A PROBABILISTIC WEYL LAW FOR TWO-DIMENSIONAL FLAT TORI

YIANNIS PETRIDIS AND JOHN A. TOTH

Abstract. We prove that the error term R(λ) in Weyl’s law for two-dimensional flat tori satisfies Hardy’s conjecture (= O(λ^{1/4+ϵ})) for all tori with the exception of a set of arbitrarily small measure. The proof is spectral in nature.

1. Introduction

Let (M^{n}, g) be a compact Riemannian manifold of dimension n with Laplace-Beltrami operator Δ and spectral counting function

N(λ) := # {λ_{j} ∈ Spec(Δ); λ_{j} ≤ λ}.

Then, a celebrated theorem of Hörmander [HO] asserts that

N(λ) = vol(M) λ^{n/2} + O(λ^{(n−1)/2}).  (1.1)

Moreover, the estimate in (1.1) is sharp as can be seen by considering the round sphere, S^{n}. The question of determining the optimal bound for the error term

R(λ) := N(λ) − vol(M)λ^{n/2}

in any given example is a difficult problem which depends on the properties of the associated geodesic flow, and is far from being understood in detail. Nevertheless, there are several important results along these lines: One of the first is a result of Duistermaat-Guillemin [DG] which asserts that in the case where the geodesic flow is clean and the set of unit-speed geodesics in S^{∗}M has null Liouville measure, then one can improve the Hörmander bound in (1.1) to

R(λ) = o(λ^{(n−1)/2}).

Subsequently, Ivrii [I1], [I2] gave a different proof of this result and extended it to manifolds with boundary. There are some additional improvements in R(λ) that are known for some specific examples. For instance, in the case of hyperbolic manifolds, a result of Bérard [B] gives

R(λ) = O (λ^{(n−1)/2}/ log λ) .

This is in all likelihood far from optimal. Indeed, it is conjectured that, at least in the non-arithmetic case R(λ) = O(λ^{ϵ}) for all ϵ > 0. In fact, even for noncompact arithmetic surfaces with cusps, such as SL_{2}(ℤ) \ ℍ, one has

R(λ) = c · λ^{1/2} log λ + O(λ^{1/2}),

The first author would like to thank the Max-Planck-Institut für Mathematik for its financial support. The second author was supported in part by an Alfred P. Sloan Research Fellowship and NSERC grant OGP0170280.

see [He, 2.21, p. 511].

In the opposite case of completely integrable geodesic flow there are several cases where improved error terms are known: For (generic) convex surfaces of revolution, Colin de Verdière [CdV] has shown that

R(λ) = O(λ^{1/3})  (1.2)

which agrees with a result of Van der Corput and Sierpinski [SI] for the classical circle problem in 2 dimensions, i.e., for the torus ℝ^{2}/ℤ^{2}.

There are also additional more general results of Volovoy [V] under long-time recurrence estimates for the geodesic flow, but they are difficult to quantify. The geometrically simplest example of an integrable geodesic flow on a surface is the 2-dimensional flat torus. In this case, Hardy’s Careful Conjecture [HA] states that ‘it is not unlikely that

R(λ) = O_{ϵ}(λ^{1/4+ϵ})  (1.3)

for all positive ϵ’. There is much evidence, both numerical and otherwise to suggest that the bound above is indeed optimal. For instance, a classical result of Cramér [CR] says that

lim_{T →∞} \frac{1}{T^{3/2}} ∫_{0}^{T} |R(λ)|^{2} dλ = c > 0  (1.4)

for a constant c.

Despite the fact that there has been much work devoted to improving the estimate for R(λ), deterministic results are still far away from (1.3). The best result that we are aware of is O(λ^{23/73} log^{315/146} λ) and is due to Huxley [HU]. The problem for general flat tori in dimensions n ≥ 4 is solved, in fact R(λ) = O(λ^{n/2−1}) for n ≥ 5, while R(λ) = O(λ log λ) for n = 4. These estimates include all tori, rational and irrational. For n = 3 estimating the sharp error term is not known and the corresponding bound to (1.2) is R(λ) = O(λ^{5/6}).

The purpose of this note is to establish the Hardy bound (1.3) in a probabilistic sense, where we average over metric deformations of 2-dimensional tori. More precisely, we prove

Theorem 1.1. (i) Fix ϵ with 0 < ϵ < 1 and let I := [1−ϵ, 1+ϵ] and N(λ; u⃗) denote the spectral counting function for the corresponding Laplace-Beltrami operator on the flat two-torus M(u⃗), u⃗ = (u_{1}, u_{2}) with metric form

H(p, σ; u_{1}, u_{2}) = u_{1} σ^{2} + u_{2} p^{2}.

Then, for any δ > 0,

∫_{I^{2}} | N(λ; u⃗) − vol (M(u⃗)) λ | du⃗ = O_{δ}(λ^{1/4+δ}).  (1.5)

(ii) Consider the general metric Hamiltonian for a flat two-dimensional torus M(u⃗), u⃗ = (u_{1}, u_{2}, u_{3}) given by:

H(p, σ; u_{1}, u_{2}, u_{3}) := u_{1}σ^{2} + u_{3}pσ + u_{2}p^{2}.  (1.6)

Then the following holds:

∫_{I^{3}} | N(λ; u⃗) − vol (M(u⃗)) λ | du⃗ = O_{δ}(λ^{1/4+δ}).  (1.7)

An immediate application of Chebyshev’s inequality gives the following corollary.

Corollary 1. For all δ, ϵ_{1} > 0 small the estimate

N(λ; u⃗) − vol(M(u⃗))λ = O_{δ,ϵ_{1}}(λ^{1/4+δ})

holds for all tori M(u⃗) in the complement of a set A_{δ,ϵ_{1}} of measure less than ϵ_{1} in the moduli space of all tori.

