Scarred eigenstates for arithmetic toral point scatterers

We investigate eigenfunctions of the Laplacian perturbed by a delta potential on the standard tori $\mathbb{R}^d/2 \pi\mathbb{Z}^d$ in dimensions $d=2,3$. Despite quantum ergodicity holding for the set of"new"eigenfunctions we show that there is scarring in the momentum representation for $d=2,3$, as well as in the position representation for $d=2$ (i.e., the eigenfunctions fail to equidistribute in phase space along an infinite subsequence of new eigenvalues.) For $d=3$, scarred eigenstates are quite rare, but for $d=2$ scarring in the momentum representation is very common --- with $N_{2}(x) \sim x/\sqrt{\log x}$ denoting the counting function for the new eigenvalues below $x$, there are $\gg N_{2}(x)/\log^A x$ eigenvalues corresponding to momentum scarred eigenfunctions.


Introduction
A basic question in Quantum Chaos is the classification of quantum limits of energy eigenstates of quantized Hamiltonians. For example, if the classical dynamics is given by the geodesic flow on a compact Riemannian manifold M, the quantized Hamiltonian is given by the positive Laplacian −∆ acting on L 2 (M). With {ψ λ } λ denoting Laplace eigenfunctions giving an orthonormal basis for L 2 (M), a quantum limit is a weak * limit of |ψ λ (x)| 2 along any subsequence of eigenvalues λ tending to infinity. More generally, given a smooth observable, i.e. a smooth function f on the unit cotangent bundle S * (M), its quantization is defined as a pseudo-differential operator Op(f ), and one wishes to understand possible limits of the distributions f → Op(f )ψ λ , ψ λ on C ∞ (S * (M)), as λ → ∞. If M has negative curvature ("strong chaos"), the celebrated Quantum Unique Ergodicity (QUE) conjecture by Rudnick and Sarnak [29] asserts that the only quantum limit is given by the uniform, or Liouville, measure on S * (M). Conversely, if the geodesic flow is integrable, many quantum limits may exist and the eigenfunctions are said to exhibit "scarring". E.g., if M = R 2 /2πZ 2 is a flat torus and a ∈ Z, then ψ a (x, y) = cos(ax) cos(y) is an eigenfunction with eigenvalue a 2 + 1, and clearly |ψ a (x, y)| 2 * → cos 2 (y)/2 as a → ∞. (For a partial classification of the set of quantum limits on R 2 /2πZ 2 , see [17].) Now, if the flow is ergodic ("weak chaos"), Schnirelman's theorem [35,41,5] asserts Quantum Ergodicity, namely that the only quantum limit, provided we remove a zero density subset of the eigenvalues, is the uniform one. However, non-uniform quantum limits may exist along the zero density subsequence of removed eigenvalues. Some interesting questions for quantum ergodic systems are thus: are there scars? If so, how large can the exceptional set of eigenvalues be? Can eigenfunctions scar in position space, i.e., is it possible that |ψ λ (x)| 2 , along some subsequence, weakly tends to something other than 1/ vol(M)? We shall address these questions for the set of "new" eigenfunctions of the Laplacian on a torus perturbed by a delta potential. The perturbation has a very small effect on the classical dynamics -only a zero measure subset of the set of trajectories is changed (hence there is no classical ergodicity), yet, as was recently shown [30,23,40], quantum ergodicity holds for the set of new eigenfunctions. (We note that this is quite different from point scatterers on tori of the form R 2 /Γ, for Γ a generic rectangular lattice. Here it was recently shown [22] that quantum ergodicity does not hold; in fact almost all new eigenfunction exhibit strong momentum scarring, cf. Section 1.2.) 1.1. Toral point scatterers. The point scatterer, or the Laplacian perturbed with a delta potential (also known as a "Fermi pseudopotential"), is a popular "toy model" for studying the transition between chaos and integrability in quantum chaos. With T d := R d /2πZ d for d = 2 or d = 3, let α ∈ R denote the "strength" of a delta potential placed at some point x 0 ∈ T d ; the formal operator −∆ + α · δ x 0 can then be realized using von Neumann's theory of self adjoint extensions. For d = 2, 3 there is a one parameter family of self adjoint extensions H ϕ , parametrized by an angle ϕ ∈ (−π, π], and the quantum dynamics we consider is generated by H ϕ . For d = 3 we will keep ϕ fixed, but in order to obtain a strong spectral perturbation for d = 2 we will allow ϕ to slowly vary with the eigenvalue; in the physics literature this is known as the "strong coupling limit", cf. Section 2 for more details.
