Quantum Unique Ergodicity for maps on the torus

0.2 MB·English

by  Lior Rosenzweig

QUANTUM UNIQUE ERGODICITY FOR MAPS ON
THE TORUS
LIOR ROSENZWEIG
5
0
0 Abstract. When a map is classically uniquely ergodic, it is ex-
2
pectedthatitsquantizationwillpossesquantumuniqueergodicity.

n In this paper we give examples of Quantum Unique Ergodicityfor
a the perturbed Kroneckermap, andanupper bound for the rate of
J
convergence.
0
3

3
v
4
1. Introduction
4
0
1.1. Background. One of the problems in Quantum Chaos is the as-
1
0 ymptotic behavior of the expectation value in eigenstates. When quan-
5 tizing classical dynamics on a phase space one constructs a Hilbert
0
space of states , , and an algebra of operators , the algebra of ”quan-
/ h
h H
tum observables”, that assigns for each smooth function on the phase
p
- space f an operator Op (f) where h implies dependence on Planck’s
h h
constant h, and the dynamics is quantized to a unitary time evolution
t
a
operator, U on . For any orthonormal basis of eigenfunctions of
m h h
H
U , ψ , the expectation value of Op (f) in the eigenstate ψ is given
v: byh {Opj}(f)ψ ,ψ . The semiclassicalhlimit of these is the limjit where
i h h j ji
X h 0. When the classical dynamics of a system is ergodic, it is known
→
r that the time average of the trajectories of the system converges to the
a
space average. An analogue of this is given by Schnirelman’s Theorem
[12],[13],[1], which states that for an ergodic system the expectation
values of Op(f) converges to the phase space average of f, for all but
possibly a zero density subsequence of eigenfunctions. This is referred
to as quantum ergodicity. The case where there are no exceptional
subsequences is referred to as quantum unique ergodicity (QUE).
When the phase space is T2 = R2/Z2 it is required that each state
will be periodic in both position and momentum and thus Planck’s
constant is restricted to be an inverse of an integer h = 1, and the
N
Hilbert space is of dimension N, namely L2(Z/NZ). The semiclassical
limit in this case is the limit where N . Given a continuous map
→ ∞
A on T2, we define its quantization as a sequence of unitary operators
1
2 LIORROSENZWEIG
on L2(Z/NZ), U (A) satisfying
N
(1) U (A)−1Op (f)U (A) Op (f A) 0 as N
k N N N − N ◦ k −→ → ∞
for all f C∞(T2), where f A(p,q) = f(A(p,q)). This is an analogue
∈ ◦
of Egorov’s Theorem, and the eigenfunctions of U (A) are analogues
N
of eigenmodes.
A first example of QUE was given on the 2-torus T2,by Marklof
and Rudnick [9], where the classical dynamics is an irrational skew
translation, that is classically uniquely ergodic. For this map they
found that for generic translations, the rate of convergence is O(N41+ǫ).
Afamousexample of a quantization ofa mapis oflinear automorphism
of T2 called the ”CAT map”,([6],[3]), that is if A SL(2,Z). If trA >
∈ | |
2 that is if A is hyperbolic, then the map is known to be ergodic, but
not uniquely ergodic. In this case it was shown that there is no QUE
([4]), butthereexists aspecialbasis(Hecke Basis) forwhich QUEholds
([7]). In this case the rate of convergence was shown to be O(N14+ǫ),
andisconjectured tobeO(N21+ǫ). (It wasshown thatinthecasewhere
N = p where p is a prime number the rate of convergence is O(p1/2)
[5]).
In this paper we will give a family of more examples of QUE on the
2-torus, all of them are also classically uniquely ergodic, and study the
rate of convergence.
1.2. QUE for maps on the torus. The map in this paper will be
the perturbed Kronecker map, that is
Φα : T2 T2
V →
p p+α
Φα : 1 mod 1
V q 7→ q +α +V(p)
2
(cid:18) (cid:19) (cid:18) (cid:19)
where α = (α ,α ), and V(p) is a smooth function of zero mean on T.
