PROBABILISTIC WEYL LAWS FOR QUANTIZED TORI 1 ...

PROBABILISTIC WEYL LAWS FOR QUANTIZED TORI T.J. CHRISTIANSEN AND M. ZWORSKI

Abstract. For the Toeplitz quantization of complex-valued functions on a 2n-dimensional torus we prove that the expected number of eigenvalues of small random perturbations of a quantized observable satisfies a natural Weyl law (1.3). In numerical experiments the same Weyl law also holds for “false” eigenvalues created by pseudospectral effects.

1. Introduction and statement of the result In a series of recent papers Hager-Sj¨ostrand [13], Sj¨ostrand [17], and Bordeaux MontrieuxSj¨ostrand [3] established almost sure Weyl asymptotics for small random perturbations of non-self-adjoint elliptic operators in semiclassical and high energy r´egimes. The purpose of this article is to present a related simpler result in a simpler setting of Toeplitz quantization. Our approach is also different: we estimate the counting function of eigenvalues using traces rather than by studying zeros of determinants. As in [13] the singular value decomposition and some slightly exotic symbol classes play a crucial rˆole. Thus we consider a quantization C ∞ (T2n ) 3 f 7−→ fN ∈ MN n (C), where T2n is a 2ndimensional torus, R2n /Z2n , and MN n (C) are N n × N n complex matrices. The general procedure will be described in §2 but if n = 1 and T = Sx × Sξ , then def

(1.1)

f = f (x) 7−→ fN = diag (f (`/N )) , ` = 0, · · · , N − 1 , def

g = g(ξ) 7−→ gN = FN∗ diag (g(k/N )) FN , k = 0, · · · , N − 1 , √ where FN = (exp(2πik`/N )/ N )0≤k,`≤0,N −1 , is the discrete Fourier transform. Let ω 7→ QN (ω) be the gaussian ensemble of complex random N n × N n matrices – see §3. With this notation in place we can state our result: Theorem. Suppose that f ∈ C ∞ (T2n ), and that Ω is a simply connected open set with a smooth boundary, ∂Ω, such that for all z a neighbourhood of ∂Ω, (1.2)

volT2n ({w : |f (w) − z| ≤ t}) = O(tκ ) , 0 ≤ t  1 ,

with 1 < κ ≤ 2. Then for any p ≥ p0 > n + 1/2
(1.3) Eω | Spec(fN + N −p QN (ω)) ∩ Ω| = N n volT2n (f −1 (Ω)) + O(N n−β ) , for any β < (κ − 1)/(κ + 1). 1

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T.J. CHRISTIANSEN AND M. ZWORSKI

100 Average for |z| (−1) δqij Mij . j6=i

Let 1lF be the characteristic function of a set F . Then       Mii σi Mii σi Mii σi (3.4) E =E 1lΣ + E 1l c det(tA + δQ) det(tA + δQ) ii det(tA + δQ) Σii since the boundary of Σii has measure 0‡. Now,  E

Mii σi 1lΣ det(tA + δQ) ii



 = E

Mii σi (tσi + δqii )Mii

 = E

σi (tσi + δqii )

∞ X k=0

Pd

j6=i (−1)

δqij Mij (tσi + δqii )Mii

1+ Pd −

j+i

j+i

j6=i (−1)

δqij Mij (tσi + δqii )Mii



!−1

1lΣii  

!k

1lΣii  .

We recall that the set Σii is chosen so that the infinite sum converges. The set Σii is invariant under the mapping (3.5)

qi1 , ..., qi,i−1 , qi,i+1 , ..., qi,d 7→ eiϕ qi1 , ..., eiϕ qi,i−1 , eiϕ qi,i+1 , ..., eiϕ qi,d

P for any real number ϕ. Since Mij ’s are independent of qij , dj6=i (−1)j+i δqij Mij is homogeneous of degree 1 under this same mapping and (tσi + δqii )Mii is independent of qij for j 6= i, we find that     σi Mii σi E 1lΣ =E 1lΣ . det(tA + δQ) ii (tσi + δqii ) ii ‡This

follows from the fact that the pushforward of the probability measure by Q (the probability 2 density) is absolutely continuous with respect to the Lebesgue measure on Cn and the set 2

{Q ∈ Cn : Q = (qij )1≤i,j≤n , |(tσi + δqii )Mii |2 = |

d X j6=i

has Lebesgue measure 0.

(−1)j+i δqij Mij |2 } ,

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T.J. CHRISTIANSEN AND M. ZWORSKI

We do a similar computation for the second term of (3.4):   !−1   Mii σi Mii σi (tσi + δqii ) 1 + Pd 1lΣcii  E 1lΣcii = E  Pd j+i δq M j+i δq M det(tA + δQ) (−1) (−1) ij ij ij ij j6=i j6=i   !k ∞ X M σ (tσ + δq ) i ii ii i = E  Pd − Pd 1lΣcii  = 0 , j+i δq M j+i δq M (−1) (−1) ij ij k=0 ij ij j6=i j6=i using, as before, the invariance properties of Σii and the homogeneity of d X

(−1)j+i δqij Mij .

j6=i

Thus we have (3.6)