Remark 1.2. (1) We should emphasize that our result in Theorem 1.1 is local. Consequently, it differs significantly from the global result of Randol [R]. Indeed, Randol shows that for G/Γ := SL(2, ℝ)/SL(2, ℤ),

∫_{G/Γ} |N(λ; g) − (ζ(2))^{−1}πλ |^{2} dg = ζ(2)^{−1}πλ + O_{N}(λ| log λ|^{−N}), (1.8)

for any integer N. Thus, when estimating over all lattices of fixed volume, one gets an asymptotic formula for R(λ) which is consistent with the Hörmander error in (1.1) and no better. This global result thus differs considerably from our Theorem 1.1. The point here is that, since our result is purely local, it does not take into account degenerate tori on the boundary of the moduli space SL(2, ℝ)/SL(2, ℤ) which appear to make a large contribution to (1.8).

Remark 1.3. Results involving spectral averaging on shorter scales than (1.4) are known. For instance, Iosevich, Sawyer, Seeger [ISS] have recently proved

∫_{T}^{T +log T} R(μ^{2})^{2} dμ ≪ T log T.

Huxley [HU2] proved the estimate

∫_{T}^{T +1} R(μ^{2})^{2} dμ ≪ T log T.

The distribution of R(λ^{2})/\sqrt{λ} for the standard torus ℝ^{2}/ℤ^{2} was studied by Heath-Brown [HB] and for shifted convex ovals has been studied by Bleher in a series of papers starting with [Bl].

Remark 1.4. A number of results are known for almost all irrational tori, where ‘almost’ is interpreted in the sense of Lebesgue measure in an appropriate parameter space. The dimension is, however, large, bigger than 4. For instance, in the case of the torus with eigenvalues given by the quadratic form

Q(p_{1}, p_{2}, …, p_{n}) = ap_{1}^{2} + Q′(p_{2}, …, p_{n})

with n ≥ 10 and with Q′ a rational quadratic form,

R(λ) = O(λ^{n/2−6/5} log^{1/4} λ)

for almost all a, see [WA]. In general, one gets only o(λ^{n/2−1}) for such forms with n ≥ 10. For diagonal forms

Q(p_{1}, p_{2}, …, p_{n}) = a_{1}p_{1}^{2} + a_{2}p_{2}^{2} + ⋯ + a_{n}p_{n}^{2}  (1.9)

Jarnik [JA1] proved

R(λ) = O(λ^{n/2−1}), n ≥ 5, R(λ) = O(λ log^{2} λ), n = 4,

while for irrational diagonal with n ≥ 5 we get R(λ) = o(λ^{n/2−1}). These bounds cannot be improved, unless one specialises even further. In a series of papers [JA2], [JA3], [JA4] Jarnik proved estimates of the form

R(λ) = O(λ^{n/2−μ+ϵ})

for an explicit μ > 1. For almost all forms of type

Q(p_{1}, p_{2}, …, p_{n}) = ap_{1}^{2} + (p_{2}^{2} + p_{3}^{2} + ⋯ + p_{n}^{2})

we can take μ = 5/4. For forms of type

Q(p_{1}, p_{2}, …, p_{n}) = b_{1}(p_{1}^{2} + ⋯ p_{k}^{2}) + b_{2}(p_{k+1}^{2} + ⋯ + p_{n}^{2})

with k, n − k ≥ 4, the μ depends on the diophantine properties of b_{2}/b_{1}. One conjectures that for almost all diagonal forms (1.9)

R(λ) = O(λ^{(n−1)/4+ϵ}).

Remark 1.5. Our method is spectral in nature and avoids estimates on exponential sums and the Hardy-Littlewood circle method. It avoids also the sophisticated techniques developed for the circle problem and (rational and irrational) tori.

Remark 1.6. It would be of interest to determine what happens to the error term R(λ) in other completely integrable situations. For example, in the case of convex surfaces of revolution, one can perhaps improve the O(λ^{1/3}) bound for R(λ) by averaging over appropriate moduli-spaces. According to [CdV], this is related to counting shifted lattice points inside a convex curve. We hope to address this point in future work. An interesting related result due to Kendall, see [KE], is the following

∫_{0}^{1} ∫_{0}^{1} R(λ, α, β)^{2} dα dβ = O(λ^{1/4}),

where (α, β) represents a shift of the standard lattice ℤ^{2}.

We first take up the case where H(p, σ; u_{1}, u_{2}) = u_{1} σ^{2} + u_{2} p^{2}. For such metrics the proof of Theorem 1.1 (i) is rather computational and so we split it up into three parts: an analysis of the averaged density of states (Section 2), mean-square density of states (Section 3) and a rescaled Tauberian argument (Section 4).

2. Averaged density of states

In this section we give an asymptotic estimate for the averaged density of states of the eigenvalues, λ_{j}, j = 1, 2, ... of the Laplacian Δ on the torus, T^{2}. We put ℏ^{−1} = \sqrt{λ}. Most of our estimates will be given in terms of ℏ but the reader should have no difficulty in expressing them in terms of λ. We define the average density of states as follows:

AV(ϕ) = ∑_{j=1}^{∞} ∫_{I^{2}} ϕ(λ_{j}(u⃗) − λ) du⃗
= ℏ ∑_{m∈ℤ} ∑_{n∈ℤ} ∫_{I^{2}} ∫_{ℝ} e^{is[H(mℏ,nℏ;u⃗)−1]/ℏ} ϕ̌(ℏs) ds du⃗,  (2.1)

where ϕ is even, belongs to the Schwartz space S(ℝ) and its Fourier transform ϕ̌ ∈ C_{0}^{∞}(ℝ). In fact, we take ϕ̌ such that its support contains only 0 among the lattice points. The second equality follows from the Fourier inversion formula. The density AV(ϕ) is a function of λ, or, equivalently, ℏ and we are interested in estimating it as ℏ → 0. Integration by parts in the s variable gives for any δ > 0 and ℏ ≤ ℏ_{0},

AV(ϕ) = ℏ ∑_{|m|,|n|=0}^{ℏ^{−1−δ}} ∫_{I^{2}} ∫_{ℝ} e^{is[H(mℏ,nℏ;u⃗)−1]/ℏ} ϕ̌(ℏs) ds du⃗ + O(ℏ^{∞}). (2.2)