1 2
The special case where V(p) = 0 (the standard Kronecker map) plays
a central role here. It is known that in this case the map is uniquely
ergodic if and only if 1,α ,α are linearly independent over Q. We
1 2
will construct a quantization of it by approximating α with rational
numbers a = (a1,a2). For rational numbers we have an exact Egorov
N N
theorem, that is
U−1 Op (f)U = Op (f τ )
a,N N a,N N ◦ a/N
and thus by the convergence of a to α we will get (1). For this map
N
we have the following theorem for polynomials:
QUANTUM UNIQUE ERGODICITY FOR MAPS ON THE TORUS 3
Theorem 1.1. Suppose 1,α ,α are linearly independent over Q. Let
1 2
f C∞(T2) be a trigonometric polynomial. Then for all eigenfunctions
∈
ψ of U (τ ) we have that for N sufficiently large
N α
Op (f)ψ,ψ = f(p,q)dpdq
h N i
T2
Z
For the more general case of smooth functions we assume a certain
restriction on α. We assume that α satisfy a certain diophantine in-
equality, that is there exists γ > 0 such that for all n ,n ,k Z
1 2
∈
(2) n α +n α +k (n ,n ) −γ (n ,n ) = (0,0)
1 1 2 2 1 2 1 2
| | ≫ k k 6
This reduces the set of numbers rather than being all α such that
1,α ,α are linearly independent over Q to a set of almost all α in
1 2
Lebesgue measure sense, and γ can be any number strictly bigger than
2 (see theorem (3.8)[10]. If α ,α are algebraic of degree d ,d respec-
1 2 1 2
tively we can choose γ to be d !d ! ([11]). For these α we have,
1 2
Theorem 1.2. Suppose 1,α ,α are linearly independent over Q and
1 2
satisfy (2) then for all f C∞(T2) ,for all eigenfunctions ψ of U (τ )
N α
∈
Op (f)ψ,ψ f(p,q)dpdq N−θ θ > 0
|h N i− | ≪ ∀
T2
Z
Ourmainresult isfortheperturbedKronecker mapΦα,forarbitrary
V
smooth V(p). We show that the map is also uniquely ergodic. In fact
we show that it is conjugate to τ and we also have QUE for it, and
α
give an upper bound for the rate of convergence:
Theorem 1.3. Suppose 1,α ,α are linearly independent over Q and
1 2
satisfy (2) then for all f C∞(T2) ,for all eigenfunctions ψ of U (Φα)
∈ N V
Op (f)ψ,ψ f(p,q)dpdq N−2
|h N i− | ≪
T2
Z
Thus for such α the rate of convergence of the matrix elements to
their classical average ismuch faster thatthe expected andknown rates
mentioned earlier on the irrational skew translation and the CAT map.
We also construct special pairs (α ,α ) and functions f(p,q) for which
1 2
the rate of convergence is arbitrarily slow (Theorems 3.3,3.16).
acknowledgments
ThisworkwassupportedinpartbytheECTMRnetworkMathemat-
ical Aspects of Quantum Chaos,EC-contract no HPRN-CT-2000-00103
and the Israel Science Foundation founded by the Israel Academy of
Sciences and Humanities. This work was carried out as part of the
author’s Msc thesis at Tel Aviv University, under the supervision of
4 LIORROSENZWEIG
Professor Zeev Rudnick. A substantial part of this work was done
during the author’s visit to the university of Bristol.
2. Background
We begin with a quantization procedure for maps on the 2-torus T2.
The procedure can be find in full description in [7],[2]. We construct a
Hilbert space of state with respect to Planck’s constant h, quantum
h
H
observables, and a quantization of our maps.
2.1. Notations. We abbreviate e(x) = e2πix, and e (x) = e(x). A
N N ≪
B or A = O(B) both means that there is a constant c such that A
| | ≤
c B .