−1

E(tr(tA + δQ) A) =

d X

 E

i=1

 σi 1lΣ . (tσi + δqii ) ii

Now, Z 1   Z 1    Z 1  σ σ σ /δ i i i E 1lΣ dt ≤ E dt = E dt (tσi + δqii ) ii |tσi + δqii | |tσi /δ + qii | 0 0 0  Z σi /δ  Z 1 1 σi /δ g(s)ds = E ds = |s + qii | π 0 0 where g is the function defined in Lemma 3.2. Using this, (3.6), and the results of Lemma 3.2 proves the proposition.  Lemma 3.3. Let F, G be d × d matrices, with F invertible, and let β = kF −1 k. Then     

1 −1 −1/4(δβd)2 −1 4 −1/4(dβδ)2 ) +O E tr (F + δQ) G = tr F G 1 + O(e kGkd e . δ The implicit constant in the error term is independent of F and G. Proof. We first note that if we replace F by its singular value decomposition, F = U SV ∗ , then  

 E tr (F + δQ)−1 G = E tr (S + δQ)−1 (U ∗ GV ) and

tr F −1 G = tr S −1 U ∗ GV . Thus we may assume that F is a diagonal matrix.

PROBABILISTIC WEYL LAWS FOR QUANTIZED TORI

19

Our proof then resembles the proof of Proposition 3.1. Let χ ∈ L∞ (R+ ) be the characteristic function of (−∞, 1/2], and, if A = (aij ), let kAksup = supij |aij |. We write   

(3.7) E tr (F + δQ)−1 G = E tr (F + δQ)−1 G χ(dkQksup δβ)  
−1 + E tr (F + δQ) G (1 − χ(dkQksup δβ)) . For the first term, ∞       X
−1 −1 −1 j E tr (F + δQ) G χ(dkQksup δβ) = E tr F (−δ(QF ) G χ(dkQksup δβ)) . 0

Using the fact that the cut-off χ(dkQksup δβ) is invariant under rotations of the qij and that the qij are complex and independent, we find  
E tr (F + δQ)−1 G χ(dkQksup δβ) = tr F −1 G)µ(Q : kQksup < 1/2δβd) (3.8)

2

= tr F −1 G)(1 + O(d2 e−1/4(δβd) )) .

Now we consider the remaining term of (3.7). In a way similar to the proof of Proposition 3.1, we denote the diagonal entries of F by fii = σi , and by Mij the (i, j) minor of F + δQ. If G = (gij ), we have    X (−1)i+j M g 
ji ji E tr (F + δQ)−1 G (1 − χ(dkQksup δβ)) = E (1 − χ(dkQksup δβ)) . det(F + δQ) i,j Just as in the proof of Proposition 3.1, to compute   Mii gii E (1 − χ(dkQksup δβ)) det(F + δQ) we write det(F + δQ) = (σi + δqii )Mii +

X

(−1)j+i δqij Mij

j6=i

and define Σii as in (3.3). Proceeding almost exactly as in the proof of Proposition 3.1, using that both Σii and the support of (1 − χ(dkQksup δβ)) are invariant under the mapping (3.5), we get that     Mii gii gii E (1 − χ(dkQksup δβ)) = E 1lΣ (1 − χ(dkQksup δβ)) . det(F + δQ) (σi + δqii ) ii But   g ii ≤ C kGk d2 e−1/4(dδβ)2 . E 1 l (1 − χ(dkQk δβ)) Σ sup ii (σi + δqii ) δ

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T.J. CHRISTIANSEN AND M. ZWORSKI

To compute  (−1)i+j M g  ji ji E (1 − χ(dkQksup δβ)) det(F + δQ) when i 6= j, we write det(F + δQ) = δqji Mji (−1)i+j + (σi + δqii )Mii +

(3.9)

X

(−1)k+i δqki Mki

k6=i,j

and define ( def

Σji =

q∈C

d2

) X (−1)k+i δqki Mki . : |δqji Mji | > (σi + δqii )Mii + i6=k6=j

Following the proof of Proposition 3.1 but treating the term δqji Mji as the distinguished one in the expansion of the determinant (3.9) and using the invariance of Σji under rotations of qji , we find that   (−1)i+j Mji gji E (1 − χ(dkQksup δβ)) det(F + δQ) ! (−1)i+j Mji gji P =E 1l c (1 − χ(dkQksup δβ)) . (σi + δqii )Mii + k6=i,j (−1)k+i δqki Mki Σji Since on the support of 1lΣcji X 1 k+i |Mji | ≤ (−1) δqki Mki ) (σi + δqii )Mii + δ|qji | k6=i,j we find   i+j (−1) M g ji ji ≤ C kGk d2 e−1/4(dδβ)2 . E (1 − χ(dkQk δβ)) sup det(F + δQ) δ  Our proof of Proposition 4.1 in the next section will use Proposition 3.5. To prove this proposition we will need several preliminary results. The first lemma below follows from well-known facts about eigenvalues of complex Gaussian ensemble. We give a direct proof suggested to us by Mark Rudelson: Lemma 3.4. Let A = (a1 , ..., ad ), with ai ∈ Cd . Then, with the notation of (3.1), Z | det A|−1 dL(A) < ∞ . kAkHS ≤1