The spectrum of H ϕ consists of two types of eigenvalues: "old" and "new" eigenvalues. The old ones are eigenvalues of the unperturbed Laplacian, i.e., integers that can be represented as sums of d integer squares, and the old eigenfunctions are the corresponding eigenfunctions of the unperturbed Laplacian that vanish at x 0 . The set of new eigenvalues, denoted by Λ, are all of multiplicity 1, and interlace between the old eigenvalues. In fact, the new eigenvalues are solutions of the spectral equation is the number of ways to represent n as a sum of d squares, and C = C(ϕ) := tan(ϕ/2) · n r d (n)/(n 2 + 1).
is allowed vary with λ when d = 2.
For λ ∈ Λ a new eigenvalue, the corresponding eigenfunction is then given by the Green's functions We remark that the delta potential introduces singularities at x 0 ; as x → x 0 , we have the asymptotic (for some a ∈ R) . Note that ϕ = π gives the unperturbed Laplacian; in what follows we will assume that ϕ ∈ (−π, π).
We can now formulate our first result, namely that some eigenfunctions strongly localize in the momentum representation in dimension three. For l ∈ N 3 let Ω(l) := ξ/|ξ| ∈ S 2 : ξ ∈ Z 3 , |ξ| 2 = l be the projection of the lattice points of distance √ l from the origin onto the unit sphere, and let δ Ω(l) denote the distribution defined by (we can view it as the uniform probability measure on the points of Ω(l)), and let ν denote the uniform measure on S 2 .
For λ ∈ Λ, let g λ ∈ L 2 (T 2 ) denote the L 2 -normalized eigenfunction with eigenvalue λ. Then for any l ∈ N 3 there exists an infinite subset Λ l ⊂ Λ, and a ∈ [ 1 2 , 1] such that for any pure momentum observable That is, the pushforward of the quantum limit along this sequence to the momentum space is a convex sum of the normalized sum of delta measures on the finite set Ω(l), and the uniform measure, with at least half the mass on the singular part -there is strong scarring in the momentum representation.
In dimension 2, when ϕ is fixed, (1) is often referred to as the "weak coupling limit", and almost all new eigenvalues remain close to the old eigenvalues (cf. [31]). To find a model which exhibits level repulsion, Shigehara [34] and later Bogomolny and Gerland [4] considered another quantization, sometimes referred to as the "strong coupling limit". One way to arrive at this quantization is by considering energy levels in a window around a given eigenvalue: e.g., for η ∈ (131/146, 1) the new eigenvalues are defined to be solutions of where n + (λ) is the smallest element of N 2 that is larger than λ. It is convenient to consider both couplings simultaneously; we may do this by letting (strong coupling) and then rewriting the spectral equation as Our next result, valid for both the weak and strong coupling limit in dimension two, is the existence of a zero density subsequence exhibiting non-uniform quantum limits in the momentum, as well as the position representation.
(1) There exists an infinite subset Λ ′ m ⊂ Λ such that the pushforward of the quantum limit along this sequence to momentum space has positive mass on a finite number of atoms ("strong momentum scarring").
(2) There exists an infinite subset Λ ′ p ⊂ Λ such that the pushforward of the quantum limit along this sequence to position space has a nontrivial non-zero Fourier coefficient ("position scarring"). Furthermore, we may take Λ ′ p = Λ ′ m . We remark that for d = 2 almost half the mass is carried on the singular part in the momentum representation -for any ǫ > 0 the singular part has mass at least 1/2 − ǫ (cf. Remark 11).
