THE ARITHMETIC THEORY OF QUANTUM MAPS

ZEÉV RUDNICK

In these lectures I describe in detail the quantization of linear symplectic maps of the torus, as a continuation of De Bievre’s lectures [4]. I will then survey the problem of quantum equidistribution for this model. This model was introduced by Hannay and Berry [7]. It turns out that it has a rich arithmetic structure, and its study uses several ingredients in modern number theory.

Contents

1. Quantum mechanics on the torus 1 1.1. Quantum states 1 1.2. Observables 2 1.3. Dynamics 3 1.4. The Stone-von Neumann theorem 4 2. Quantizing cat maps 5 3. Quantum ergodicity 7 4. Quantum Unique Ergodicity 9 5. Arithmetic QUE 10 5.1. Symmetries and Hecke operators 10 5.2. Equidistribution of Hecke eigenfunctions 10 5.3. Higher dimensions 11 References 11

1. Quantum mechanics on the torus

We review the basics of quantum mechanics on the torus 𝕋, viewed as a phase space, compare De Bievre’s lectures in this volume [4]. In what follows we use the abbreviations e(z) := e^{2πiz}, e_{N}(z) := e^{2πiz/N}.

1.1. Quantum states. We start with a description of the Hilbert space of states of such a system. In brief, Planck’s constant is restricted to be an inverse integer: h = 1/N, and the Hilbert space of states H_{N} is N-dimensional. The “state vectors” are distributions on the line which are periodic in both momentum and position representations: ψ(q + 1) = ψ(q), [F_{h}ψ](p + 1) = [F_{h}ψ](p), where [F_{h}ψ](p) = h^{−1/2} ∫ ψ(q) e(−pq/h) dq. The space of such distributions is finite dimensional, of dimension precisely N = 1/h, and consists of periodic point-masses at the coordinates q = Q/N, Q ∈ ℤ. We may then identify H_{N} with the N-dimensional vector space L^{2}(ℤ/Nℤ), with the inner product ⟨ · , · ⟩ defined by

Date: June 29, 2006.

⟨ϕ, ψ⟩ = \frac{1}{N} ∑_{Q mod N} ϕ(Q) \overline{ψ}(Q) .

1.2. Observables. Classical observables, that is real-valued functions f ∈ C^{∞}(𝕋), give rise to quantum observables, that is self-adjoint operators Op_{N}(f) on H_{N}. To define these, one starts with the translation operators

[Q̂ψ](Q) = e_{N}(Q) ψ(Q), position

and

[P̂ψ](Q) = ψ(Q + 1), momentum

which may be viewed as the analogues of differentiation and multiplication (respectively) operators. In fact in terms of the usual position and momentum operators on the line q̂ψ(q) = qψ(q) and p̂ψ(q) = \frac{h}{2πi} \frac{d}{dq}ψ(q), they are given by Q̂ = e(q̂), P̂ = e(p̂). In this context, Heisenberg’s commutation relations read

(1.1)  P̂^{a}Q̂^{b} = Q̂^{b}P̂^{a}e_{N}(ab) ∀a, b ∈ ℤ.

More generally, mixed translation operators are defined for n = (n_{1}, n_{2}) ∈ ℤ^{2} by

T_{N}(n) = e_{N}(\frac{n_{1}n_{2}}{2} )Q̂^{n_{2}} P̂^{n_{1}}.

These are unitary operators on H_{N}, whose action on a wave-function ψ ∈ H_{N} is given by:

T_{N}(n)ψ(Q) = e^{\frac{iπn_{1}n_{2}}{N}} e(\frac{n_{2}Q}{N})ψ(Q + n_{1}) .

As follows from the commutation relation (1.1), we have

(1.2)  T_{N}(m) T_{N}(n) = e_{N}(\frac{ω(m, n)}{2}) T_{N}(m + n)

where ω is the symplectic form ω(m, n) = m_{1}n_{2} − m_{2}n_{1}.