PROBABILISTIC WEYL LAWS FOR QUANTIZED TORI

21

Proof. We begin by introducing some more notation. For p ≤ d, p ∈ N, v ∈ Cd denote by Pp v projection onto the subspace spanned (over the complex numbers) by a1 , ..., ap . This of course depends on a1 , ..., ap , but we omit this in our notation for simplicity. Using the Graham-Schmidt process, we can, if A is invertible (as it is off a set of measure 0), write the matrix A = U R, with U a unitary matrix and R being upper triangular. The diagonal entries of R are then given by ka1 k and k(1 − Pp−1 )ap k, p = 2, ..., d. Thus | det A| = ka1 kk(1 − P1 )a2 kk(1 − P2 )a3 k · · · k(1 − Pd−1 )ad k. Note that ka1 kk(1 − P1 )a2 kk(1 − P2 )a3 k · · · k(1 − Pd−2 )ad−1 k is independent of ad , that is, independent of a1d , a2d , ..., add . Therefore Z | det A|−1 dL(A) kAkHS ≤1

Z

1 dL(ad )dL(ad−1 )...dL(a1 ) k(1 − Pd−1 )ad k kAk ≤1 ka1 kk(1 − P1 )a2 k · · · k(1 − Pd−2 )ad−1 k Z HS Z dL(ad−1 )...dL(a1 ) dL(ad ) . ≤ ··· ka1 k≤1 kad k≤1 k(1 − Pd−1 )ad k ka1 kk(1 − P1 )a2 k · · · k(1 − Pd−2 )ad−1 k R The value of kad k≤1 1/k(1 − Pd−1 )ad kdL(ad ) depends only on d and the rank of the space spanned by a1 , ..., ad−1 . We find 1/k(1 − Pd−1 )ad k is locally integrable over R2d ' Cd , because ad ∈ Cd and the space spanned by a1 , ..., ad−1 has complex dimension at most d − 1.Therefore Z 1 (3.10) dL(ad ) ≤ C < ∞. kad k≤1 k(1 − Pd−1 )ad k =

Here the constant C can be chosen independent of a1 , ..., ad−1 , as the maximum of the integral in (3.10) occurs when a1 , ..., ad−1 span a d − 1 dimensional vector space. The proof follows by iterating the above argument.  Proposition 3.5. Let A(s, t) be a d × d matrix depending smoothly on (s, t) ∈ U ⊂ C2 . Let Q denote a d × d random matrix, with each entry an independent complex N (0, 1) random variable. Then for δ > 0, (s, t) ∈ U , E(tr((A(s, t) + δQ)−1 ∂t A) is smooth on U , and

∂s E tr((A(s, t) + δQ)−1 ∂t A) = ∂t E tr((A(s, t) + δQ)−1 ∂s A) . This proposition has the following corollary.

22

T.J. CHRISTIANSEN AND M. ZWORSKI

Corollary 3.6. Let M , B, be d × d matrices independent of s and t. Then Z 1  Z 1 

 −1 E tr (sB + M + δQ) B ds = E tr (B + tM + δQ)−1 M dt 0 0 Z 1  Z 1  

 −1 −1 − E tr (tM + δQ) M dt + E tr (sB + δQ) B ds. 0

0

Proof. Using the previous proposition, this follows from the Fundamental Theorem of Calculus: Z 1  Z 1 

 −1 −1 E tr (sB + M + δQ) B ds − E tr (sB + δQ) B ds 0 0 Z 1 Z 1
= ∂t E tr (sB + tM + δQ)−1 B dsdt 0 0 Z 1 Z 1
= ∂s E tr (sB + tM + δQ)−1 M dtds 0 0 Z 1  Z 1 

 −1 = E tr (B + tM + δQ) M dt − E tr (tM + δQ)−1 M dt. 0

0

 Proposition 3.5 follows from the subsequent two lemmas. Lemma 3.7. Let A(s, t), B(s, t) be d × d matrices depending smoothly on (s, t) ∈ U ⊂ C2 . With Q a random matrix as in Proposition 3.5 and δ > 0,

E tr (A(s, t) + δQ)−1 B(s, t) ∈ C ∞ (U ). Proof. We prove the lemma by writing the expected value as an integral: Z
2 −1 E tr((A + δQ) B) = tr((A + δQ)−1 B)e−kQkHS dL(Q) Z 1 2 = tr((δQ)−1 B)e−kQ− δ AkHS dL(Q). d−1 ˜ | tr((δQ)−1 B)| ˜ ≤ C| det Q|−1 kBkkQk ˜ Now, for a d × d matrix B, /δ, where the constant C depends on d. Moreover,  j+k X 1 kQk 2 j 0 k0 j k −kQ− 1δ Ak2HS |∂s ∂t e | ≤ Cj,k,d ( k∂s ∂t Ak) e−kQ− δ AkHS . 2 δ j 0 ≤j,k0 ≤k

R 1 2 Since, using Lemma 3.4 | det Q|−1 (1 + kQk)m e−kQ− δ AkHS dL(Q) < ∞, for any finite m, the smoothness of A and B proves the lemma. 