In order to quantify how common scars are we need some further notation. For d = 2, 3, let N d (x) denote the counting function ("Weyl's law") for the number of new eigenvalues λ ≤ x. For d = 3, N 3 (x) ∼ x and, as the eigenvalues that give rise to scars are essentially powers 4 l , the exceptional subset is of size x o(1) and thus very sparse. For d = 2, (1) , and our construction of eigenfunctions that scar both in position and momentum is a subset with counting function of size x 1/2−o(1) -hence fairly rare. However, if we restrict ourselves to scarring only in the momentum representation, we can use some recent results by Maynard [27] to show that scarred eigenvalues are in fact quite common.
1.2. Discussion. In [32]Šeba proposed quantum billiards on rectangles with irrational aspect ratio, perturbed with a delta potential, as a solvable singular model exhibiting wave chaos; in particular that the level spacings should be given by random matrix theory (GOE).Šeba andŻyczkowski later noted [33] that the level spacings were not consistent with GOE, in particular large gaps are much more frequent (essentially having a Poisson distribution tail.) Shigehara subsequently found [34] that level repulsion is only present in the strong coupling limit. Recently Rudnick and Ueberschär proved [31], in dimension two, that the level spacing for the weak coupling limit is the same as the level spacings of the unperturbed Laplacian (after removing multiplicities). This in turn is conjectured to be Poissonian, and we note that a natural analogue of the prime k-tuple conjecture for integers that are sums of two squares can be shown to imply Poisson gaps [10].
In [31] the three dimensional case was also investigated and the mean displacement between new and old eigenvalues was shown to equal half the mean spacing.
In [30], Rudnick and Ueberschär proved a position space analogue of Quantum Ergodicity for the new eigenfunctions: there exists a full density subset of the new eigenvalues such that as λ → ∞ along this subset, the only weak limit of |ψ λ (x)| 2 is the uniform measure on T 2 . Further, in [23] the first author and Ueberschär proved an analogue of Quantum Ergodicity: there exists a full density subset of the new eigenvalues such that the only quantum limit along this subset is the uniform measure on the full phase space (i.e., the unit cotangent bundle S * (T d ).) This result was later shown to hold also for d = 3 by Yesha [40]; already in [39] he showed that all eigenfunctions equidistribute in the position representation.
For irrational tori, Keating, Marklof and Winn proved in [18] that there exist non-uniform quantum limits (in fact, strong momentum scarring was already observed in [3]), assuming a spectral clustering condition implied by the old eigenvalues having Poisson spacings (which in turn follows from the Berry-Tabor conjecture.) Recently the first author and Ueberschär unconditionally showed [22] that for tori having diophantine aspect ratio, essentially all new eigenfunctions strongly scar in the momentum representation. Recently Griffin showed [13] that similar results hold for Bloch eigenmodes (i.e., non-zero quasimomentum) for periodic point scatterers in three dimensions, provided a certain Diophantine condition on the aspect ratio holds.
1.3. Scarring and QUE for some other models. For Quantum Ergodic systems almost all eigenfunctions equidistribute, but in general not much is known about the (potential) subset of expectional eigenfunctions giving non-uniform quantum limits. In some cases Quantum Unique Ergodicity is known to hold; notable examples are Hecke eigenfunctions on modular surfaces [24,36] and "quantized cat maps" [21,19]. For these models there exist large commuting families of "Hecke symmetries" that also commute with the quantized Hamiltonian, and it is then natural to consider joint eigenfunctions of the full family of commuting operators. Other examples arise when the underlying classical dynamics is uniquely ergodic, QUE is then "automatic", e.g., see [28,26].
On the other hand there are Quantum Ergodic systems exhibiting scarring. For example, if Hecke symmetries are not taken into account, quantized cat maps can have very large spectral degeneracies. Using this, Faure, Nonnenmacher and de-Bievre ( [9]) proved that scars occur in this model. For higher dimensional analogues of cat maps, Kelmer found a scar construction not involving spectral degeneracies, but rather certain invariant rational isotropic subspaces [19,20].