For any smooth function f ∈ ℂ^{∞}(𝕋), define a quantum observable Op_{N}(f), called the Weyl quantization of f, by

Op_{N}(f) = ∑_{n∈ℤ^{2}} f̂(n)T_{N}(n)

where f̂(n) are the Fourier coefficients of f.

We list some basic properties of the quantized observables Op_{N}(f):

(1) The adjoint is given by

(1.3)  Op_{N}(f)^{∗} = Op_{N}(f̄) .

(2) The composition of operators satisfies:

(1.4)  Op_{N}(f) Op_{N}(g) = Op_{N}(fg) + O_{f,g}(\frac{1}{N})

for f, g ∈ C^{∞}(𝕋).

(3) For any orthonormal basis of H_{N} we have

(1.5)  \frac{1}{N} ∑_{j=1}^{N}⟨Op_{N}(f)ψ_{j}, ψ_{j}⟩ = ∫_{𝕋} f + O_{f}(\frac{1}{N}) .

That is, the mean of the expectation values is asymptotic to the classical average of the the observable f.

Note: There is a straightforward extension of this model to quantum mechanics on tori 𝕋^{2g} of even dimension.

1.3. Dynamics. To introduce dynamics, we consider a smooth, areapreserving (symplectic) map Φ of the torus. Iterating Φ we get a discrete dynamical system. Our primary example is that of a linear automorphism Φ ∈ SL_{2}(ℤ). Then the system is ergodic and in fact mixing if Φ is hyperbolic, that is | tr Φ| > 2. Such a map is called a “cat map” in the physics literature.

Definition 1. A quantization of Φ is a sequence of unitary maps U_{N} : H_{N} → H_{N} such that

(1.6)  U_{N}^{∗} Op_{N}(f)U_{N} − Op_{N}(f ∘ Φ) → 0, N → ∞ .

The operator U_{N} is called the quantum propagator, whose iterates give the evolution of the quantum system, and we require the quantum evolution to be asymptotic to the classical evolution as N → ∞ (this is an analogue of “Egorov’s theorem”). In this case we say that the map Φ is “quantizable”.

In the example of the linear map Φ, one can construct a unitary operator U_{N}(Φ) which satisfies an exact version of Egorov’s theorem:

(1.7)  U_{N}(Φ)^{∗} Op_{N}(f)U_{N}(Φ) = Op_{N}(f ∘ Φ) .

This will be done in §2 below.

1.4. The Stone-von Neumann theorem. We recall the Canonical Commutation Relations (CCRN): For ζ_{N} = e^{2πi/N},

P̂Q̂ = ζ_{N} Q̂P̂ (1.8)

P̂^{N} = I = Q̂^{N} (1.9)

Lemma 1.1. P̂, Q̂ act irreducibly on H_{N}

Proof. first note that the action of P̂, Q̂ on δ_{j} is given by

(1.10)  Q̂δ_{j} = ζ_{N}^{j}δ_{j},  P̂δ_{j} = δ_{j−1}

Now take a subspace 0 ≠ V ⊆ H_{N} preserved by P̂, Q̂. Then Q̂ has an eigenvector in V. Since the only eigenvectors of Q̂ are δ_{j} (since they have distinct eigenvalues!), there is some j so that δ_{j} ∈ V. But then P̂^{i}δ_{j} = δ_{j−i} is still in V, so as i runs through ℤ/Nℤ we get all the vectors δ_{k} in V, hence V = H_{N}. □

Theorem 1.2. If Q′, P′ are unitary operators on H_{N} satisfying the canonical commutation relations (1.8) then there is a unitary operator U for which

(1.11)  Q′ = U^{∗}Q̂U,  P′ = U^{∗}P̂U

and U is unique up to a scalar multiple.

