On the distribution of matrix elements for the quantum cat map

0.24 MB·English

by  P. Kurlberg

ON THE DISTRIBUTION OF MATRIX ELEMENTS
FOR THE QUANTUM CAT MAP
4
0
0 PA¨R KURLBERG AND ZEE´V RUDNICK
2

n
a Abstract. Formanyclassicallychaoticsystemsitisbelievedthat
J the quantumwavefunctions become uniformlydistributed, thatis

0 thematrixelementsofsmoothobservablestendtothephasespace
1
average of the observable. In this paper we study the fluctuations

of the matrix elements for the desymmetrized quantum cat map.
]
T We presentaconjectureforthedistributionofthe normalizedma-
N trix elements, namely that their distribution is that of a certain
. weighted sum of traces of independent matrices in SU(2). This
h
is in contrast to generic chaotic systems where the distribution
t
a is expected to be Gaussian. We compute the second and fourth
m
moment of the normalized matrix elements and obtain agreement
[ with our conjecture.

3
v
7
7
1. Introduction
2
2
Afundamentalfeatureofquantumwavefunctionsofclassicallychaotic
0
3 systems is that the matrix elements of smooth observables tend to the
0
phase space average of the observable, at least in the sense of conver-
/
h gence in the mean [15, 2, 17] or in the mean square [18]. In many
t
a systems it is believed that in fact all matrix elements converge to the
m
micro-canonical average, however this has only been demonstrated for
:
v a couple of arithmetic systems: For “quantum cat maps” [10], and
Xi conditional on the Generalized Riemann Hypothesis1 also for the mod-
ular domain [16], in both cases assuming that the systems are desym-
r
a
metrized by taking into account the action of “Hecke operators”.
As for the approach to the limit, it is expected that the fluctuations
of the matrix elements about their limit are Gaussian with variance
given by classical correlations of the observable [7, 5]. In this note we
Date: January 10, 2004.
This work was supported in part by the EC TMR network “Mathematical
aspects of Quantum Chaos” (HPRN-CT-2000-00103). P.K. was also supported in
part by the NSF (DMS 0071503), the Royal Swedish Academy of Sciences and
the Swedish Research Council. Z.R. was also supported in part by the US-Israel
Bi-National Science Foundation.
1
An unconditional proof was recently announced by Elon Lindenstrauss.
1
2 PA¨R KURLBERGANDZEE´V RUDNICK
study these fluctuations for the quantum cat map. Our finding is that
for this system, the picture is very different.
We recall the basic setup [8, 3, 4, 10] (see section 2 for further back-
ground and any unexplained notation): The classical mechanical sys-
tem is the iteration of a linear hyperbolic map A SL(2,Z) of the
∈
torus T2 = R2/Z2 (a “cat map”). The quantum system is given by
specifying an integer N, which plays the role of the inverse Planck con-
stant. In what follows, N will be restricted to be a prime. The space
of quantum states of the system is = L2(Z/NZ). Let f C∞(T2)
N
H ∈
be a smooth, real valued observable and Op (f) : its quan-
N HN → HN