We also note that Berkolaiko, Keating, and Winn has shown [3, 2] that simultaneous momentum and position scarring can occur for quantum star graphs, e.g., for certain star graphs with a fixed (but arbitrarily large) number of bonds, there exists quantum limits supported only on two bonds.
Another way to construct scars is to use "bouncing ball quasimodes". For example, functions of the form ψ n (x, y) = f (x) sin(ny) are approximate Laplace eigenfunctions on a stadium shaped domain (say with Dirichlet boundary conditions), and semiclassically localize on vertical periodic trajectories. Hassell showed [16] that for a generic aspect ratio stadium, there are few eigenvalues near n and hence ψ n overlaps strongly with an eigenfunction φ n with eigenvalue near n, which then also must partially localize on vertical periodic trajectories. The number of "bouncing ball eigenfunctions" having eigenvalue at most E grows (at most) as E 1/2+o (1) , to be compared with the Weyl asymptotic c · E; hence these scarred eigenstates are fairly rare. In [1,37,25] the asymptotic behaviour of sets of bouncing ball eigenfunctions for some ergodic billiards were considered. Interestingly, for the stadium billiard it was argued that the number of scarred bouncing ball eigenfunctions, with eigenvalue at most E, are much more numerous, namely of order E 3/4 (again to be compared the Weyl asymptotic c · E.) In fact, in [1] it was argued that given any δ ∈ (1/2, 1), there exists a Sinai type billiard whose bouncing ball eigenfunction count is of order c δ · E δ .

1.4.
Outline of the proofs. The proofs are based on finding new eigenvalues λ that are quite near certain old eigenvalues. After rewriting equation (5) as we show that for any m ∈ N d there exists a new eigenvalue λ such that |m − λ| ≪ r d (m)/H ′ m (m) (though it should be emphasized that we do not know whether λ > m or λ < m). We then find a sequence of integers m such that both r d (m) and r d (m)/H ′ m (m) are bounded, and thus get a control on the distance of a new eigenvalue from these m.
(To find such m we use the lower bound sieve methods when d = 2; for d = 3 we find integers m for which the representation number r 3 (m) is very small.) We conclude by using an explicit description for the relevant eigenfunctions to compute the limits in the theorems.
The paper is organized as follows: In Section 2 we set the necessary background for the point scatterer model, then give some number theoretic background, and in Section 3 we prove some auxiliary analytic and number theoretic results needed in the proofs of our main theorems. In Sections 4 and 5 we prove Theorems 1 and 2, and Section 6 contains the proof of Theorem 3.
Acknowledgements. We would like to thank Zeév Rudnick and Henrik Ueberschär for helpful discussions about this work.

Background
In this section we briefly review some results and definitions about point scatterers and give a short number theoretic background.
2.1. Point scatterers on the flat torus. We begin with the point scatterers, and recall the definition and properties of the quantization of observables (see [30,23] for more details; further background can be found in [38,40].) 2.1.1. Basic definitions and properties. For d = 2, 3 we consider the restriction of the Laplacian −∆ on The restriction is symmetric though not self-adjoint, but by von Neumann's theory of self adjoint extensions there exists a one-parameter family of self-adjoint extensions; for ϕ ∈ (−π, π] there exists a selfadjoint extension H ϕ , where the case ϕ = π corresponds to the unperturbed Laplacian. The spectrum of H ϕ consists of two types of eigenvalues and eigenfunctions: (1) Eigenvalues of the unperturbed Laplacian, and the corresponding eigenfunctions that vanish at x 0 . The multiplicities of the new eigenvalues are reduced by 1, due to the constraint of vanishing at x 0 . (2) New eigenvalues λ ∈ R satisfying the equation For λ ∈ R satisfying (7), the corresponding Green's function is an eigenfunction, and is an L 2 -normalized eigenfunction.