PROBABILISTIC WEYL LAWS FOR QUANTIZED TORI

23

If M is an invertible matrix depending smoothly on s and t, then ∂t det M (3.11) tr(M −1 ∂t M ) = and ∂s tr(M −1 Mt ) = ∂t tr(M −1 Ms ). det M The lemma below shows that something similar is true when taking expected values, even though the matrices under consideration are not invertible for some values of the random variable. Lemma 3.8. Let A(s, t) be a d × d matrix depending smoothly on (s, t) ∈ U ⊂ C2 , and Q a random matrix as in Proposition 3.5. Then for δ > 0

∂s E tr((A + δQ)−1 ∂t A) = ∂t E tr((A + δQ)−1 ∂s A) . Proof. Let χ ∈ C ∞ (R) satisfy χ (x) = 1 for |x| < /2 and χ (x) = 0 for |x| > . Then

∂s E tr((A + δQ)−1 ∂t A) = ∂s E χ (det(A + δQ)) tr((A + δQ)−1 ∂t A)


(3.12) + ∂s E 1 − χ (det(A + δQ)) tr (A + δQ)−1 ∂t A . Now


∂s E 1 − χ (det(A + δQ)) tr (A + δQ)−1 ∂t A Z

2 = 1 − χ (det(A + δQ)) ∂s tr (A + δQ)−1 ∂t A e−kQkHS dL(Q) Z

2 − χ0 (det(A + δQ)) ∂s det(A + δQ) tr (A + δQ)−1 ∂t A e−kQkHS dL(Q) where we can freely interchange differentiation and integration since the integrand is smooth and it and its derivatives are integrable. But using (3.11), we get


∂s E 1 − χ (det(A + δQ)) tr (A + δQ)−1 ∂t A Z

2 = 1 − χ (det(A + δQ)) ∂t tr (A + δQ)−1 ∂s A e−kQkHS dL(Q) Z
2 − χ0 (det(A + δQ))∂t det(A + δQ) tr (A + δQ)−1 ∂s A e−kQkHS dL(Q)


= ∂t E 1 − χ (det(A + δQ)) tr (A + δQ)−1 ∂s A . On the other hand, the first term on the right in (3.12) satisfies
lim ∂s E χ (det(A + δQ))(tr((A + δQ)−1 ∂t A) ↓0 Z 2 = lim ∂s χ (det(A + δQ))(tr((A + δQ)−1 ∂t A)e−kQkHS dL(Q) ↓0 Z 1 2 = lim ∂s χ (det(δQ))(tr((δQ)−1 ∂t A)e−kQ− δ AkHS dL(Q) = 0 ↓0

1

2

since (tr((δQ)−1 ∂t A)e−kQ− δ AkHS and its s derivative are both in L1 , using Lemma 3.4. 

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T.J. CHRISTIANSEN AND M. ZWORSKI

4. Reduction to a deterministic problem In this section we will show how to reduce the random problem problem to a deterministic one. That will be done using the singular value decomposition of the matrix fN . Let A be a square matrix, and let U SV ∗ be a singular value decomposition for A. We make the following simple observation: for ψ ∈ Cc∞ (R, R) equal to 1 on [−1, 1], (A + αψ(AA∗ /α2 )U V ∗ )−1 = O(1/α) : `2 −→ `2 ,

(4.1)

which becomes totally transparent by writing ψ(AA∗ /α2 )U V ∗ = U ψ((S/α)2 )V ∗ . The random problem is reduced to a deterministic one by using an operator of the form (4.1). Proposition 4.1. For a smooth curve γ define Z def (4.2) IN (γ) = E tr(fN + δQN − z)−1 dz γ n

n

where QN is a complex N × N matrix, with entries indepent N (0, 1) random variables. Let fN = UN SN VN∗ be a singular value decomposition of fN , and let ψ ∈ Cc∞ (R; [0, 1]) be equal to 1 on [−1, 1]. If 0 ∈ γ , |γ| < α/4 ,

(4.3)

δ  α,

then Z

E tr(fN + δQN (ω) − z)−1 dz = γ Z E tr(fN + αψ(fN fN∗ /α2 )UN VN∗ + δQN (ω) − z)−1 dz + E1 = γ Z tr(fN + αψ(fN fN∗ /α2 )UN VN∗ − z)−1 dz + E2 ,

(4.4)

γ

where  E1 , E2 = O d log

(4.5)

α δ

N 4n −α2 /4(3N n δ)2 e + δ

 ,

and d = rank 1lsupp ψ (fN fN∗ /α2 ). The proof of this proposition will use the following lemma. Lemma 4.2. Let fN , UN , SN , VN , ψ, δ, d, and α be as in the statement of Proposition 4.1. Let χ ∈ L∞ (R) be the characteristic function for the support of ψ. Then, if |z| ≤ α/4, Z 1
∗ 2 ∗ −1 ∗ 2 ∗ E tr (fN + sαχ(fN fN /α )UN VN − z + δQ) αχ(fN fN /α )UN VN ds 0

satisfies the bound (4.5).

PROBABILISTIC WEYL LAWS FOR QUANTIZED TORI

Proof. First suppose that for a  A˜11 ˜ (4.6) A= A˜21

˜ m × m matrix A,   A˜12 −1 ˜ and A = A˜22

˜11 B ˜12 B ˜21 B ˜22 B

25



˜11 d × d matrices and A˜11 , B ˜11 (m − d) × (m − d) matrices. Then if A˜22 is with A˜11 , B invertible, we have the Schur complement formula,  −1 −1 ˜ ˜ ˜ ˜ ˜ (4.7) B11 = A11 + A12 A22 A21 , see [18] for a review of some of its applications in spectral theory. We note, using ψ(AA∗ /α2 )U V ∗ = U ψ((S/α)2 )V ∗ and the unitarity of UN , VN ,
(4.8) E tr (fN + sαχ(fN fN∗ /α2 )UN VN∗ − z + δQN )−1 αχ(fN fN∗ /α2 )UN VN∗
∗ ∗ = E tr (SN + sαχ(SN SN /α2 ) − UN∗ zVN + δQN )−1 αχ(SN SN /α2 ) . The main idea of the proof will be to effectively reduce the dimension of the matrices we work with, from N n to d. We can assume that UN , VN , SN are chosen so that the diagonal elements σ1 , ..., σN n of SN satisfy σ1 ≤ σ2 · ·· ≤ σN n . Let J denote projection onto the 2 2 range of χ(SN /α2 ), which is the same as projection off of the kernel of χ(SN /α2 ). Then   Id 0 J = . 0 0 2 and αχ(SN /α2 ) takes the form



αId 0 0 0

 .