Proof. Since Q^{′N} = I, all eigenvalues of Q′ are N-th roots of unity. Let

V_{j} = {v ∈ H_{N} : Q′v = ζ_{N}^{j}v}

be the eigenspace corresponding to eigenvalue ζ_{N}^{j} (possibly V_{j} = 0). If V_{j} ≠ 0, then we claim that P′V_{j} ⊂ V_{j−1}: Indeed, if v ∈ V_{j} then

Q′(P′v) = ζ_{N}^{−1}P′Q′v = ζ_{N}^{−1}P′ζ_{N}^{j}v = ζ_{N}^{j−1}(P′v)

which means P′v ∈ V_{j−1}. Since P′ is unitary, it is in particular one-to-one so P′ : V_{j} → V_{j−1} is one-to-one. If 0 ≠ v_{j} ∈ V_{j} is a unit vector then we find v_{j−k} := (P′)^{k}v_{j} ≠ 0 is a unit vector and lies in V_{j−k}, hence for all i ∈ ℤ/Nℤ we found a unit eigenvector v_{i} ∈ V_{i} for Q′: Q′v_{i} = ζ_{N}^{i}v_{i}. Now define a unitary map U : H_{N} → H_{N} by taking

Uv_{i} := \sqrt{N}δ_{i}, i mod N

(U is unitary since it takes one orthonormal basis to another). We have UQ′ = Q̂U and UP′ = P̂U since they have the same effect on the basis vectors v_{i}. Thus we have constructed the required unitary map. Uniqueness follows from Lemma 1.1 by Schur’s lemma. □

2. Quantizing cat maps

Let Φ ∈ SL_{2}(ℤ) and suppose Φ satisfies the parity condition

(2.1)  Φ = (\begin{matrix}a & b \\ c & d \end{matrix}) ,  ab ≡ cd ≡ 0 mod 2

Theorem 2.1. For every such Φ, there is a unitary operator U = U_{N}(Φ) on H_{N}, unique up to a phase, so that

U^{∗} Op_{N}(f)U = Op_{N}(f ∘ Φ)

Proof. We will use the Stone-von Neumann Theorem 1.2 to construct U_{N}(Φ). Recall that P̂ = Op_{N}(e_{(1,0)}), Q̂ = Op_{N}(e_{(0,1)}). Define

(2.2) P′ := Op_{N}(e_{(1,0)} ∘ Φ) = Op_{N}(e_{(1,0)Φ}) = T_{N}((1, 0)Φ),
(2.3) Q′ := Op_{N}(e_{(0,1)} ∘ Φ) = Op_{N}(e_{(0,1)Φ}) = T_{N}((0, 1)Φ)

We will show that these satisfy the canonical commutation relations (1.8). Firstly, we have

(P′)^{N} = T_{N}((1, 0)Φ)^{N} = T_{N}(N(1, 0)Φ) = T_{N}((Na, Nb))

Now

T_{N}((Na, Nb))ψ(Q) = e^{\frac{iπ}{N} N^{2}ab}e_{N}(NbQ)ψ(Q + Na) = e^{iπNab}ψ(Q)

which equals ψ(Q) since ab is even. Thus (P′)^{N} = I. Likewise we have (Q′)^{N} = I since cd ≡ 0 mod 2.

Secondly, we have by (1.2)

P′Q′ = T_{N}((1, 0)Φ)T_{N}((0, 1)Φ) = e^{\frac{iπ}{N} ω((1,0)Φ,(0,1)Φ)}T_{N}((1, 1)Φ)

Now since Φ ∈ SL_{2}(ℝ), we have ω(vΦ, wΦ) = ω(v, w) and so we find

ω((1, 0)Φ, (0, 1)Φ) = ω((1, 0), (0, 1)) = 1

and so

P′Q′ = e^{\frac{iπ}{N}} T_{N}((1, 1)Φ)

Likewise we find that

Q′P′ = e^{− \frac{iπ}{N}} T_{N}((1, 1)Φ)

and so we get Q′P′ = ζ_{N}^{−1}P′Q′. Thus we have the commutation relations (1.8).