We decompose the last expression in (2.2) into a sum of the two terms:

dρ_{0}(ϕ) := ℏ ∑_{|m|,|n|=0}^{ℏ^{−1−δ}} ∫_{I^{2}} ∫_{ℝ} e^{is[H(mℏ,nℏ;u⃗)−1]/ℏ} ζ(s) ϕ̌(ℏs) ds du⃗,  (2.3)

where ζ(s) is ≡ 1 close to 0 and is in C_{0}^{∞}(ℝ), and

dρ_{+}(ϕ) := ℏ ∑_{|m|,|n|=0}^{ℏ^{−1−δ}} ∫_{I^{2}} ∫_{ℝ} e^{is[H(mℏ,nℏ;u⃗)−1]/ℏ} (1 − ζ(s)) ϕ̌(ℏs) ds du⃗. (2.4)

Using (6.5) from the Appendix (see also [U, Section 1.3]), we get

dρ_{0}(ϕ) = c_{0} ϕ̌(0) + O(ℏ^{∞}),  (2.5)

and it remains to estimate the long-time term, dρ_{+}(ϕ). We can assume that either |mℏ| ≥ C > 0 or that |nℏ| ≥ C > 0. Otherwise λ_{j}(u) − λ ≫ λ. Since ϕ ∈ S(ℝ), we have ϕ(x) ≪ |x|^{−N} for |x| large and every N > 0, while the number of such summands in (2.2) is O(λ). So such terms contribute O(ℏ^{∞}). Assume first that |mℏ| ≥ C > 0. Then, by integrating by parts in the u_{1} variable in (2.4), we get that

ℏ ∑_{|m|=Cℏ^{−1}}^{ℏ^{−1−δ}} ∑_{|n|=0}^{ℏ^{−1−δ}} ∫_{I^{2}} ∫_{ℝ} e^{is[H(mℏ,nℏ;u⃗)−1]/ℏ} (1 − ζ(s)) ϕ̌(ℏs) ds du⃗

≪ ℏ ∑_{|m|=Cℏ^{−1}}^{ℏ^{−1−δ}} ∑_{|n|=0}^{ℏ^{−1−δ}} ∫_{ℝ} (\frac{1}{m^{2}ℏ}) \frac{1}{s}(1 − ζ(s)) ϕ̌(ℏs) ds ≪ | log ℏ| ℏ^{−δ},  (2.6)

since the summation over m contributes ≪ ℏ and the integration in the s variable contributes ≪ log ℏ. When |nℏ| ≥ C > 0 we repeat the above argument, except that we integrate by parts with respect to the u_{2} variable. Since δ > 0 is arbitrarily small, a combination of (2.6) with the stationary phase estimate for dρ_{0}(ϕ) gives the following proposition.

Proposition 2.1. Let ϕ ∈ S(ℝ) with ϕ̌ ∈ C_{0}^{∞}(ℝ). Then, given the two-parameter family of metric forms H(p, σ; u_{1}, u_{2}) := u_{1}σ^{2} + u_{2}p^{2} on the torus, we have that for ℏ sufficiently small and any δ > 0,

|AV(ϕ; ℏ)| = O_{δ}(ℏ^{−δ}).

Remark 2.2. The result of Proposition 2.1, i.e., the estimate |AV(ϕ; ℏ)| = O_{δ}(ℏ^{−δ}) also holds for the three-parameter family of metric forms H(p, σ; u_{1}, u_{2}, u_{3}) := u_{1}σ^{2} + u_{3}pσ + u_{2}p^{2}. The proof is the same.

3. Mean-square density of states

The mean square density of states is given by the expression

MS(ϕ) = ∫_{I^{2}} |dρ(ϕ; u⃗, ℏ) − c_{0}(u⃗)ϕ̌(0)|^{2} du⃗
= ∫_{I^{2}} |dρ_{+}(ϕ; u⃗, ℏ)|^{2} du⃗ + O(ℏ^{2−δ}| log ℏ|).  (3.1)

Here, the last estimate in (3.1) is an immediate consequence of (2.5) and (2.6). By introducing χ(y) a function which is ≥ 1 on [1 − ϵ, 1 + ϵ] and with Fourier transform χ̂ of compact support, we are thus reduced to estimating

∫_{I^{2}} |dρ_{+}(ϕ; u⃗, ℏ)|^{2} du⃗ ≪ ℏ^{2} ∑_{m_{i},n_{i}} ∫_{ℝ^{4}} e^{i Φ(m_{1},n_{1},m_{2},n_{2};s⃗,ℏ)/ℏ} a(s⃗; ℏ) χ(u_{1}) χ(u_{2}) ds⃗du⃗, (3.2)

where, by the integration by parts argument in section 2 it suffices, modulo O(ℏ^{∞}) errors, to sum over quadruples with max(|m_{j}|, |n_{j}|) ≤ ℏ^{−1−δ}; j = 1, 2. We set s⃗ = (s_{1}, s_{2}). To simplify the writing in (3.2) we have put

Φ(m_{1}, n_{1}, m_{2}, n_{2}; s⃗, ℏ) = H(m_{1}ℏ, n_{1}ℏ; u⃗) s_{1} − H(m_{2}ℏ, n_{2}ℏ; u⃗) s_{2},

and

a(s⃗; ℏ) := (1 − ζ(s_{1})) (1 − ζ(s_{2})) ϕ̌(ℏs_{1}) ϕ̌(ℏs_{2}) e^{i(s_{2}−s_{1})/ℏ}.

To estimate the integral in (3.2), we fix δ > 0 and consider the set

Ω(m_{1}, n_{1}, m_{2}, n_{2}; ℏ) := {u ∈ I^{2}; |H(m_{j}ℏ, n_{j}ℏ; u⃗) − 1| ≤ ℏ^{1−δ}; j = 1, 2 }.

By an integration by parts in the s_{1}, s_{2} variables in (3.2), it suffices to assume, modulo O(ℏ^{∞}) errors, that we only sum over quadruples (m_{1}, n_{1}, m_{2}, n_{2}) with the property that for ℏ ≤ ℏ_{0},

Ω(m_{1}, n_{1}, m_{2}, n_{2}; ℏ) ≠ ∅.