tization. The quantization of the classical map A is a unitary map
U (A) of .
N N
H
In [10] we introduced Hecke operators, a group of commuting uni-
tary maps of , which commute with U (A). The space has an
N N N
H H
orthonormal basis consisting of joint eigenvectors ψ N of U (A),
{ j}j=1 N
whichwecallHeckeeigenfunctions. Thematrixelements Op (f)ψ ,ψ
h N j ji
converge2 to the phase-space average f(x)dx [10]. Our goal is to
T2
understand their fluctuations around their limiting value.
R
Ourmainresultistopresentaconjectureforthelimitingdistribution
of the normalized matrix elements
F(N) := √N Op (f)ψ ,ψ f(x)dx .
j h N j ji−
T2
(cid:18) Z (cid:19)
For this purpose, define a binary quadratic form associated to A by
a b
Q(x,y) = cx2 +(d a)xy by2, A =
− − c d
(cid:18) (cid:19)
For an observable f C∞(T2) and an integer ν, set
∈
f#(ν) := ( 1)n1n2f(n)
−
n=(nX1,n2)∈Z2
Q(n)=ν b
where f(n) are the Fourier coefficients of f.
Conjecture 1. As N through primes, the limiting distribution
b → ∞
of the normalized matrix elements F(N) is that of the random variable
j
X := f#(ν)tr(U )
f ν
ν6=0
X
where U are independently chosen random matrices in SU(2) endowed
ν
with Haar probability measure.
2For arbitrary eigenfunctions, that is ones which are not Hecke eigenfunctions,
this need not hold, see [6].
MATRIX ELEMENTS FOR QUANTUM CAT MAPS 3
This conjecture predicts a radical departure from the Gaussian fluc-
tuations expected to hold for generic systems [7, 5]. Our first result
confirms this conjecture for the variance of these normalized matrix
elements.
Theorem 2. As N through primes, the variance of the normal-
→ ∞
ized matrix elements F(N) is given by
j
N
1
(1.1) F(N) 2 E(X2) = f#(ν) 2 .
N | j | → f | |
j=1 ν6=0
X X
For a comparison with the variance expected for the case of generic
systems, seeSection6.1. Asimilar departurefromthisbehaviour ofthe
variance was observed recently by Luo and Sarnak [12] for the modular
domain. For another analogy with that case, see section 6.2.
(N)
We also compute the fourth moment of F and find agreement
j
with Conjecture 1:
Theorem 3. The fourth moment of the normalized matrix elements is
given by
N
1
F(N) 4 E( X 4) = 2 f#(ν) 4
N | j | → | f| | |
j=1 ν6=0
X X
as N through primes.
→ ∞
Acknowledgements: We thank Peter Sarnak for discussions on his
work with Wenzhi Luo [12].
2. Background
The full details on the cat map and its quantization can be found
in [10]. For the reader’s convenience we briefly recall the setup: The
classical dynamics are given by a hyperbolic linear map A SL(2,Z)
∈
so that x = (p) T2 Ax is a symplectic map of the torus. Given
q
∈ 7→
an observable f C∞(T2), the classical evolution defined by A is
∈
f f A, where (f A)(x) = f(Ax).
7→ ◦ ◦
For doing quantum mechanics on the torus, one takes Planck’s con-
stant to be 1/N and as the Hilbert space of states one takes :=
N
H
L2(Z/NZ), where the inner product is given by
1
φ,ψ = φ(Q)ψ(Q).
h i N
QmodN
X
4 PA¨R KURLBERGANDZEE´V RUDNICK
The basic observables are given by the operators T (n), n Z2,
N
∈
acting on ψ L2(Z/NZ) via:
∈