2.1.2. Strong coupling. In [30] Rudnick and Ueberschär showed that for d = 2 the set of "new" eigenvalues "clump" with the Laplace eigenvalues, and in fact the eigenvalue spacing distribution coincides with that of the Laplacian. In [34] Shigehara, and in [4] Bogomolny, Gerland and Schmit considered another type of quantization, with the intent of finding a model that exhibits level repulsion. This quantization is sometimes referred to as the "strong coupling" (compared to the "weak coupling" given by equation (7)). One way of arriving at this quantization is by truncating the summation in (7) outside an energy window of size O(λ η ) for any fixed η > 131/146. This leads to the following spectral equation for the new eigenvalues: we define the quantization of it as a pseudodifferential operator Op(a) : We refer the reader to [23] for details on the 2 dimensional case, and [40] for the 3 dimensional case. We are mainly interested in either pure momentum, or pure position observables, that is a( respectively; this considerably simplifies the discussion of quantizing observables. Namely, given f (x) ∈ C ∞ (T d ), the action of a pure position observable a = a(x) ∈ C ∞ (T d ) is given by whereas the action of a pure momentum observable a = a(ξ) ∈ C ∞ (S d−1 ) is given by in particular, for pure momentum observables we have 2.2. Number theoretic background.

2.2.1.
Integers that are sums of 2 or 3 squares. We begin with a short summary about integers that can be represented as sums of d squares for d = 2 or 3. Sums of 2 squares: It is well known (e.g., see [8]) that r 2 (n) is determined by the prime factorization of n. If we write n = 2 a 0 p a 1 1 . . . p ar r q b 1 1 . . . q b l l , where the p i 's are primes all ≡ 1 (mod 4), and the q i 's are primes all ≡ 3 (mod 4), then n is a sum of two squares if and only if all the b i are even, and r 2 (n) = 4d(p a 1 1 . . . p ar r ), where d(·) is the divisor function.
Sums of 3 squares: For d = 3, any number n that is not of the form n = 4 a n 1 , where 4 ∤ n 1 and n 1 ≡ 7 (mod 8) can be represented as a sum of 3 squares. Moreover, r 3 (4n) = r 3 (n) for any n ∈ Z, and if we let R 3 (n) denote the number of primitive representation of n as a sum of 3 squares (that is the number of ways to write n = x 2 + y 2 + z 2 with x, y, z coprime), we can relate r 3 (n) to class numbers of quadratic imaginary fields as follows (cf. [14, Theorem 4, p. 54]): 0 n ≡ 0, 4, 7 (mod 8) 16 n ≡ 3 (mod 8) 24 n ≡ 1, 2, 5, 6 (mod 8) .
and χ n (m) = (−4n/m) is the Kronecker symbol. By a celebrated theorem of Siegel, for any ǫ > 0, L(1, χ n ) ≫ ǫ n −ǫ and thus, for n ≡ 0, 4, 7 mod 8, Further, given an integer n that is a sum of 3 squares, let Fomenko-Golubeva and Duke showed (see [12,6], or [7, Lemma 2]) that the sets Ω(n) equidistribute in S 2 as r 3 (n) → ∞ (or equivalently n 1 → ∞) inside N 3 . Namely, there exists α > 0, such that for any spherical harmonic Y (x), there is significant cancellation in the Weyl sum in the sense that where the implied constant is independent of n.

2.3.
Sieve method results. We list below some sieve results that show the existence of various infinite sequences of integers with bounded number of prime divisors. We first recall a few definitions. A positive integer n is called r-almost prime if n has at most r prime divisors. We denote by P r the set of all r-almost prime integers. A finite set of polynomials has no fixed prime divisors, that is the equation F (x) ≡ 0 (mod p) has less that p solutions for any prime p. The followoing theorem combines results from [15,11,27]:

Auxiliary results
Before proceeding to the proofs of the main theorems, we begin with a few auxiliary results. For the benefit of the reader we note that §3.1 is relevant for all theorems, Lemma 6 is relevant for the proof of Theorem 2, and Lemma 7, and Proposition 8 are relevant for the proof of Theorem 3.