So, we have that ∫_{I^{2}} |dρ_{+}(ϕ; u⃗, ℏ)|^{2} du⃗ is bounded by

Cℏ^{2} ∑_{Ω(m_{1},n_{1},m_{2},n_{2};ℏ) ≠ ∅} ∫_{ℝ^{4}} e^{i Φ(m_{1},n_{1},m_{2},n_{2};s⃗,ℏ)/ℏ} a(s⃗; ℏ) χ(u_{1}) χ(u_{2}) ds⃗du⃗ + O(ℏ^{∞}).

Furthermore, by integrating by parts in u_{1}, u_{2}, we can assume, modulo O(ℏ^{∞}) errors, that we can find (s_{1}, s_{2}) ∈ supp a such that

\frac{∂}{∂u_{j}} Φ(m_{1}, n_{1}, m_{2}, n_{2}; s⃗, ℏ) ≪ ℏ^{1−δ}, j = 1, 2.  (3.3)

Written out explicitly, the inequality in (3.3) reads

|m_{1}ℏ|^{2} s_{1} − |m_{2}ℏ|^{2} s_{2} ≪ ℏ^{1−δ}
|n_{1}ℏ|^{2} s_{1} − |n_{2}ℏ|^{2} s_{2} ≪ ℏ^{1−δ},  (3.4)

Since min(|s_{1}|, |s_{2}|) ≫ 1 on supp a(s⃗; ℏ), for a given (m_{1}, n_{1}, m_{2}, n_{2}) we need to be able to solve (3.4) for some s_{1}, s_{2} with min(|s_{1}|, |s_{2}|) ≫ 1. By inverting the matrix equation in (3.4) using Cramer’s rule and the estimate max(|m_{j}|, |n_{j}|) ≪ ℏ^{−1−δ}, we get

|m_{1}ℏ|^{2} |n_{2}ℏ|^{2} − |m_{2}ℏ|^{2} |n_{1}ℏ|^{2} ≪ ℏ^{1−3δ}.  (3.5)

On the other hand, the condition Ω(m_{1}, n_{1}, m_{2}, n_{2}; ℏ) ≠ ∅ means that for some u_{1} = u_{1}(m_{1}, n_{1}, m_{2}, n_{2}; ℏ), u_{2} = u_{2}(m_{1}, n_{1}, m_{2}, n_{2}; ℏ),

(u_{1}, u_{2}) · (|m_{1}ℏ|^{2}, |n_{1}ℏ|^{2}) = 1 + O(ℏ^{1−δ})
(u_{1}, u_{2}) · (|m_{2}ℏ|^{2}, |n_{2}ℏ|^{2}) = 1 + O(ℏ^{1−δ}),  (3.6)

where · is the standard inner product in ℝ^{2}. Resubstituting (3.6) back into (3.5) or using Cramer’s rule, we get

\frac{1}{u_{2}} (|m_{1}ℏ|^{2} − |m_{2}ℏ|^{2}) = O(ℏ^{1−3δ}),  (3.7)

and

\frac{1}{u_{1}} (|n_{1}ℏ|^{2} − |n_{2}ℏ|^{2}) = O(ℏ^{1−3δ}).  (3.8)

Since δ is arbitrarily small, we will abuse notation somewhat and will write δ instead of 3δ. Consequently, the two estimates in (3.8) and (3.7) imply that:

|m_{1}| − |m_{2}| ≪ \frac{ℏ^{−1−δ}}{|m_{1}| + |m_{2}|} and |n_{1}| − |n_{2}| ≪ \frac{ℏ^{−1−δ}}{|n_{1}| + |n_{2}|}.  (3.9)

Thus, we have shown that

MS(ϕ) ≪ ℏ^{2} ∑_{m_{2},n_{2}} ∑_{m_{1}∈β(m_{2},ℏ)} ∑_{n_{1}∈γ(n_{2},ℏ)} ∫_{ℝ^{2}} ϕ̌(ℏs_{1}) ϕ̌(ℏs_{2})

×χ̂(ℏ[s_{1}m_{1}^{2} − s_{2}m_{2}^{2}]) χ̂(ℏ[s_{1}n_{1}^{2} − s_{2}n_{2}^{2}]) ds⃗,  (3.10)

where

β(m_{2}, ℏ) := \{m_{1}; |m_{1}| − |m_{2}| ≪ \frac{ℏ^{−1−δ}}{|m_{1}| + |m_{2}|}\} ,  (3.11)

and

γ(n_{2}, ℏ) := \{n_{1}; |n_{1}| − |n_{2}| ≪ \frac{ℏ^{−1−δ}}{|n_{1}| + |n_{2}|}\} .  (3.12)

We make either the change of variables

S = ℏ(m_{1}^{2}s_{1} − m_{2}^{2}s_{2}), T = s_{1} + s_{2},

or, alternatively,

S = ℏ(n_{1}^{2}s_{1} − n_{2}^{2}s_{2}), T = s_{1} + s_{2}.

We notice that, since ϕ̌ has compact support, ℏ(s_{1} +s_{2}) ≪ 1, which gives T ≪ ℏ^{−1}. Equation (3.10) gives:

|MS(ϕ)| ≪ ℏ^{2−2−2δ} ∑_{m_{2},n_{2}≠0} \frac{1}{m_{2}n_{2}} min (\frac{1}{ℏm_{2}^{2}} , \frac{1}{ℏn_{2}^{2}} ) ∫_{|T |≪ℏ^{−1}} dT
+ℏ^{2−1−1/2−2δ} ∑_{m_{2}≠0,n_{2}=0} \frac{1}{m_{2}} \frac{1}{ℏm_{2}^{2}} ∫_{|T |≪ℏ^{−1}} dT  (3.13)

+ ℏ^{2−1−1/2−2δ} ∑_{n_{2}≠0,m_{2}=0} \frac{1}{n_{2}} \frac{1}{ℏn_{2}^{2}} ∫_{|T |≪ℏ^{−1}} dT

+ℏ^{2} ∑_{m_{1}≠0,m_{2}=0}^{ℏ^{−1/2−δ}} ∑_{n_{1}≠0,n_{2}=0}^{ℏ^{−1/2−δ}} min (\frac{1}{ℏm_{1}^{2}} , \frac{1}{ℏn_{1}^{2}} ) ∫_{|T |≪ℏ^{−1}} dT.