| |
2.2. Hilbert space of state. Our classical phase space is T2. The
elements of the Hilbert space are thus, distribution on the line R that
are periodic in both position and momentum. Using the momentum
representation of a wave-function ψ by the Fourier transform
1 ∞ qp
ψ(p) = ψ(q)e(− )dq
h
F √h h
Z−∞
we find that the requirements
ψ(q +1) = ψ(q) ψ(p) = ψ(p+1)
h h
F F
restricts planck’s constant h to be an inverse of integer h = 1, and
N Hh
consists of periodic point-mass distributions at the coordinates Q =
q . We therefore find that the Hilbert space is of dimension N, and
N
therefore denote , and we may identify it with L2(Z/NZ), with the
N
H
inner product
1
¯
ψ,φ = ψ(Q)φ(Q)
h i N
Q modN
X
The Fourier transform is given by
1
ˆ
ψ(P) = [ ψ](P) = ψ(Q)e ( QP)
N N
F √N −
Q modN
X
and its inverse formula is
1
ψ(Q) = −1ψˆ (Q) = ψˆ(P)e (PQ)
FN √N N
h i P XmodN
QUANTUM UNIQUE ERGODICITY FOR MAPS ON THE TORUS 5
2.3. Quantum observables. We now assign each classical observable
,smooth functions f C∞(T2), a quantum observable, that is an op-
∈
erator Op (f) on that satisfy,
N HN
(1) Op (f¯) = Op (f)∗
N N
(2) Op (f)Op (g) Op (fg) as N
N N ∼ N → ∞
(3) 1 [Op (f),Op (g)] Op ( f,g ) as N
2πiN N N ∼ N { } → ∞
where [A,B] = AB BA is the commutator, and f,g = ∂f ∂g ∂g∂p
− { } ∂p∂q −∂p∂q
are the Poisson bracket. The norm used is the induced norm from the
inner product on .
N
H
The translation operators
[t ψ](Q) = ψ(Q+1)
1
and
[t ψ](Q) = e (Q)ψ(Q)
2 N
play a special role, they are analogues of the of the differentiation and
multiplication operators. Heisenberg’s commutation relations are
tatb = tbtae (ab) a,b Z
1 2 2 1 N ∀ ∈
Notice that
t = t
N 1 N 2
F F
and
t = t−1
FN 2FN 1
With these operators we construct
n n
T (n) = e ( 1 2)tn2tn1,n = (n ,n ) Z2
N N 2 2 1 1 2 ∈
whose action on a wave-function ψ is
N
∈ H
iπn1n2
TN(n)ψ(Q) = e N eN(n2Q)ψ(Q+n1)
Notice that
T (n)∗ = T ( n)
N N
−
ω(m,n)
(3) T (m)T (n) = e ( )T (m+n)
N N N N
2
where, ω(m,n) = m n m n , and that T is a unitary operator.
1 2 2 1 N
−
Finally for a general smooth function
ˆ
f(x) = f(n)e(n x)
·
n∈Z2
X
6 LIORROSENZWEIG
where x = (p,q). we define its quantization Op (f)
N
ˆ
(4) Op (f) = f(n)T (n)
N N
n∈Z2
X
and the conditions mentioned are all satisfied.
3. Quantization of maps and rate of convergence
When quantizing a map, we look for a sequence of unitary opera-
tors U (A) on , the quantum propagator, whose iterates give the
N N
H
evolution of the quantum system, and that in the semiclassical limit,
(the limit as N or h ), the quantum evolution follows the
→ ∞ → ∞
classical evolution as described in the following definition.
Definition 3.1 (”Egorov’s Theorem”). A quantization of a continuous
map A : T2 T2 is a sequence of unitary operators , U , satisfying:
N
→ { }
(5) U−1Op (f)U Op (f A) 0 as N
k N N N − N ◦ k → → ∞
The stationary states of the quantum system are given by the eigen-
functions ψ of U (A). We will find that for the maps studied in this
N
paperthelimitingexpectationvalueofobservablesinnormalizedeigen-
states converges to the classical average of the observable, that is
Op (f)ψ,ψ f as N
h N i → → ∞
T2
Z
3.1. Quantizing Kronecker map. In this section we will construct
a quantization of the Kronecker map.
τ : T2 T2
α
→
p p+α
1 mod 1
q 7→ q +α +V(p)
2
(cid:18) (cid:19) (cid:18) (cid:19)
Lemma 3.1. suppose (a1,a2) is a sequence of rational numbers such
N
that (a1N,a2) = N~a−N−→−−→∞α~ then the sequence UN(τα) := TN(−a2,a1) is a
quantization of Kronecker’s map.