Proof. Define
where the exponents E p are chosen as follows: let E p = 1 if p ≡ 3 mod 4, otherwise let E p be the minimal integer so that p Ep > H 2 . Further, let q 1 < q 2 < . . . < q H be primes congruent to 1 mod 4, chosen so that q 1 > 2H, and given integer exponents e 1 , . . . , e H ≥ 1, define Q 2 = Q 2 (e 1 , e 2 , . . . , e H ) := i≤H q e i i and finally let Q := Q 1 · Q 2 . By the Chinese remainder theorem we may find γ mod Q such that the following holds: (19) γ ≡ 0 mod p Ep if p ≡ 1, 2 mod 4 and p < 2H, (20) γ ≡ 1 mod p if p ≡ 3 mod 4 and p < 2H and for each prime q i |Q 2 so that by G i (t) := ((Qt + γ) 2 + i 2 )/(q e i i d i ), i = 1, 2, . . . , H. By definition, d i |γ 2 + i 2 , and (21) implies that q e i i |γ 2 + i 2 . Thus, since (Q 1 , Q 2 ) = 1 implies that (q i , d i ) = 1, we find that q e i i d i |γ 2 + i 2 and consequently G i (t) ∈ Z[t] for all i.
is an admissible set of polynomials (i.e., H i=1 G i (x) does not have any fixed prime divisors).
To prove the claim we argue as follows: If p > 2H and all G i are nonconstant modulo p (i.e., p ∤ Q) there are at most 2H residues n (modulo p) for which G i (n) ≡ 0 mod p for some i. Hence there exist n ∈ Z such that H i=1 G i (n) ≡ 0 mod p. On the other hand, if p > 2H and p|Q then p = q i for some i, and by the definition of G i (in particular, recall (21)), we find that G i (n) ≡ 0 mod q i for all n ∈ Z. Moreover, if j = i, (as 0 < |i − j| < H, 0 < i + j < 2H and q i > 2H), and thus G j (n) ≡ 0 mod q i for all n ∈ Z.
For p < 2H we argue as follows: if p ≡ 3 mod 4, (20) gives that γ 2 +i 2 ≡ 0 mod p for all i ∈ Z. Otherwise, i 2 ≤ H 2 < p Ep by our choice of E p , and since γ was chosen so that γ ≡ 0 mod p Ep (recall (19)), we find that γ 2 + i 2 ≡ i 2 ≡ 0 mod p 2Ep , as i 2 ≤ H 2 and p Ep > H 2 . Consequently (γ 2 + i 2 )/d i is not divisible by p. The proof of the claim is concluded. Now, given an integer r > 0, let P r denote the set of integers that can be written as a product of at most r primes, as in §2.3. Since the polynomials {G i (x)} H i=1 form an admissible set, part (1) of Theorem 4 implies that there exists some r > 0 (only depending on H) such that for infinitely many n. Given such an n, let m i = G i (n); then each m i is a sum of squares that in addition has at most r prime factors. Consequently, if we set a i = m i · q e i i · d i , we find that a i ∈ N 2 for all 1 ≤ i ≤ H, and that where C = C(H) ≥ 1 is independent of the exponents e 1 , . . . , e H . Choosing e 1 , . . . , e H appropriately we can ensure that r 2 (a i+1 ) > R · r 2 (a i ) holds for all i, as well as that r 2 (a H ) ≪ H C H R H ≪ H R H (a somewhat better C-dependency can be obtained but we shall not need it.) Finally, since a i = m i q e i i d i = G i (n)q e i i d i = (Qn + γ) 2 + i 2 we find that a H − a 1 = H 2 − 1 < H 2 and the proof of Lemma 7 is concluded.