We may now use Theorem 1.2 to deduce the existence of U, unique up to a scalar, so that

P′ = U^{∗}P̂U, Q′ = U^{∗}Q̂U

that is that

(2.4)  T_{N}(nΦ) = U^{∗}T_{N}(n)U

holds for the basis vectors n = (1, 0) and (0, 1). Using the relation for the product of T_{N}(n)T_{N}(m) it then follows that (2.4) holds for all n ∈ ℤ^{2}: Indeed, T_{N}((n_{1}, n_{2})) = e^{\frac{iπ}{N} n_{1}n_{2}} Q̂^{n_{2}} P̂^{n_{1}} and so

U^{∗}T_{N}(n)U = U^{∗}e^{\frac{iπ}{N} n_{1}n_{2}}T_{N}(0, 1)^{n_{2}}T_{N}(1, 0)^{n_{1}}U
= e^{\frac{iπ}{N} n_{1}n_{2}}T_{N}((0, 1)Φ)^{n_{2}}T_{N}((1, 0)Φ)^{n_{1}}
= e^{\frac{iπ}{N} n_{1}n_{2}}T_{N}(n_{2}(0, 1)Φ)T_{N}(n_{1}(1, 0)Φ)
= e^{\frac{iπ}{N} n_{1}n_{2}}e^{\frac{iπ}{N} ω(n_{2}(0,1)Φ,n_{1}(1,0)Φ)}T_{N}(nΦ)

Now since Φ is symplectic we have

e^{\frac{iπ}{N} ω(n_{2}(0,1)Φ,n_{1}(1,0)Φ)} = e^{\frac{iπ}{N} n_{1}n_{2}ω((0,1),(1,0))} = e^{− \frac{iπ}{N} n_{1}n_{2}}

and so U^{∗}T_{N}(n)U = T_{N}(nΦ).

Hence by linearity we have

Op_{N}(f ∘ Φ) = U^{∗} Op_{N}(f)U

for all f ∈ C^{∞}(𝕋^{2}). □

Note: There is a similar quantization in higher dimensions, for linear symplectic maps Φ ∈ Sp_{2g}(ℤ). Examples:

1. For Φ = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) we may take the inverse Fourier transform

U_{N}((\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}))ψ(Q) = \frac{1}{\sqrt{N}} ∑_{P mod N} ψ(P)e_{N}(QP)

2. Consider the map Φ = (\begin{matrix}1 & 2 \\ 0 & 1 \end{matrix}), that is the map of the torus given

by

p ↦ p + 2q mod 1
q ↦ q

The quantization is the multiplication operator Uψ(Q) = e_{N}(Q^{2})ψ(Q).

It is easy to find eigenfunctions: Indeed Uδ_{j} = e_{N}(j^{2})δ_{j} so the position eigenstates are also eigenstates of U. To compute the matrix elements for this basis of eigenstates, note that

T_{N}(n)δ_{j} = e_{N}(\frac{n_{1}n_{2}}{2} + n_{2}j)δ_{j−n_{1}}

and so for the normalized eigenstates δ̂_{j} := \sqrt{N}δ_{j} we have

⟨T_{N}(n)δ̂_{j}, δ̂_{j}⟩ = \begin{cases}0, & n_{1} ≠ 0 mod N \\ e_{N}(\frac{n_{1}n_{2}}{2} + n_{2}j), & otherwise \end{cases}

Therefore as N → ∞,

⟨Op_{N}(f)δ̂_{j}, δ̂_{j}⟩ = ∑_{n_{1}≡0 mod N} f̂(n_{1}, n_{2})e(\frac{n_{1}n_{2}}{2N} + \frac{n_{2}j}{N})

∼ ∑_{n_{2}} f̂(0, n_{2})e(\frac{n_{2}j}{N})

= ∫_{0}^{1} f(p, \frac{j}{N})dp = ∫_{𝕋^{2}} fdp ⊗ δ_{j/N}(q)dq

Thus if j/N → q_{0} we get dp⊗δ(q−q_{0})dq as the corresponding quantum limit. These are delta-functions on invariant curves q = q_{0} of the classical map.