We also write SN + and

∗ sαχ(SN SN /α2 )

−

UN∗ zVN

 =

sαId + A11 A12 A21 A22

 ,



 Q11 Q12 QN = Q21 Q22 where A11 , Q11 are d × d-dimensional matrices, and A22 , Q22 are (N n − d) × (N n − d)dimensional. Since SN is diagonal and |z| ≤ α/4, we have kA12 k ≤ α/4, kA21 k ≤ α/4. Using this notation, we have that A22 is invertible, with norm at most 4/3α. Now restrict QN to the set with (4.9)

δkQN − J QN J ksup ≤ αN −n /4.

Note that poses no restriction on Q11 . For such QN , A22 + δQ22 is invertible, with norm at most 2/α. Restricting to this set of QN and using (4.7), we find  
∗ 2 ∗ −1 ∗ 2 tr (SN + sαχ(SN SN /α ) − UN zVN ) αχ(SN SN /α ) = trd α(sαId + Md + δQ11 )−1 ,

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T.J. CHRISTIANSEN AND M. ZWORSKI

where we use the notation trd to emphasize we are taking the trace of a d × d matrix, and where Md = A11 − (A12 + δQ12 )(A22 + δQ22 )−1 (A21 + δQ21 ) is a d × d matrix depending on Q12 , Q21 , and Q22 , but not on Q11 . Since kA11 k = kJ (SN − zUN∗ VN )J k ≤ Cα and kAk12 ≤ α/4, kAk21 ≤ α/4, we have kMd k ≤ Cα, for a new constant C independent of N , δ, and QN satisfying (4.9). Next we take the expected value in the Q11 variables only: Z 2 EQ11 (F (QN )) = F (QN )e−kQ11 kHS dL(Q11 ). Q11 ∈Cd2

Still requiring QN to satisfy (4.9), which is not a restriction on Q11 , and using Corollary 3.6, we get Z 1 
 EQ11 α trd Md + sαId + δQ11 )−1 ds = 0 Z 1 Z 1  

 −1 −1 EQ11 trd tMd + αId + δQ11 ) Md dt − EQ11 trd sαId + δQ11 ) α ds 0 0 Z 1 
 + EQ11 trd tMd + δQ11 )−1 Md dt. 0

Recalling that kMd k ≤ Cα we see from Proposition 3.1 that the second and third terms on the right are O(d log(α/δ)), if α/δ > e. Moreover, kMd − J SN J k ≤

α , 2

and SN ≥ 0. Therefore, for 0 ≤ t ≤ 1, αId + tMd is invertible, with the inverse having norm at most 3/α. Thus from Lemma 3.3 we see that Z 1   4 
 d 2 /4(3dδ)2 −α −1 e . EQ11 trd tMd + αId + δQ11 ) Md dt = O(d) + O δ 0 The implicit constants in both cases are independent of Q − J QJ satisfying (4.9). Thus we get (4.10) Z 1  
2 2 ∗ −1 2 2 α E tr (SN + sαχ(SN /α ) + δQ − zUN VN ) αχ(SN /α ) 1l{δkQ−J QJ ksup ≤ 4N n } ds 0   2 2 = O(d log(α/δ)) + O d4 δ −1 e−α /4(d3δ) where for a set E, 1lE is the characteristic function of E.

PROBABILISTIC WEYL LAWS FOR QUANTIZED TORI

27

Exactly as in the proof of the Lemma 3.3, we can show that  
2 2 2 2 E tr (SN + sαχ(SN α ) + δQN − zUN∗ VN )−1 αχ(SN α ) 1l{δkQ−J QJ ksup >α/(4N n )}   (4.11) 4n −1 −α2 /(4N n δ)2 =O N δ e . Using (4.8), (4.10), and (4.11), we prove the lemma.



We now use Lemma 4.2 in a preliminary step towards proving Proposition 4.1. Lemma 4.3. Let fN , UN , SN , VN , ψ, δ, d, and α be as in the statement of Proposition 4.1, and set χ = 1lsupp ψ . Then Z Z −1 E tr(fN + δQN − z) dz = E tr(fN + αχ(fN fN∗ /α2 )UN VN∗ + δQN − z)−1 dz γ γ   4n   α  N −α2 /4(3N n δ)2 e + O d log . +O δ δ Proof. The proof uses the same type of argument as Corollary 3.6. Using the Fundamental Theorem of Calculus, Z Z −1 E tr(fN + δQN − z) dz − E tr(fN + αχ(fN fN∗ /α2 )UN VN∗ + δQN − z)−1 dz γ

γ

Z =−

1

Z ∂s

0

Z =−

Z ∂z

E tr(fN + sαχ(fN fN∗ /α2 )UN VN∗ + δQN − z)−1 dzds

γ 1


E tr(fN + sαχ(fN fN∗ /α2 )UN VN∗ + δQN − z)−1 αχ(fN fN∗ /α2 ) dsdz

0

γ

where we use Proposition 3.5. The right hand side is X Z 1
∓ E tr (fN + sαχ(fN fN∗ /α2 )UN VN∗ + δQN − z± )−1 αχ(fN fN∗ /α2 ) ds ±