iπn1n2 n2Q
(2.1) (TN(n1,n2)ψ)(Q) = e N e( )ψ(Q+n1).
N
where e(x) = e2πix.
For any smooth classical observable f C∞(T2) with Fourier ex-
∈
pansion f(x) = f(n)e(nx), its quantization is given by
n∈Z2
P Op (f) := f(n)T (n) .
bN N
n∈Z2
X
b
2.1. Quantum Dynamics: ForAwhich satisfies a certain parity con-
dition, we can assign unitary operators U (A), acting on L2(Z/NZ),
N
having the following important properties:
“Exact Egorov”: For all observables f C∞(T2)
• ∈
U (A)−1Op (f)U (A) = Op (f A).
N N N N ◦
The quantization depends only on A modulo 2N: If A B
• ≡
mod 2N then U (A) = U (B).
N N
The quantization is multiplicative: if A,B are congruent to the
•
identity matrix modulo 4 (resp., 2) if N is even (resp., odd),
then [10, 13]
U (AB) = U (A)U (B)
N N N
2.2. Heckeeigenfunctions. Letα,α−1 betheeigenvaluesofA. Since
A is hyperbolic, α is a unit in the real quadratic field K = Q(α). Let
O = Z[α], which is an order of K. Let v = (v ,v ) O2 be a vector
1 2
∈
a b
such that vA = αv. If A = , we may take v = (c,α a). Let
c d −
(cid:18) (cid:19)
I := Z[v ,v ] = Z[c,α a] O. Then I is an O-ideal, and the matrix
1 2
− ⊂
of α acting on I by multiplication in the basis v ,v is precisely A. The
1 2
choice of basis of I gives an identification I = Z2 and the action of O
∼
on the ideal I by multiplication gives a ring homomorphism
ι : O Mat (Z)
2
→
with the property that det(ι(β)) = (β), where : Q(α) Q is the
N N →
norm map.
Let C(2N) be the elements of O/2NO with norm congruent to 1
mod 2N, and which congruent to 1 modulo 4O (resp., 2O) if N is even
(resp.,odd). Reducing ι modulo 2N gives a map
ι : C(2N) SL (Z/2NZ).
2N 2
→
MATRIX ELEMENTS FOR QUANTUM CAT MAPS 5
Since C(2N) is commutative, the multiplicativity of our quantization
implies that
U (ι (β)) : β C
N 2N
{ ∈ }
forms a family of commuting operators. Analogously with modular
forms, we call these Hecke operators, and functions ψ that are si-
N
∈ H
multaneous eigenfunctions ofalltheHecke operatorsaredenoted Hecke
eigenfunctions. Note that a Hecke eigenfunction is an eigenfunction of
U (ι (α)) = U (A).
N 2N N
The matrix elements are invariant under the Hecke operators:
Op (f)ψ ,ψ = Op (f B)ψ ,ψ , B C(2N)
h N j ji h N ◦ j ji ∈
Thisfollowsfromψ beingeigenfunctionsoftheHeckeoperatorsC(2N).
j
In particular, taking f(x) = e(nx) we see that
(2.2) T (n)ψ ,ψ = T (nB)ψ ,ψ .
N j j N j j
h i h i
2.3. The quadratic form associated to A: We define a binary qua-
a b
dratic form associated to A = by
c d
(cid:18) (cid:19)
Q(x,y) = cx2 +(d a)xy by2
− −
This, uptosign,isthequadraticform (xc+y(α a))/ (I)induced
N − N
by the norm form on the ideal I = Z[c,α a] described in Section 2.2,
where (I) = #O/I. Indeed, since I =−Z[c,α a] and O = Z[1,α]
N −
we have (I) = c . A computation shows that the norm form is then
N | |
sign(c)Q(x,y).
By virtue of the definition of Q as a norm form, we see that A
and the Hecke operators are isometries of Q, and since they have unit
norm they actually land in the special orthogonal group of Q. That is