3. Quantum ergodicity

One of the fundamental questions in the subject of “quantum chaos” is the limiting behaviour of eigenfunctions. In particular, given an observable f ∈ C^{∞}(𝕋^{2}), we would like to study the possible limits of matrix elements ⟨Op_{N}(f), ψ, ψ⟩ where ψ ∈ H_{N} are eigenfunctions of U_{N}(Φ). It turns out that these limits (as functionals of the observable f) have to be probability measures on the torus, invariant under the map Φ. A very general result says that at least for almost all choices of eigenfunctions, the limit must be Lebesgue measure. This is known as the quantum ergodicity theorem. In the context of ergodic geodesic flows this was formulated by Schnirelman [18] and proved in [19, 3]. For quantum maps, the form that quantum ergodicity assumes is the following:

Theorem 3.1 ([2, 20, 21]). Let Φ ∈ SL_{2}(ℤ) be ergodic. Then for any orthonormal basis ψ_{j} of H_{N} consisting of eigenfunctions of U_{N}(Φ), there is a subset J(N) ⊂ \{1, 2, …, N\}, with \frac{#J(N)}{N} → 1, so that for j ∈ J(N) we have:

⟨Op_{N}(f)ψ_{j}, ψ_{j}⟩ → ∫_{𝕋} f, as N → ∞

for all observables f ∈ C^{∞}(𝕋).

Theorem 3.1 is a consequence, using positivity and a standard diagonalization argument, of the following estimate for the variance due to Zelditch [20]

Theorem 3.2. Let Φ ∈ SL_{2}(ℤ) be ergodic. For any orthonormal basis ψ_{j}, j = 1, …, N of of H_{N} consisting of eigenfunctions of U_{N}(Φ), we have

\frac{1}{N} ∑_{j=1}^{N} |⟨Op_{N}(f)ψ_{j}, ψ_{j}⟩ − ∫_{𝕋} f|^{2} → 0

for all observables f ∈ C^{∞}(𝕋).

Proof. Without loss of generality, we may assume that ∫_{𝕋} f = 0. We fix T ≥ 1. By Egorov’s theorem (1.7), we have

\frac{1}{T} ∑_{j=1}^{T}(U_{N}(Φ)^{t})^{∗} Op_{N}(f)U_{N}(Φ)^{t} = \frac{1}{T} ∑_{t=1}^{T} Op_{N}(f ∘ Φ^{t}) = Op_{N}(f^{T})

where f^{T} := \frac{1}{T} ∑_{t=1}^{T} f ∘ Φ^{t} is the ergodic average of f. Moreover, if ψ_{j} is an eigenfunction: U_{N}(Φ)ψ_{j} = e^{iλ_{j}}ψ_{j}, then

⟨Op_{N}(f)ψ_{j}, ψ_{j}⟩ = ⟨Op_{N}(f)U_{N}(Φ)ψ_{j}, U_{N}(Φ)ψ_{j}⟩
= ⟨U_{N}(Φ)^{∗} Op_{N}(f)U_{N}(Φ)ψ_{j}, ψ_{j}⟩
= ⟨Op_{N}(f ∘ Φ)ψ_{j}, ψ_{j}⟩ .

Consequently, if ψ_{j} is an eigenfunction then for all T ≥ 0,

(3.1)  ⟨Op_{N}(f)ψ_{j}, ψ_{j}⟩ = ⟨Op_{N}(f^{T})ψ_{j}, ψ_{j}⟩ .