ˆ
Proof. First assume f(x) = e (z) := e(n z) in this case we get f(n) =
n
·
ˆ
1,f(m) = 0 for m = n ,and therefore Op (f) = T (n).
6 N N
Denote a˜ := ( a ,a ),and notice that n a = ω(n,˜a). Now
2 1
− ·
U (τ )−1T (n)U (τ ) = T ( a˜)T (n)T (a˜)
N α N N α N N N
−
which due to (3) and the linearity and antisymmetry of ω(m,n)
(6) e (ω(n,˜a))T (n) = e (n a))T (n)
N N N N
·
QUANTUM UNIQUE ERGODICITY FOR MAPS ON THE TORUS 7
on the other hand,we have
(e τ )(x) = e(n (p+α )+n (q +α )) = e(n α~)e (x)
n α 1 1 2 2 n
◦ ·
and so
(7) Op (e τ ) = e(n α~)T (n)
N n ◦ α · N
From (6),(7) we get that
U−1(τ )T (n)U (τ ) e(n α)T (n) = e (n ~a) e (n α~) T (n)
k N α N N α − · N k | N · − N · |·k N k
T is a unitary operator so T (n) = 1 we get
N N
k k
~a ~a
e (n ) e (n α~) n α~
N N
| · N − · | ≪ k k| − N|
Therefore we established (5) for f = e (x). By linearity we also have
n
(5) for trigonometric polynomials. suppose now that f(x) is a general
function of C∞(T2) and therefore
ˆ
f(x) = f(n)e (x)
n
n∈Z2
X
Consider
U−1(τ )Op (f)U (τ ) Op (f A) =
k N α N N α − N ◦ k
U−1(τ ) fˆ(n)T (n) U (τ ) fˆ(n)e(n α)T (n) =
k N α { N } N α − · N k
n∈Z2 n∈Z2
X X
ˆ ˆ
f(n) e (n a) e(n α) T (n) f(n) e(n a) e(n α) T (n)
N N N
k { · − · } k ≤ | |·| · − · |·k k
n∈Z2 n∈Z2
X X
and therefore
~a ~a
U−1(τ )Op (f)U (τ ) Op (f A) = α~ n fˆ(n) = O( α~ )
k N α N N α − N ◦ k | − N| k k | − N|
n∈Z2
X
which goes to zero since α~ ~a 0 as N implying that U is
| − N| → → ∞ N
a quantization of τ .
α
Remark 3.1. Notice that for each N, we have exact Egorov for τ ,
a/N
that is
U−1(τ )Op (f)U (τ ) = Op (f τ )
N a/N N N a/N N ◦ a/N
(cid:3)
8 LIORROSENZWEIG
3.2. Convergence of eigenstates. We now wish to give an upper
bound for the remainder
(8) Op (f)ψ,ψ f
|h N i− |
ZT2
where ψ is an eigenfunction of U . Actually we will prove the following
N
two theorems:
Theorem3.2. Suppose1,α ,α are linearlyindependentoverQ. Then
1 2
For any eigenfunction ψ(Q) of U
N
(1) If f is a polynomial then for N large enough,
Op (f)ψ,ψ = f
h N i
ZT2
(2) if α = (α ,α ) is diophantine (see definition 3.2) and α~ ~a
1 2 | −N| ≪
1 then for all f C∞(T2)
N ∈
1
Op (f)ψ,ψ f = O( ) , θ > 0
h N i− Nθ ∀
ZT2
Theorem 3.3. For any positive increasing function g(x), there exists
α = (α ,α ) such that 1,α ,α are linearly independent over the ratio-
1 2 1 2
nals ,f C∞(T2), and a basis of eigenfunctions ψ N such that
∈ { j}j=1
1
Op (f)ψ ,ψ f
|h N j ji− | ≫ g(N)
ZT2
Remark 3.2. The set of all diophantine pairs is of Lebesgue measure
1 (see theorem 3.8). An example for such pairs are α = (α ,α ) such
1 2
that α ,α are algebraic and 1,α ,α are linearly independent over Q
1 2 1 2
(see theorem 3.7).
To prove these theorems we will start with the following lemma:
Lemma 3.4. Let ψ(Q) to be an eigenfunctions of U .