we find that under the above identifications, C(2N) is identified with
B SO(Q,Z/2NZ) : B I mod 2 .
{ ∈ ≡ }
2.4. A rewriting of the matrix elements. We now show that when
ψ is a Hecke eigenfunction, the matrix elements Op (f)ψ,ψ have a
h N i
modified Fourier series expansion which incorporates some extra in-
variance properties.
Lemma 4. If m,n Z2 are such that Q(m) = Q(n), then for all
∈
sufficiently large primes N we have m nB mod N for some B
≡ ∈
SO(Q,Z/NZ).
Proof. We may clearly assume Q(m) = 0 because otherwise m = n = 0
6
since Q is anisotropic over the rationals. We take N a sufficiently
large odd prime so that Q is non-degenerate over the field Z/NZ. If
N > Q(m) then Q(m) = 0 mod N and then the assertion reduces
| | 6
6 PA¨R KURLBERGANDZEE´V RUDNICK
to the fact that if Q is a non-degenerate binary quadratic form over
the finite field Z/NZ (N = 2 prime) then the special orthogonal group
6
SO(Q,Z/NZ) acts transitively on the hyperbolas Q(n) = ν , ν = 0
mod N. { } 6 (cid:3)
Lemma 5. Fix m,n Z2 such that Q(m) = Q(n). If N is a suffi-
∈
ciently large odd prime and ψ a Hecke eigenfunction, then
( 1)n1n2 T (n)ψ,ψ = ( 1)m1m2 T (m)ψ,ψ
N N
− h i − h i
Proof. For ease of notation, set ǫ(n) := ( 1)n1n2. By Lemma 4 it
−
suffices to show that if m nB mod N for some B SO(Q,Z/NZ)
≡ ∈
then ǫ(n) T (n)ψ,ψ = ǫ(m) T (m)ψ,ψ .
N N
h i h i
By the Chinese Remainder Theorem,
SO(Q,Z/2NZ) SO(Q,Z/NZ) SO(Q,Z/2Z)
≃ ×
(recall N is odd) and so
C(2N) B SO(QZ/2NZ) : B I mod 2 SO(Q,Z/NZ) I
≃ { ∈ ≡ } ≃ ×{ }
Thus if m nB mod N for B SO(Q,Z/NZ) then there is a unique
≡ ∈
B˜ C(2N) so that m nB˜ mod N.
∈ ≡
We note that ǫ(n)T (n) has period N, rather than merely 2N for
N
T (n) as would follow from (2.1). Then since m = nB˜ mod N,
N
ǫ(m)T (m) = ǫ(nB˜)T (nB˜) = ǫ(n)T (nB˜)
N N N
˜ ˜ ˜
(recall that B C(2N) preserves parity: nB n mod 2, so ǫ(nB) =
∈ ≡
ǫ(n)). Thus for ψ a Hecke eigenfunction,
ǫ(m) T (m)ψ,ψ = ǫ(n) T (nB˜)ψ,ψ = ǫ(n) T (n)ψ,ψ
N N N
h i h i h i
(cid:3)
the last equality by (2.2).
Define for ν Z
∈
f#(ν) := ( 1)n1n2f(n)
−
n∈Z2:Q(n)=ν
X
and b
(2.3) V (ψ) := √N( 1)n1n2 T (n)ψ,ψ
ν N
− h i
where n Z2 is a vector with Q(n) = ν (if it exists) and set V (ψ) = 0
ν
∈
otherwise. By Lemma 5 this is well-defined, that is independent of the
choice of n. Then we have
Proposition 6. If ψ is a Hecke eigenfunction, f a trigonometric poly-
nomial, and N N (f), then
0
≥
√N Op (f)ψ,ψ = f#(ν)V (ψ)
h N i ν
ν∈Z
X
MATRIX ELEMENTS FOR QUANTUM CAT MAPS 7
To simplify the arguments, in what follows we will restrict ourself to
dealing with observables that are trigonometric polynomials.
3. Ergodic averaging
We relate mixed moments of matrix coefficients to traces of certain
averages of the observables: Let
1
(3.1) D(n) = T (nB)
N
C(2N)
| | B∈C(2N)
X
The following shows that D(n) is essentially diagonal when expressed
in the Hecke eigenbasis.
Lemma 7. Let D˜ be the matrix obtained when expressing D(n) in
terms of the Hecke eigenbasis ψ N . If N is inert in K, then D˜ is
{ i}i=1
diagonal. If N splits in K, then D˜ has the form
D D 0 0 ... 0
11 12
D D 0 0 ... 0
21 22
 