For the estimate on the first line of (3.13) we used the range of m_{1} and m_{2} in (3.11) and (3.12). For the second estimate we notice that, when n_{2} = 0, (3.12) gives n_{1}^{2} ≪ ℏ^{−1−δ}. The third estimate is deduced using (3.12) and the fourth using (3.11) and (3.12).

Thus, since |m_{2}ℏ|^{2} + |n_{2}ℏ|^{2} ≥ C > 0 and |m_{1}ℏ|^{2} + |n_{1}ℏ|^{2} ≥ C > 0, it follows that:

min (\frac{1}{ℏm_{2}^{2}} , \frac{1}{ℏn_{2}^{2}} ) ≪ ℏ and min (\frac{1}{ℏm_{1}^{2}} , \frac{1}{ℏn_{1}^{2}} ) ≪ ℏ,

and so,

|MS(ϕ)| ≪ ℏ^{−2δ} ∑_{m_{2},n_{2}≠0}^{ℏ^{−1−δ}} \frac{1}{m_{2}n_{2}} + ℏ^{1/2−2δ} ∑_{m_{2}≠0}^{ℏ^{−1−δ}} \frac{1}{m_{2}}

+ℏ^{1/2−2δ} ∑_{n_{2}≠0}^{ℏ^{−1−δ}} \frac{1}{n_{2}} + ℏ^{2−1/2−δ−1/2−δ}

≪ ℏ^{−2δ}| log ℏ|^{2} + ℏ^{1/2−2δ}| log ℏ| + ℏ^{1−δ}.  (3.14)

Since δ > 0 in (3.14) is arbitrarily small, we have proved:

Proposition 3.1. Consider the two-parameter family of flat metrics on T^{2} = ℝ^{2}/ℤ^{2} given by the Hamiltonian functions H(p, σ; u_{1}, u_{2}) := u_{1}σ^{2} + u_{2}p^{2}. Then, for any δ > 0,

MS(ϕ) := ∫_{I^{2}} |dρ(ϕ; u⃗, ℏ) − c_{0}(u⃗)ϕ̌(0)|^{2} du⃗ = O_{δ}(ℏ^{−δ}).

We now turn to the proof of our main theorem. For this we will need to combine the estimates for the averaged and mean-square density of states with a rescaled, well-known Tauberian decomposition, see [DG], [V].

4. The Tauberian Decomposition: Proof of Theorem 1.1(i)

Let ϕ ∈ 𝕊(ℝ) with ϕ(λ) > 0, and ϕ̌ ∈ C_{0}^{∞}(ℝ) with ϕ̌(0) = 1. We start with a rescaled Tauberian argument used by Duistermaat and Guillemin [DG], but applied to the eigenvalues of Δ rather than \sqrt{Δ}. Taking into account this rescaling we will naturally encounter semiclassical density of states on scales of order ∼ ℏ^{2} where ℏ^{−1} = \sqrt{λ}. Our starting point is the following basic Tauberian decomposition (see [DG], [V]):

∫_{−∞}^{∞} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) = ∫_{λ′≥λ+1} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) (4.1)
+ ∫_{|λ−λ′|<1} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗)
+ ∫_{λ′<λ−1} ∫_{−∞}^{∞} ϕ(x − λ′) dx dN(λ′; u⃗)
− ∫_{λ′≤λ−1} ∫_{λ}^{∞} ϕ(x − λ′) dx dN(λ′; u⃗).

In the case at hand, all of the quantities in (4.1) depend on the external parameters (u_{1}, u_{2}) ∈ I^{2} and our task is to estimate the (integrated) asymptotics of both sides of (4.1). First, we turn to the simplest term, which is the third term on the right-hand side of (4.1). Indeed, since 1 = ∫_{−∞}^{∞} ϕ(s)ds, it follows that

∫_{λ′<λ−1} ∫_{−∞}^{∞} ϕ(x − λ′) dx dN(λ′; u⃗) = N(λ − 1; u⃗).

To estimate the other terms in (4.1), we will need the following result which hinges on the estimates for the averaged and mean-square density of states in sections 2 and 3:

Proposition 4.1. Let H(p, σ; u_{1}, u_{2}) = σ^{2}u_{1} + p^{2}u_{2}. Then, for any ϕ ∈ S(ℝ) as above, we have that:

∫_{I^{2}} |∫_{−∞}^{∞} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) − (2π)^{−2} λ vol(M(u⃗)) | du⃗ = O_{δ}(λ^{\frac{1}{4} +δ}).

Proof. First, by the Fubini Theorem,

∫_{−∞}^{∞} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) = ∫_{−∞}^{λ} ∫_{−∞}^{∞} ϕ(x − λ′) dN(λ′; u⃗) dx.

Since ϕ ∈ S(ℝ) and 0 ≤ λ_{j} ∈ Spec(Δ),

∫_{−∞}^{∞} ϕ(x − λ′) dN(λ′; u⃗) = O(|x|^{−∞}),  (4.2)

as x → −∞. Thus, from (4.2) it follows that

∫_{−∞}^{∞} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) = ∫_{−∞}^{∞} ∫_{0}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) + O(1). (4.3)

Put ℏ^{−1} = \sqrt{λ} and consider the rescaled operator H := ℏ^{2}Δ with eigenvalues λ_{j}(ℏ) = ℏ^{2}λ_{j} : j = 1, 2, .... Then, making the change of variables Λ := ℏ^{2}x, and taking (4.3) into account, it follows that, modulo O(1) errors, we are reduced to estimating