0

where z± are the endpoints of γ. Then using Lemma 4.2 finishes the proof. We are now able to give a straightforward proof of Proposition 4.1. Proof of Proposition 4.1. We begin by noting that, with χ = 1lsupp ψ k(fN + αχ(fN fN∗ /α2 )UN VN∗ − z)−1 k = O(1/α) and k(fN + αψ(fN fN∗ /α2 )UN VN∗ − z)−1 k = O(1/α)



28

T.J. CHRISTIANSEN AND M. ZWORSKI

when |z| ≤ α/4. Moreover, the rank of χ(fN fN∗ /α2 ) is d and the rank of ψ(fN fN∗ /α2 ) is at most d, and both operators have norm at most 1. Then Z
∗ 2 ∗ −1 ∗ 2 ∗ −1 tr(fN + αχ(fN fN /α )UN VN − z) − tr(fN + αψ(fN fN /α )UN VN − z) dz γ Z
= α tr (fN + αχ(fN fN∗ /α2 )UN VN∗ − z)−1 χ(fN fN∗ /α2 ) − ψ(fN fN∗ /α2 ) γ
×(fN + αψ(fN fN∗ /α2 )UN VN∗ − z)−1 dz Z Cd ≤ dz = O(d). γ α Thus, applying Lemmas 4.3 and 3.3 proves the Proposition.



5. Proof of Theorem The proof of Theorem will be deduced from the following local result: Proposition 5.1. Under the assumption of the main theorem, let γ ⊂ ∂Ω be a connected segment of length |γ| ≤

(5.1)

1 1 α , h= , α = hρ , 0 < ρ < C 2πN 2

and let IN (γ) be as defined by (4.2). Then for exp (−h− ) < δ < hp0 , we have Z Z n (5.2) IN (γ) = N (f (w) − z)−1 dL(w)dz + O(|γ|h−n+ρ(κ−1)−2 ) + O(|γ|h−n+1−2ρ ) , γ

T2n

where we note that (1.2) with κ > 1 implies that (f (w) − z)−1 ∈ L1 (T2n ) so that the first term on the right hand side makes sense. Assuming the proposition we easily give the Proof of Theorem. We divide ∂Ω into J = C 0 /α disjoint segments γj , |γj | ≤ α/C. Proposition 5.1 implies that  Z  X J −1 E tr (fN + δQN − z) = IN (γj ) = ∂Ω

Nn

Z ∂Ω

Z

j=1

(f (w) − z)−1 dL(w)dz + O(h−n+ρ(κ−1)−2 ) + O(h−n+1−2ρ ) .

T2n

We now choose ρ = 1/(κ + 1), to optimize the error, that is to arrange, ρ(κ − 1) = 1 − 2ρ. That means that the error is O(N n−β ) for any β < 1 − 2ρ = (κ − 1)/(κ + 1).

PROBABILISTIC WEYL LAWS FOR QUANTIZED TORI

29

Hence Eω |Spec (fN + N

−p

Z
1 QN (ω)) ∩ Ω| = E tr(fN + N −p QN (ω) − z)−1 dz 2πi ∂Ω Z Z 1 dL(w) n = N dz + O(N n−β ) 2πi ∂Ω f (w) − z 2n Z ZT 1 dz = Nn dL(w) + O(N n−β ) 2πi f (w) − z 2n T ∂Ω = N n volT2n (f −1 (Ω)) + O(N n−β ) ,

which is the statement of the theorem. Proof of Proposition 5.1. Without loss of generality we can assume that 0 ∈ γ. From Proposition 4.1 we already know that IN (γ) can be approximated by a deterministic expression Z def (5.3) IeN (γ) = tr(fN + αψ(fN fN∗ /α2 )UN VN∗ − z)−1 dz , γ n

with, if α/δN  0  α   −co α/N n δ ˜ IN (γ) − IN (γ) = O e + d log , δ for some c0 > 0, where d is the rank of ψ(fN fN∗ /α2 ). We choose α as in (5.1), α = hρ , where 1 1 h= , 0 n + 1/2, IN (γ) − I˜N (γ) = O(h−n++κρ + exp(−c0 hn−p0 +ρ ) = O(|γ|h−n+(κ−1)ρ− ) . Thus we will prove (5.2) by showing that (5.4)

tr(fN +

αψ(fN fN∗ /α2 )UN VN∗

−1

− z)

=N

n

Z

(f (w) − z)−1 dL(w)

T2n

+ O(h−n+1−2ρ ) + O(h−n+ρ(κ−1) ) . We first show that it is enough to consider z = 0. In fact, let UN (z)SN (z)VN (z)∗ be the singular value decomposition of fN − z, and put def

BN (z, w) = (fN − w + αψ((fN − z)(fN − z)∗ /α2 )UN (z)VN∗ (z))−1 . Then tr (BN (z, z) − BN (0, z)) =

α tr BN (0, z) ψ(fN fN∗ /α2 )UN VN∗ − ψ((fN − z)(fN − z)∗ /α2 ))UN (z)VN∗ (z) BN (z, z) .