0 0 D 0 ... 0
D˜ = 33
0 0 0 D ... 0
 44 
 ... ... ... ... ... ... 
 
 0 0 0 0 ... D 
 NN
 
where ψ ,ψ correspond to the quadratic character of C(2N). More-
1 2
over, in the split case, we have
D N−1/2
ij
| | ≪
for 1 i,j 2.
≤ ≤
Proof. If N is inert, then the Weil representation is multiplicity free
when restricted to C(2N) (see Lemma 4 in [9].) If N is split, then
C(2N) is isomorphic to (Z/NZ)∗ and the trivial character occurs with
multiplicity one, the quadratic character occurs with multiplicity two,
and all other characters occur with multiplicity one (see [11], section
˜
4.1). This explains the shape of D.
As for the bound on in the split case, it suffices to take f(x,y) =
e(n1x+n2y) for some n ,n Z. We may give an explicit construction of
N 1 2 ∈
theHecke eigenfunctions asfollows(see [11], section4 formoredetails):
there exists M SL (Z/2NZ) such that the eigenfunctions ψ ,ψ can
2 1 2
∈
be written as
N
ψ = √N U (M)δ , ψ = U (M)(1 δ )
1 N 0 2 N 0
· N 1 · −
r −
8 PA¨R KURLBERGANDZEE´V RUDNICK
where δ (x) = 1 if x 0 mod N, and δ (x) = 0 otherwise. Setting
0 0
≡
φ = √Nδ and φ = N (1 δ ), exact Egorov gives
1 0 2 N−1 − 0
q
D = T ((n ,n ))ψ ,ψ = T ((n′,n′))φ ,φ
ij h N 1 2 i ji h N 1 2 i ji
where (n′,n′) (n ,n )M mod N. Since we may assume n not to
1 2 ≡ 1 2
be an eigenvector of A modulo N, we have n′ 0 mod N and n′ 0
1 6≡ 2 6≡
mod N. Hence
n′n′
D = T ((n′,n′))φ ,φ = e( 1 2)δ (0+n′) = 0
11 h N 1 2 1 1i 2N 0 1
since n′ 0 mod N. The other estimates are analogous. (cid:3)
1 6≡
Remark: In the split case, it is still true that D N−1/2 for all
ij
≪
i,j, but this requires the Riemann hypothesis for curves, whereas the
above is elementary.
Lemma 8. Let ψ N be a Hecke basis of , and let k,l,m,n Z2.
{ i}i=1 HN ∈
Then
N
T (m)ψ ,ψ T (n)ψ ,ψ = tr D(m)D∗(n) +O(N−1)
N i i N i i
h ih i
i=1
X (cid:0) (cid:1)
Moreover,
N
T (k)ψ ,ψ T (l)ψ ,ψ T (m)ψ ,ψ T (n)ψ ,ψ
N i i N i i N i i N i i
h ih ih ih i
i=1
X
= tr D(k)D∗(l)D(m)D∗(n) +O(N−2)
Proof. By definition(cid:0) (cid:1)
N N
T (m)ψ ,ψ T (n)ψ ,ψ = D(m) D(n)
N i i N i i ii ii
h ih i
i=1 i=1
X X
On the other hand, by lemma 7,
N
tr D(m)D(n)∗ = D (m)D (n)+D (m)D (n)+ D (m)D (n)
12 21 21 12 ii ii
i=1
(cid:0) (cid:1) X
where D (m),D (m),D (n) and D (n) are all O(N−1/2). Thus
12 21 12 21
N
T (m)ψ ,ψ T (n)ψ ,ψ = tr D(m)D(n)∗ +O(N−1)
N i i N i i
h ih i
i=1
X (cid:0) (cid:1)
(cid:3)
The proof of the second assertion is similar.
MATRIX ELEMENTS FOR QUANTUM CAT MAPS 9
4. Proof of Theorem 2
In order to prove Theorem 2 it suffices, by Proposition 6, to show
that as N ,
→ ∞
N
1 1 if µ = ν,
V (ψ )V (ψ ) E trU trU =
ν j µ j ν µ
N → 0 if µ = ν,
(
j=1 6
X (cid:0) (cid:1)
where U ,U SU are random matrices in SU , independent if ν = µ.
µ ν 2 2
∈ 6