I(u⃗, ℏ) = ℏ^{−2} ∑_{j=1}^{∞} ∫_{0}^{1} ϕ (\frac{λ_{j}(ℏ; u⃗) − Λ}{ℏ^{2}}) dΛ

= ℏ^{−2} ∑_{m,n}^{ℏ^{−1−δ}} ∫_{0}^{1} ϕ (\frac{H(mℏ, nℏ; u⃗) − Λ}{ℏ^{2}}) dΛ + O(ℏ^{∞}). (4.4)

Next, we apply the Fourier inversion formula to right-hand side in (4.4) and split the resulting integral into two pieces corresponding to the zero and non-trivial period spectrum. More precisely, let ζ ∈ C_{0}^{∞}(ℝ) be non-negative, equal to 1 close to 0. Its support should be small enough as explained in the Appendix. Then,

I(u⃗, ℏ) = I_{0}(u⃗, ℏ) + I_{+}(u⃗, ℏ),  (4.5)

where

I_{0}(u⃗, ℏ) = ℏ^{−1}Trace ∫_{0}^{1} ∫_{−∞}^{∞} e^{is[H(Q_{1},Q_{2};u⃗)−Λ]/ℏ}ζ(s)ϕ̌(ℏs) ds dΛ,

and

I_{+}(u⃗, ℏ) = ℏ^{−1}Trace ∫_{0}^{1} ∫_{−∞}^{∞} e^{is[H(Q_{1},Q_{2};u⃗)−Λ]/ℏ}(1 − ζ(s))ϕ̌(ℏs) ds dΛ,

and Q_{1} = mℏ, Q_{2} = nℏ. First, we estimate I_{0}(u⃗, ℏ) on the right-hand side of (4.5) by exactly the same method as in the Appendix, where we estimated dρ_{0}(ϕ). We get

I_{0}(u⃗, ℏ) = (2π)^{−2}vol(M(u⃗)) ℏ^{−2} + O(ℏ^{∞}).  (4.6)

To prove Proposition 4.1, we need to estimate

∫_{I^{2}} |I(u⃗, ℏ) − I_{0}(u⃗, ℏ)| du⃗ = ∫_{I^{2}} |I_{+}(u⃗, ℏ)| du⃗.

To do this, we appeal to Proposition 3.1 on the mean-square density of states and apply the Cauchy-Schwartz inequality. First,

\begin{matrix}∫_{I^{2}} |I_{+}(u⃗, ℏ)|^{2}du⃗ \\ = ℏ^{−2} ∑_{m_{1},n_{1}} ∑_{m_{2},n_{2}} ∫ e^{iΦ(m_{1},n_{1},m_{2},n_{2};s⃗,ℏ)/ℏ} a(s⃗; Λ_{1}, Λ_{2}, ℏ)dΛ_{1} dΛ_{2} ds⃗ du⃗, \end{matrix} (4.7)

and

a(s⃗, Λ_{1}, Λ_{2}; ℏ) = e^{i[Λ_{1}s_{1}−Λ_{2}s_{2}]/ℏ}(1 − ζ(s_{1}))ϕ̌(ℏs_{1})(1 − ζ(s_{2}))ϕ̌(ℏs_{2}).

Note that since ϕ̌ ∈ C_{0}^{∞}(ℝ) the summations in (4.7) are restricted to indices with max(|m_{i}|, |n_{i}|) ≪ ℏ^{−1−δ}. Moreover, since min(|s_{1}|, |s_{2}|) ≥ 1 on supp a, by an iterated integration in the Λ_{1}, Λ_{2} variables, we get

∫_{0}^{1} ∫_{0}^{1} a(s⃗, Λ_{1}, Λ_{2}; ℏ) dΛ_{1} dΛ_{2} = (\frac{ℏ^{2}}{s_{1}s_{2}} ) · b(s_{1}, s_{2}; ℏ),

where b ∈ C^{∞} with b(s_{1}, s_{2}; ℏ) = O(1) as for ℏ ≤ ℏ_{0}. So, the end result is that

\begin{matrix}∫_{I^{2}} |I_{+}(u⃗, ℏ)|^{2}du⃗ \\ = ∑_{|m_{i}|,|n_{i}|>0}^{Cℏ^{−1−δ}} ∫ e^{iΦ(m_{1},n_{1},m_{2},n_{2};s⃗,ℏ)/ℏ} b(s_{1}, s_{2}; ℏ)\frac{1}{s_{1}s_{2}} χ(u_{1}) χ(u_{2})ds⃗ du⃗. \end{matrix} (4.8)

Since

s_{1}s_{2} ≥ \frac{1}{2} (s_{1} + s_{2}) when min(s_{1}, s_{2}) ≥ 1,

it follows by the argument in Proposition 3.1 that the right-hand side in (4.8) is

≪ ∑_{m_{2},n_{2}} ∑_{m_{1}∈β(m_{2},ℏ)} ∑_{n_{1}∈γ(n_{2},ℏ)} ∫_{|s_{1}|,|s_{2}|≥C} \frac{ϕ̌(ℏs_{1})ϕ̌(ℏs_{2})}{|s_{1}| + |s_{2}|}

×χ̂(ℏ[s_{1}m_{1}^{2} − s_{2}m_{2}^{2}])χ̂(ℏ[s_{1}n_{1}^{2} − s_{2}n_{2}^{2}]) ds⃗,

which is bounded by

ℏ^{−2−2δ} ∑_{m_{2},n_{2}} \frac{1}{m_{2}n_{2}} min (\frac{1}{ℏm_{2}^{2}} , \frac{1}{ℏn_{2}^{2}} ) ∫_{C<|T |≪ℏ^{−1}} \frac{dT}{T} ≪ ℏ^{−1−2δ} | log ℏ|^{3}.

The sets β(m_{2}, ℏ) and γ(n_{2}, ℏ) are defined in (3.11), (3.12) and the argument follows as in (3.13) with the only change being the appearance of the denominator |s_{1}|+|s_{2}|. Now, by the Cauchy-Schwartz inequality, we have

∫_{I^{2}} |I_{+}(u⃗, ℏ)|du⃗ ≪ (∫_{I^{2}} |I_{+}(u⃗, ℏ)|^{2} du⃗)^{1/2}
≪ (ℏ^{−1−3δ})^{1/2} ≪ λ^{1/4+3δ}.  (4.9)

This completes the proof of Proposition 4.1.