Now we look at the sum (recall that ∫_{𝕋} f = 0)

S_{2}(f, N) := \frac{1}{N} ∑_{j=1}^{N} |⟨Op_{N}(f)ψ_{j}, ψ_{j}⟩|^{2} .

We will show that lim_{N→∞} S_{2}(f, N) = 0.

By (3.1), we have S_{2}(f, N) = S_{2}(f^{T} , N) for all T ≥ 1. By Cauchy-Schwartz, we have

|⟨Op_{N}(f^{T})ψ_{j}, ψ_{j}⟩|^{2} ≤ || Op_{N}(f^{T})ψ_{j}||^{2}||ψ_{j}||^{2} = ⟨Op_{N}(f^{T})^{∗} Op_{N}(f^{T})ψ_{j}, ψ_{j}⟩ .

Moreover, by (1.3), (1.4),

Op_{N}(f^{T})^{∗} Op_{N}(f^{T}) = Op_{N}(|f^{T} |^{2}) + O_{f,T} (\frac{1}{N})

and so

S_{2}(f, N) ≲ \frac{1}{N} ∑_{j=1}^{N}⟨Op_{N}(|f^{T} |^{2})ψ_{j}, ψ_{j}⟩ + O_{f,T} (\frac{1}{N}) .

By (1.5) we thus find that for fixed T ≥ 1,

lim sup S_{2}(f, N) ≤ ∫_{𝕋} |f^{T} |^{2} .

So far we have used nothing about the cat map except Egorov’s theorem. Now we use the fact that it is ergodic, in particular the mean ergodic theorem holds: For F ∈ L^{2}(𝕋), the ergodic averages F^{T} converge to ∫_{𝕋} F in L^{2}. Thus we have ∫_{𝕋} |f^{T} |^{2} → 0 as T → ∞. Therefore given ϵ > 0, we can find T = T (f, ϵ) for which ∫_{𝕋} |f^{T} |^{2} < ϵ and consequently

lim sup S_{2}(f, N) < ϵ

which shows that S_{2}(f, N) → 0 as required. □

Note that the argument used nothing more than Egorov’s theorem and the ergodicity of the map. In fact it gives the result for any symplectic ergodic (quantizable) map of the torus.

4. Quantum Unique Ergodicity

A key problem is:

Problem 4.1. Is it true that all eigenfunctions become equidistributed as N → ∞?

In the analogous situation of the quantization of the geodesic flow on a surface of negative curvature, it is conjectured [17] that all eigenfunctions are equidistributed. This is the Quantum Unique Ergodicity conjecture (QUE), see Lindenstrauss’ lectures [15]. It was thus a surprise when DeBievre, Faure and Nonnenmacher [8] discovered that this is false for cat maps. For instance, there is a (sparse) sequence of inverse Planck constants N_{k} and eigenfunctions ψ_{k} ∈ H_{N_{k}} for which the matrix coefficients converge to \frac{1}{2}(δ_{0} + dx).

The existence of these “scars” is related to the fact that eigenspaces of U_{N}(Φ) may have large dimension. In fact, the mean degeneracy is N/ ord(Φ, N) where ord(Φ, N) the order (or period) of Φ modulo N, that is the least integer k ≥ 1 for which Φ^{k} = I mod N. It can be shown that the mean degeneracy can be as large as N/ log N for arbitrarily large N. It is precisely for these N’s that the authors of [8] constructed “scars”, that is eigenfunctions which are not uniformly distributed.

These “scars” are rare; for instance, Kurlberg-Rudnick show [14] that for almost all values of N (that is the exceptional set has density zero), all eigenfunctions are uniformly distributed. On GRH, this holds for almost all prime values of N [12]. This has recently been improved by Bourgain [1] who showed unconditionally that the result holds for almost all primes and that the exceptional set is sparse. Bourgain’s main new ingredient are his estimates on incomplete exponential sums, encountered already in Friedlander’s lectures [6].