Proposition 9. Let ψ N be a Hecke basis of . If N N (µ,ν)
{ i}i=1 HN ≥ 0
is prime and µ,ν 0 mod N, then
6≡
1 N 1+O(N−1) if µ = ν,
V (ψ )V (ψ ) =
N ν j µ j O(N−1) otherwise.
(
j=1
X
Proof. Choose m,n Z2 such that Q(m) = µ and Q(n) = ν. By (2.3)
∈
and Lemma 8 we find that
N N
1
V (ψ )V (ψ ) = ( 1)m1m2+n1n2 T (n)ψ ,ψ T (m)ψ ,ψ
ν j µ j N j j N j j
N − h ih i
j=1 j=1
X X
= ( 1)m1m2+n1n2tr D(n)D(m)∗ +O(N−1)
−
By definition of D(n) we have
(cid:0) (cid:1)
1
D(n)D(m)∗ = T (nB )T (mB )∗ .
C(2N) 2 N 1 N 2
| | B1,BX2∈C(2N)
We now take the trace of both sides and apply the following easily
checked identity (see (2.1)), valid for odd N and B ,B C(2N):
1 2
∈
( 1)m1m2+n1n2N if nB mB mod N,
tr(T (nB )T (mB )∗) = − 1 ≡ 2
N 1 N 2
0 otherwise.
(
We get
N
1
(4.1) V (ψ )V (ψ ) =
ν j µ j
N
j=1
X
( 1)m1m2+n1n2
= − ( 1)m1m2+n1n2N +O(N−1)
C(2N) 2 −
| | B1,BX2∈C(2N)
nB1≡mB2 modN
N
= B C(2N) : n mB mod N +O(N−1)
C(2N) ·|{ ∈ ≡ }|
| |
which, since C(2N) = N 1, equals 1+O(N−1) if there exists B
| | ± ∈
C(2N) such thatn mB mod N, andO(N−1) otherwise. Finally, for
≡
10 PA¨R KURLBERGANDZEE´V RUDNICK
N sufficiently large (i.e., N N (µ,ν)), Lemma 4 gives that n mB
0
mod N for some B C(2N)≥is equivalent to µ = ν. ≡ (cid:3)
∈
5. Proof of theorem 3
5.1. Reduction. In order to prove Theorem 3 it suffices to show that
N
1
(5.1) V (ψ )V (ψ )V (ψ )V (ψ ) E trU trU trU trU
κ j λ j µ j ν j κ λ µ ν
N →
j=1
X (cid:0) (cid:1)
where U ,U ,U and U are independent random matrices in SU .
κ λ µ ν 2
Let S Z4 be the set of four-tuples (κ,λ,µ,ν) such that κ = λ,µ =
⊂
ν, or κ = µ,λ = ν, or κ = ν,λ = µ, but not κ = λ = µ = ν.
Proposition10. Let ψ N be a Hecke basis of and let κ,λ,µ,ν
{ i}i=1 HN ∈
Z. If N is a sufficiently large prime, then
2+O(N−1) if κ = λ = µ = ν,
N
1
V (ψ )V (ψ )V (ψ )V (ψ ) = 1+O(N−1) if (κ,λ,µ,ν) S,
κ j λ j µ j ν j 
N ∈
Xj=1 O(N−1/2) otherwise.
Given Proposition 10 it is straightforward to deduce (5.1), we need

only to note that E (trU)4 = 2, E (trU)2 = 1, and E trU = 0.
TheproofofProposition10willoccupytheremainder ofthissection.
(cid:0) (cid:1) (cid:0) (cid:1) (cid:0) (cid:1)
For the reader’s convenience, here is a brief outline:
(1) Express the left hand side of (5.1) an exponential sum.
(2) Show that the exponential sum is quite small unless pairwise
equality of κ,λ,µ,ν occurs, in which case the exponential sum
is given by the number of solutions (modulo N) of a certain
equation.
(3) Determine the number of solutions.
5.2. Ergodic averaging.
Lemma 11. Choosek,l,m,n Z2 such thatQ(k) = κ,Q(l) = λ,Q(m) =
∈
µ, and Q(n) = ν. Then
1 N N2
(5.2) V (ψ )V (ψ )V (ψ )V (ψ ) =
N κ j λ j µ j ν j C(2N) 4·
j=1 | |
X
t(ω(kB , lB )+ω(mB , nB ))
1 2 3 4
e − −
· N
B1,B2,BX3,B4∈C(N) (cid:18) (cid:19)
kB1−lB2+mB3−nB4≡0 modN