By reshuffling the terms in the Tauberian decomposition, it is clear that

|N(λ − 1; u⃗) − I_{0}(u⃗, ℏ)| ≤ |I_{+}(u⃗, ℏ)|
+ |∫_{λ′≥λ+1} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) |

+ |∫_{|λ−λ′|<1} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) |
+ |∫_{λ′≤λ−1} ∫_{λ}^{∞} ϕ(x − λ′) dx dN(λ′; u⃗) | . (4.10)

Integrating both sides of (4.10) over the deformation parameters u_{1}, u_{2}, we get

∫_{I^{2}} |N(λ − 1; u⃗) − I_{0}(u⃗, ℏ) | du⃗ ≤ ∫_{I^{2}} |I_{+}(u⃗, ℏ)|du⃗

+ ∫_{I^{2}} |∫_{λ′≥λ+1} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) | du⃗

+ ∫_{I^{2}} |∫_{|λ−λ′|<1} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) | du⃗

+ ∫_{I^{2}} |∫_{λ′≤λ−1} ∫_{λ}^{∞} ϕ(x − λ′) dx dN(λ′; u⃗) | du⃗

= T_{0} + T_{1} + T_{2} + T_{3}  (4.11)

From Proposition 4.1 we have that for any δ > 0,

T_{0} = ∫_{I^{2}} |I_{+}(u⃗, ℏ)| du⃗ = O_{δ}(ℏ^{−1/2−δ}).

Consequently, it remains to estimate the terms T_{1}, T_{2}, T_{3} on the right-hand side of the inequality (4.11). We start with T_{1}:

|T_{1}| = ∫_{I^{2}} |∫_{λ′≥λ+1} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) | du⃗

≤ C_{N} ∑_{m=1}^{∞} ( ∫_{I^{2}} ∫_{m≤λ′−λ≤m+1} dN(λ′, u⃗) ) m^{−N}

≤ C_{N} ∑_{m=1}^{∞} ( ∫_{I^{2}} N(λ + m + 1; u⃗) − N(λ + m; u⃗) du⃗ ) m^{−N}. (4.12)

One gets the first inequality above by splitting up the range of λ′ − λ ∈ [0, ∞) and using the fact ϕ ∈ S(ℝ). We let λ̃ = λ + m. It follows from Proposition 2.1 for the averaged density of states that for any δ > 0,

∫_{I^{2}} N(λ + m + 1; u⃗) − N(λ + m; u⃗) du⃗ ≪ (m + λ)^{δ},

so that the last line in (4.12) is bounded by

C_{N}λ^{δ} ∑_{m=1}^{∞} m^{−N+δ} = O_{δ}(λ^{δ}).

The end result is that

|T_{1}| = ∫_{I^{2}} |∫_{λ′≥λ+1} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗) | du_{1}du_{2} = O_{δ}(λ^{δ}). (4.13)

The term T_{3} can be estimated in the same way as T_{1}. So, in particular it follows that:

|T_{3}| = O(λ^{δ})  (4.14)

for any δ > 0. We now turn to the estimating the term T_{2}. Here we also need the bound for the averaged density of states given in Proposition 2.1:

T_{2} = ∫_{I^{2}} |∫_{|λ−λ′|<1} ∫_{−∞}^{λ} ϕ(x − λ′) dx dN(λ′; u⃗)| du⃗

≤ ∫_{I^{2}} |∫_{|λ−λ′|<1} ∫_{−∞}^{∞} ϕ(x − λ′) dx dN(λ′; u⃗)| du⃗
= ∫_{I^{2}} |N(λ + 1; u⃗) − N(λ − 1; u⃗) | du⃗,

since ϕ(x) ≥ 0 and ∫_{−∞}^{∞} ϕ(x)dx = 1.

Now, let Θ ∈ S(ℝ) with Θ ≥ 0 and Θ(x) ≥ 1 when |x| ≤ 1. Then

∫_{I^{2}} |N(λ + 1; u⃗) − N(λ − 1; u⃗) | du⃗ ≤ ∑_{j=1}^{∞} ∫_{I^{2}} Θ(λ_{j} − λ) du⃗ = O_{δ}(ℏ^{−δ}),

where the estimate is a consequence of Proposition 2.1 for the averaged density of states. Therefore, we have that

|T_{2}| = O_{δ}(ℏ^{−δ}).  (4.15)

Combining the estimates (4.9), (4.13), (4.14), (4.15) and (4.6) with the integrated Tauberian decomposition in (4.11) gives

∫_{I^{2}} |N(λ − 1; u⃗) − c_{0}(u⃗) λ| du⃗ = O_{δ}(λ^{1/4+δ}) + O_{δ}(λ^{δ}) + O(1)

for any δ > 0. This completes the proof of Theorem 1.1 (i).

5. Generic flat tori: Proof of Theorem 1.1 (ii)

Here we take up the case where

H(p, σ; u_{1}, u_{2}, u_{3}) := u_{1}σ^{2} + u_{3}pσ + u_{2}p^{2}.

Then, just as in the proof of Theorem 1.1 (i), we get from the eigenfunction equation that for some u⃗ = u⃗(m_{1}, n_{1}, m_{2}, n_{2}; ℏ),

u_{1}m_{1}^{2} + u_{3}m_{1}n_{1} + u_{2}n_{1}^{2} = ℏ^{−2} + O(ℏ^{−1−δ}),
u_{2}m_{2}^{2} + u_{3}m_{2}n_{2} + u_{2}n_{2}^{2} = ℏ^{−2} + O(ℏ^{−1−δ}),  (5.1)

Furthermore, since there are now three parameters, there are three inequalities analogous to the estimates in (3.5) and (3.6). These inequalities come from solving the system of three equations that we get by computing the Fourier transforms in the u_{1}, u_{2}, u_{3} variables in the expression for MS(ϕ; ℏ). These are

m_{1}^{2} n_{2}^{2} − m_{2}^{2} n_{1}^{2} ≪ ℏ^{−3−δ},
m_{1} m_{2} ( m_{2}n_{1} − m_{1}n_{2} ) ≪ ℏ^{−3−δ},
n_{1} n_{2} ( m_{2}n_{1} − m_{1}n_{2} ) ≪ ℏ^{−3−δ}.  (5.2)

Multiplying the first equation in (5.1) by m_{2}^{2} , the second by m_{1}^{2} and subtracting gives

ℏ^{−2}(m_{2}^{2} − m_{1}^{2}) = u_{3}m_{1}m_{2} ( m_{2}n_{1} − m_{1}n_{2} )
+ u_{2}(n_{1}^{2}m_{2}^{2} − n_{2}^{2}m_{1}^{2}) + O(ℏ^{−3−δ})
= O(ℏ^{−3−δ}),  (5.3)

where the last inequality in (5.3) follows from the first two estimates in (5.2).