N
(1)
(9) Op (f)ψ,ψ = Op (fT)ψ,ψ
h N i h N i
where
T−1
1
fT(p,q) = f τt
T ◦ (a/N)
t=0
X
(2) For f(x) = e (x), Op (f)ψ,ψ is identically zero for large
n h N i
enough N.
QUANTUM UNIQUE ERGODICITY FOR MAPS ON THE TORUS 9
Proof. (1) Since ψ is an eigenfunction of U then U ψ = e(φ)ψ,
N N
and therefore for all t
Op (f)Ut ψ,Ut ψ = e(tφ)Op (f)ψ,e(tφ)ψ = Op (f)ψ,ψ
h N N N i h N i h N i
Now,
Op (f)Ut ψ,Ut ψ = U−tOp (f)Ut ψ,ψ
h N N N i h N N N i
and since
U−tOp (f)Ut = Op (f τt )
N N N N ◦ a/N
we have (9).
(2) fix ~n = (n ,n ) Z2 , f(x) = e (x) and therefore
1 2 n
∈
Op (f) = T (n). Notice that for f = e we have,
N N n
T−1 T−1
1 1
fT = e τt = e(n (p+ta /N)+n (q +ta /N)) =
T n ◦ (a/N) T 1 1 2 2
t=0 t=0
X X
T−1
1
e (p,q) e ((n a +n a )t)
n N 1 1 2 2
T
t=0
X
and for T = N we have,
f if n a +n a = 0 (mod N)
(10) fN = 2 2 1 1
0 else
(
and therefore,
Op (f) if n a +n a = 0 (mod N)
(11) Op (fN) = N 2 2 1 1
N 0 else
(
but
a a
n a +n a = Nk n 2 +n 1 = k Z
2 2 1 1 2 1
⇐⇒ N N ∈
~a ~a
n α +O( α~ ) +n α +O( α~ ) = k Z
2 2 1 1
⇐⇒ { | − N| } { | − N| } ∈
and so we get
~a
(12) n α +n α +O( n α~ ) = k Z
2 2 1 1
k k| − N| } ∈
α ,α arelinearly independent over Q so we candenote 0 < δ =
1 2
dist(n α + n α ,Z). Now assume that there exists infinitely
1 1 2 2
many pairs ~a = (a ,a ) such that (8) is nonzero i.e. n a +
1 2 2 2
n a = Nk .From (12) we get that
1 1 ~a
~a
(13) O( n α~ ) = k +n α +n α > δ > 0,N
2 2 1 1
k k| − N| | | → ∞
10 LIORROSENZWEIG
now since n is fixed and α~ ~a 0 as N we get a
| − N| → → ∞
contradiction! so we can deduce that for N n
≫ k k
Op (f)ψ,ψ f 2 = T (n)ψ,ψ = 0
|h N i− | |h N i|
T2
Z
(cid:3)
Corollary 3.5 (QUE for Kronecker map). For any eigenfunction ψ of
U ,
N
(1) if f is a trigonometric polynomial, Op (f)ψ,ψ = forlarge
h N i T2
enough N.
R
(2) For any f C∞(T2),
∈
Op (f)ψ,ψ f 0 as N
|h N i− | → → ∞
ZT2
Proof. (1) From the previous lemma we get that every trigonomet-
ric function has N such that (8) is identically zero so for a finite
linear combination
m
a e(n x)
n
·
n=1
X
simply choose the largest N given from e (x),n = 1,...,m
n
(2) For a general f C∞(T2), we have
∈
ˆ
Op (f) = f(n)T (n)
N N
n∈Z2
X
For ǫ > o, there exists R , such that R > R ,
0 0
∀
ˆ
f(n) < ǫ
| |
knk>R
X
For the polynomial
ˆ
P = f(n)e(n x)
R
·
knk<R
X
there exists N , such that for all N > N
0 0
Op (P )ψ,ψ = 0
h N R i
and so we have ,
Op (f)ψ,ψ 6
|h N i|
Op (P )ψ,ψ + fˆ(n) T (n)ψ,ψ 6 ǫ
|h N R i| | h N i|
knk>R
X
for N > N .
0