Lemma 10.3 will be given in "Appendix A". It will be handy to derive an explicit formula for the function F, which we will now do. Let ζ = ζ (z) be the solution to the differential equation which for z ≥ 1 is given by (see [24,Eq. 10.20.3]). The interval z ≥ 1 is bijectively mapped to the interval ζ ≤ 0; denote by z = z (ζ ) the inverse function. Additionally, for y ≥ 0 we have F (0, y) = π y, and for (−x, y) ∈ S we have Proof. Let x > 0, and denote t = F(x,y) x . Then F 1 t , y t x = 1 so that the point 1 t , y t x lies on the graph of g, and therefore The other claims are also straightforward from the definitions.
We proceed towards the proof of Proposition 10.2 by following the ideas of [8,Sec. 3]. Let where we used (10.9) and the relation κ −n,k = κ n,k . We first compare N 1 D (μ) and N 1 disk,σ μ 2 : Lemma 10.5. There exists a constant C = C c,σ > 0 such that Proof. Assume that |n| < c · k. By (10.9) and the homogeneity of F we have Hence, if F n, k + max(0, −n) − 3 4 ≥ μ, then k c μ. Combining this with Lemma 10.3, we see that The proof of the other inequality is similar.
We will now compare between N 2 D (μ) and N 2 disk,σ μ 2 . To this end, for fixed k ≥ 1, we denote Lemma 10.6. Given a sufficiently large c > 0, there exists a constant C = C c,σ > 0 such that
x , and therefore (note that when c is taken sufficiently large. In particular, ) is strictly increasing for x ≥ c · k, and so A k (μ) is bounded above by the number of integer points in the interval which in turn is bounded above by length(I ) + 1; by the mean value theorem, keeping in mind that (F −1 ) x =F −1 x , we conclude that The proof of the other inequality is similar.
For large values of k we will use the following estimate: Lemma 10.8. There exists a constant C = C c,σ > 0 such that for k > μ 4/7 , we have Proof. By Lemma 10.3, This, together with Lemma 10.5 gives the claim.
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