5. Arithmetic QUE

5.1. Symmetries and Hecke operators. It transpires that there is a commutative group of unitary operators on the state-space H_{N} which commute with the quantized map and therefore act on its eigen-spaces. These operators were discovered by Kurlberg and the author [13], where we called them “Hecke operators”, in analogy with the setting of the modular surface.

To understand their origin, one needs to note that the definition of U_{N}(Φ) given in §2 is such that it only depends on the remainder of Φ mod 2N. From the fact that the Egorov property (1.7) holds one thus gets a projective representation Φ ↦ U_{N}(Φ) of the subgroup of “quantizable” elements in the finite modular group SL(2, ℤ/2Nℤ). It turns out that it can be made into an ordinary representation if we further restrict to the subgroup Γ(4, 2N) given by g = I mod 4 for N even, g = I mod 2 for N odd. Thus for Φ, Φ′ ∈ Γ(4, 2N) we have U_{N}(ΦΦ′) = U_{N}(Φ)U_{N}(Φ′) (see [16, 10] for improvements on the congruence condition). Consequently, if ΦΦ′ = Φ′Φ mod 2N then their propagators commute. This is the basic principle that we use to form the Hecke operators. Remark. The congruence ΦΦ′ = Φ′Φ mod 2N is much less restrictive than the equation ΦΦ′ = Φ′Φ. The latter has as its solutions in SL(2, ℤ) essentially only ± powers of Φ (at least for Φ “primitive”).

5.2. Equidistribution of Hecke eigenfunctions. Since the Hecke operators commute with U_{N}(Φ), they act on its eigenspaces, and since they commute with each other there is a basis of H_{N} consisting of joint eigenfunctions of U_{N}(Φ) and the Hecke operators, whose elements we call Hecke eigenfunctions.

Theorem 5.1 ([13]). Let Φ ∈ SL(2, ℤ) be hyperbolic, Φ = I mod 4, and f ∈ C^{∞}(𝕋) a smooth observable. Then for all normalized Hecke eigenfunctions ψ ∈ H_{N} of U_{N}(Φ), the expectation values ⟨Op_{N}(f)ψ, ψ⟩ converge to the phase-space average of f as N → ∞. Moreover, for all ϵ > 0 we have

⟨Op_{N}(f)ψ, ψ⟩ = ∫_{𝕋} f(x)dx + O_{f,ϵ}(N^{−1/4+ϵ}), as N → ∞ .

The exponent of 1/4 in this theorem is not optimal. What is in fact shown is that if ψ_{i}, i = 1, …, N is an orthonormal basis of H_{N} consisting of Hecke eigenfunctions then

(5.1)  ∑_{i=1}^{N} |⟨Op_{N}(f)ψ_{i}, ψ_{i}⟩ − ∫_{𝕋} f(x)dx|^{4} ≪ N^{−1+ϵ}

(compare Theorem 3.2). We deduce Theorem 5.1 from (5.1) by taking an orthonormal basis with ψ_{1} = ψ and omitting all but one term on the LHS. If all terms on the LHS of (5.1) are of roughly the same size then we would expect this to give the exponent 1/2.

For prime values of N, this is in fact known; for primes where Φ is diagonalizable modulo N (this is 50% of the primes), this is shown in [5], while for the remaining 50% of primes this was shown by Gurevich and Hadani [9]. A key ingredient here is the Weil bound on complete exponential sums.

5.3. Higher dimensions. There is an extension of the above theory for linear symplectic maps of higher dimensional tori, including the existence of Hecke operators. Kelmer [11] discovered that the analogue of arithmetic quantum unique ergodicity fails in certain cases, and has shown that this happens if and only if there are isotropic rational subspaces, invariant under the linear map. In the case that there are invariant isotropic rational subspaces, Kelmer constructed “super-scars”: For each prime value of N, there are Hecke eigenfunctions ψ_{N} ∈ H_{N} that localize on a co-isotropic invariant subtorus.

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