30

T.J. CHRISTIANSEN AND M. ZWORSKI

Since rank ψ((fN − z)(fN − z)∗ /α2 )) = O(h−n+κρ ) for z ∈ γ, and B(z, w) = O`2 →`2 (1/α) for |z − w| ≤ α/C 0 , we obtain tr (BN (z, z) − BN (0, z)) = O(h−n+ρ(κ−1) ) , which can be absorbed in the error on the right hand side of (5.4). Thus we only need to prove (5.4) with the left hand side replaced by B(z, z) and we can simply take z = 0. In other words we now want to prove (5.5) tr(fN +

αψ(fN fN∗ /α2 )UN VN∗ )−1

=N

n

Z T2n

dL(w) + O(h−n+1−2ρ ) + O(h−n+ρ(κ−1) ) . f (w)

The difficulty lies in the fact that the operators fN + αψ(fN fN∗ /α2 )UN VN∗ do not seem to have a nice microlocal characterization. We are helped by the following identity: if ψ˜ ∈ Cc∞ (R, [0, 1]) is equal to 1 on the support of ψ then (5.6)

˜ ∗ fN /α2 ))(fN + αψ(fN f ∗ /α2 )UN V ∗ )−1 = (1 − ψ(f N N N ∗ 2 ∗ ∗ 2 ∗ ˜ (1 − ψ(f fN /α ))f (fN f + α ψ(fN f /α2 ))−1 . N

N

N

N

This is a consequence of an identity from linear algebra: Lemma 5.2. Let A be a matrix and U SV ∗ be its singular value decomposition. If ψ, ψ˜ ∈ Cc∞ (R; [0, 1]), ψ is equal to 1 on [−1, 1], and ψ˜ is equal to 1 on the support of ψ, then (5.7)

˜ ∗ A))(A + ψ(AA∗ )U V ∗ )−1 = (1 − ψ(A ˜ ∗ A))A∗ (AA∗ + ψ(AA∗ ))−1 . (1 − ψ(A

Proof. We first note that ˜ ∗ A) = V ψ(S ˜ 2 )V ∗ , A∗ A = V S 2 V ∗ , ψ(A ˜ ≡ 0, we and similarly ψ(AA∗ ) = U ψ(S 2 )U ∗ . Since S is a diagonal matrix, and (1 − ψ)ψ get ˜ ∗ A))(A + ψ(AA∗ )U V ∗ )−1 = V (1 − ψ(S ˜ 2 ))V ∗ V (S + ψ(S 2 ))−1 U ∗ (1 − ψ(A ˜ 2 ))(S + ψ(S 2 ))−1 U ∗ = V (1 − ψ(S ˜ 2 ))S(S 2 + ψ(S 2 ))−1 U ∗ = V (1 − ψ(S  
2 ∗ ˜ = V (1 − ψ(S ))V (V SU ∗ ) U (S 2 + ψ(S 2 ))−1 U ∗ ˜ ∗ A))A∗ (AA∗ + ψ(AA∗ ))−1 , = (1 − ψ(A concluding the proof.



The identity (5.6) follows from (5.7) by putting A = fN /α, U = UN , and V = VN . Using this we we will find a new expression for the left hand side of (5.5) so that the identification with the right hand side will follow from a suitable semiclassical operator calculus.

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31

Lemma 5.3. We have the following approximation for the left hand side of (5.5): (5.8) tr(fN + αψ(fN fN∗ /α2 )UN VN∗ )−1 = tr fN∗ (fN fN∗ + α2 ψ(fN fN∗ /α2 ))−1 + O(h−n+ρ(κ−1) ) . Proof. We use (5.6) and first note that 1 − ψ˜ can be removed from the left hand side since (5.9)

˜ ∗ fN /α2 )(fN + αψ(fN f ∗ /α2 )UN V ∗ )−1 = tr ψ(f N N N  

∗ 2 ∗ 2 ∗ −1 ˜ O (rank ψ fN fN /α )k(fN + αψ(fN fN /α )UN VN ) k = O h−n+ρ(κ−1) .

The same argument works for the right hand side once we observe that fN∗ (fN fN∗ + α2 ψ(fN fN∗ /α2 ))−1 = O`2 →`2 (1/α) , and this follows from using the singular value decomposition since for non-negative diagonal matrices 2 2 SN (SN + α2 ψ(SN /α2 ))−1 ≤ 1/α .  In view of (5.5) and the lemma we have to prove Z dL(w) ∗ ∗ 2 ∗ 2 −1 n (5.10) tr fN (fN fN + α ψ(fN fN /α )) = N + O(h−n+1−2ρ ) + O(h−n+(κ−1)ρ ) , f (w) 2n T but that follows from the calculus developed in §2. In fact, with the α-order function m(w, α) = α2 + |f (w)|2 , given in Lemma 2.6, fN fN∗ + α2 ψ(fN fN∗ /α2 ) = TN , T ∈ S(m, α) , T = T0 + h1−2ρ T1 , T0 (w) = |f (w)|2 + α2 ψ(|f (w)|2 /α2 ) , T1 ∈ S(m, α) . where we also applied Lemma 2.8. We also have T0 ≥ m/2 and hence 1/T0 ∈ S(1/m, α) , 1/T ∈ S(1/m, α) . √ Since f ∈ S( m, α), we conclude that √ fN∗ (fN fN∗ + α2 ψ(fN fN∗ /α2 ))−1 = PN , P ∈ S(1/ m, α) , √ f¯(w) . P = P0 + h1−2ρ P1 , P1 ∈ S(1/ m) , P0 (w) = |f (w)|2 + α2 ψ(|f (w)|2 /α2 ) We now apply Lemma 2.5 and obtain (with k  n) Z Z ∗ ∗ 2 ∗ 2 −1 n −k+n tr fN (fN fN + α ψ(fN fN /α )) = N P (w)dL(w) + O(N ) sup |∂ β P |dL |β|≤k