Similarily, multiplying the first equation in (5.1) by n_{2}^{2}, the second by n_{1}^{2} and subtracting gives

ℏ^{−2}(n_{2}^{2} − n_{1}^{2}) = u_{3}n_{1}n_{2} ( m_{2}n_{1} − m_{1}n_{2} )
+ u_{2}(n_{1}^{2}m_{2}^{2} − n_{2}^{2}m_{1}^{2}) + O(ℏ^{−3−δ})
= O(ℏ^{−3−δ}),  (5.4)

which follows from the first and last estimates in (5.2). Combining the inequalities in (5.3) and (5.4) gives

|m_{1}| − |m_{2}| ≪ \frac{ℏ^{−1−δ}}{|m_{1}| + |m_{2}|} and |n_{1}| − |n_{2}| ≪ \frac{ℏ^{−1−δ}}{|n_{1}| + |n_{2}|}.

Restricting to quadruples satisfying the inequalities (3.9), the proof of Theorem 1.1 (ii) then proceeds exactly as before.

6. Appendix: Proof of (2.5)

We set

χ̌(s, ℏ) = ζ(s) · ϕ̌(ℏs).  (6.1)

From (2.3) it is clear that, up to O(ℏ^{∞}) errors, we have

dρ_{0}(ϕ) = ℏ · Trace ∫_{ℝ} e^{is[−ℏ^{2}Δ−1]/ℏ}χ̌(s, ℏ) ds.

We consider the operator P = (−ℏ^{2}Δ−1)/ℏ. The one-parameter group of operators U(s) = e^{isP} satisfies the equation

\frac{∂}{∂s}U(s) = (−iℏΔ − iℏ^{−1})U(s)

and its Schwartz kernel Ũ(s, x, y; ℏ) is given by

Ũ(s, x, y; ℏ) = \frac{1}{(2πℏ)^{2}} ∫ e^{i[(x−y)·ξ+s|ξ|^{2}−s]/ℏ} dξ.

This calculation holds for ℝ^{2}. Since the torus is covered by ℝ^{2}, we have

U(s, x, y; ℏ) = ∑_{γ∈Γ} Ũ(s, x + γ, y; ℏ).

The trace formula

∑_{j} χ(λ_{j}) = ∫ ∫ U(s, x, x; ℏ)χ̌(s) ds dvol(x)

gives

dρ_{0}(ϕ) = c_{0}ℏ^{−1} ∑_{γ∈Γ} ∫_{ℝ^{2}} ∫_{ℝ} e^{iγ·ξ/ℏ}e^{is(|ξ|^{2}−1)/ℏ}χ̌(s, ℏ) ds dξ.  (6.2)

We choose the support of χ̌(s; ℏ), or, equivalently ζ(s) to be sufficiently small, as explained in (6.3) below. The main contribution in (6.2) comes from the range |ξ|^{2} ∼ 1, because on the complement we can integrate by parts in s as many times as needed and get O(ℏ^{∞}) error. We look first at the summands with γ ≠ 0. We integrate by parts in the (ξ_{1}, ξ_{2}) variables. Since |ξ_{j}| ≤ 2, say, and γ ≠ 0, we can find a constant c > 0, independent of u⃗, such that max(|γ_{j}|) ≥ c. We take

supp(χ̌) ⊂ (−c/5, c/5).  (6.3)

Then

|\frac{∂}{∂ξ_{j}} (γ · ξ + |ξ|^{2}s)| = |γ_{j} + 2ξ_{j}s| ≥ c/5.

With the choice (6.3) for the support of χ̌ we can ignore the contribution of γ ≠ 0, up to O(ℏ^{∞}) errors. We get

dρ_{0}(ϕ) = c_{0}ℏ^{−1} ∫_{ℝ} ∫_{ℝ^{2}} e^{is(|ξ|^{2}−1)/ℏ}χ̌(s; ℏ) ds dξ + O(ℏ^{∞}).  (6.4)

We set ϕ_{ℏ}(τ) = ℏ^{−1}ϕ(τ/ℏ) and from (6.1) it follows that χ = ζ̂∗ ϕ_{ℏ}. We change to polar coordinates in (6.4) and substitute r^{2} − 1 = ℏx to get

dρ_{0}(ϕ) = c_{0}ℏ^{−2} ∫_{0}^{∞} ∫_{ℝ} ζ̂((r^{2} − 1)/ℏ − y)ϕ(y/ℏ) r dydr + O(ℏ^{∞})

= c_{0}ℏ^{−1} ∫_{−1/ℏ}^{∞} ∫_{ℝ} ζ̂(x − y)ϕ(y/ℏ) dy dx + O(ℏ^{∞}).

By Fubini, we do the x-integration first using

∫_{−1/ℏ}^{∞} ζ̂(x − y) dx = ∫_{ℝ} ζ̂(x − y) dx + O(ℏ^{∞}) = ζ(0) + O(ℏ^{∞}).

Since ζ(0) = 1, we get finally

dρ_{0}(ϕ) = c_{0} ∫_{ℝ} ϕ(u) du + O(ℏ^{∞}).  (6.5)

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Department of Mathematics and Statistics, McGill University, Montreal, CANADA