T2n

= Nn

Z T2n

P0 (w)dL(w) + O(h−n+(1−2ρ) + h−n+k(1−ρ) )

Z T2n

m(w, α)−1/2 dL(w)

32

T.J. CHRISTIANSEN AND M. ZWORSKI

We have m(w, α)−1/2 ≤ |f (w)|−1 and (1.2) at z = 0 with κ > 1 implies that |f (w)|−1 is integrable (κ = 1 would mean that |f (w)|−1 is in weak L1 ): Z Z ∞ Z ∞ −1 −2 |f (w)| dL(w) = L({|f (w)| < t})t dt = O(min(tκ , 1))t−2 dt < ∞ . T2n

0

It remains to show that Z (5.11)

0

|P0 (w) − f (w)−1 |dL(w) = O(hρ(κ−1) ) .

T2n def

Putting ϕ(x) = ψ(x2 ), we rewrite the left hand side above as   Z ∞ Z ∞ −α2 (t/α)ϕ0 (t/α) −α2 ϕ(t/α) dt = dt L({|f (w)| < t}) L({|f (w)| < t})∂t t(t2 + α2 ϕ(t/α) t2 (t2 + α2 ϕ(t/α)) 0 0 Z ∞ α2 ϕ(t/α)(3t2 + α2 ϕ(t/α) + α2 (t/α)ϕ0 (t/α)) + L({|f (w)| < t}) dt t2 (t2 + α2 ϕ(t/α))2 0 Z 2α ≤ C tκ−2 dt = C 0 ακ−1 , 0

which is (5.11). Since we have now established (5.10) this also completes the proof of Proposition 5.1. References ´ 67(1988), 5–42. [1] E. Bierstone and P.D. Milman, Semianalytic and subanalytic sets, Publ. IHES, [2] W. Bordeaux Montrieux, personal communication. [3] W. Bordeaux Montrieux and J. Sj¨ ostrand, Almost sure Weyl asymptotics for non-self-adjoint elliptic operators on compact manifolds, arXiv:0903.2937. [4] D. Borthwick, Introduction to K¨ ahler quantization, In First Summer school in analysis and mathematical physics, Cuernavaca, Mexico. Contemporary Mathematics series 260, AMS(2000), 91–132. [5] D. Borthwick and A. Uribe, On the pseudospectra of Berezin-Toeplitz operators, Methods and Applications of Analysis 10 (2003), 31–65. [6] A. Bouzouina and S. De Bi`evre, Equipartition of the eigenfunctions of quantized ergodic maps on the torus, Commun. Math. Phys. 178 (1996) 83–105. [7] S.J. Chapman and L.N. Trefethen, Wave packet pseudomodes of twisted Toeplitz matrices, Comm. Pure Appl. Math. 57 (2004), 1233–1264. [8] N. Dencker, J. Sj¨ ostrand, and M. Zworski, Pseudospectra of semi-classical (pseudo)differential operators, Comm. Pure Appl. Math., 57 (2004), 384–415. [9] M. Dimassi and J. Sj¨ ostrand, Spectral Asymptotics in the semi-classical limit, Cambridge University Press, 1999. [10] L.C. Evans and M. Zworski, Lectures on Semiclassical Analysis, http://math.berkeley.edu/∼zworski/semiclassical.pdf [11] P. Forrester, N. Snaith, and V. Verbaarschot, Introduction Review to Special Issue on Random Matrix Theory, Jour. Physics A: Mathematical and General 36 (2003), R1-R10. [12] M. Hager, unpublished, 2007. [13] M. Hager and J. Sj¨ ostrand, Eigenvalue asymptotics for randomly perturbed non-selfadjoint operators. Math. Ann. 342 (2008), no. 1, 177–243.

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[14] N. Mehta, Random Matrices, 3rd Edition, Elsevier, Amsterdam, 2004. [15] S. Nonnenmacher and M. Zworski, Distribution of resonances for open quantum maps, Comm. Math. Phys. 269 (2007), 311–365. [16] E. Scheck, Weyl laws for partially open quantum maps, Annales Henri Poincar´e, 10 (2009), 714–747. [17] J. Sj¨ ostrand, Eigenvalue distribution for non-self-adjoint operators on compact manifolds with small multiplicative random perturbations, arXiv:0809.4182. [18] J. Sj¨ ostrand and M. Zworski, Elementary linear algebra for advanced spectral problems, Ann. Inst. Fourier (Grenoble) 57 (2007), 2095-2141. [19] J. Sj¨ ostrand and M. Zworski, Fractal upper bounds on the density of semiclassical resonances, Duke Math. J. 137 (2007), 381–459. [20] M. Zworski, Numerical linear algebra and solvability of partial differential equations, Comm. Math. Phys. 229 (2002), 293–307. Department of Mathematics, University of Missouri, Columbia, Missouri 65211, USA E-mail address: [email protected] Mathematics Department, University of California, Evans Hall, Berkeley, CA 94720, USA E-mail address: [email protected]