The following proposition might be of independent interest -using the full power of [27], the method of the proof in fact gives the following: given k ≥ 2 and R > 1 there exists A > 0 such that, as x → ∞, there are ≫ x/(log x) A integers n ≤ x such that r 2 (n + h i+1 ) ≥ Rr 2 (n + h i ) holds for i = 1, . . . , k − 1 and 0 < h 1 < h 2 < . . . < h k ≪ k 1. For simplicity we only state and prove it for k = 2. Proposition 8. There exist an integer H ≥ 1 with the following property: for all sufficiently large R there exist an integer h ∈ (0, H 2 ) such that |{n ∈ N 2 : n ≤ x, 0 < r 2 (n) ≪ R H , r 2 (n + h) ≥ R · r 2 (n)}| Proof. By part (3) of Theorem 4 there exists integers i, j such that 0 < i < j ≤ H with the property that . . , F H } any admissible set of H linear forms, provided H is sufficiently large. For such an H, and a given (large) R, Lemma 7 shows there exists a 1 , . . . , a H > 0 such that for 1 ≤ i ≤ H we obtain a set of H admissible linear forms (here admissibility is trivial since F i (0) ≡ 0 mod p for any prime p), hence there exists i, j with j > i such that Further, given primes p = F i (n) and p ′ = F j (n), define m = a j · p and m ′ = a i · p ′ . Now, since a i ≡ 0 mod 4 for all i, F i (n) ≡ 1 mod 4 for all n, hence p, p ′ ≡ 1 mod 4 and consequently m, m ′ ∈ N 2 . Further, and thus 0 < h < H 2 . Moreover, r 2 (m) = r 2 (p · a j ) = 2 · r 2 (a j ) and similary r 2 (m ′ ) = 2 · r 2 (a i ). Since r 2 (a j ) ≥ R · r 2 (a i ) we find that r 2 (m ′ ) ≥ R · r 2 (m), and that r 2 (m) = 2 · r 2 (a i ) ≪ R H . Taking n = m ′ and h = m − m ′ we find that the number of n ≪ R x with the desired property is ≫ x/(log x) H , thus concluding the proof.

Proof of Theorem 1
We prove Theorem 1 by calculating the Fourier coefficients of the measure (or more precisely, the coefficents in the spherical harmonics expansion.) Let Y (x) be a spherical harmonic on S 2 . Then by the definition of g λ = G λ G λ (cf. (9)), and the action of Op(Y ) (cf. (13)) we get and rewrite the "new" eigenvalue equation (7) as We can now apply Lemma 5. Setting λ = m + δ, let Then Notice that for |δ| < 1 2 there exists an absolute constant C > 1 such that To find m for which the above bound is valid, we proceed as follows. For l ∈ N 3 fixed, define Ω(l) := ξ ξ : ξ 2 = l, ξ ∈ Z 3 and let M l := 4 k l : k ∈ N . For m ∈ M l we then have r 3 (m) = r 3 (l), hence r 3 (m) is uniformly bounded; we also note that Ω(m) = Ω(l).
Since for any integer m there exists an integer m ′ ≡ 0, 4, 7 (mod 8) of bounded distance from m, (15) implies that (27) Since r 3 (m) is uniformly bounded for m ∈ M l , we find that for all sufficiently large m ∈ M l . By the above argument, we have thus found infinitely many m for which there exist a nearby new eigenvalue λ satisfying |m−λ| < Cr 3 (l)/f ′ m (0) < m −1/4+ǫ . In fact, using (27) we can apply Lemma 5 again, to get that (25) holds for C = 1+O(m −1/4+ǫ ) and δ = O(m −1/4+ǫ ). Let Λ l be the sequence of these eigenvalues; for λ ∈ Λ l we then find, upon recalling the equality in (24), and that (25) is valid since |m − λ| = O(m −1/4+ǫ ), that which is bounded. From now on we restrict Λ l to a subsequence such that the limit exists, and hence by (28) is bounded by r 3 (l). Furthermore, for any spherical harmonic Y , We claim that the RHS converges to 0 as m → ∞. To see this, write For the first sum, using that |λ − n| > 1/2 for n = m together with the bound W Y (n) ≪ m 1/2−α (using (16)) we find that for all n ≤ m + m 1/3 in the summand, the first sum is ≪ m 1/2−α . For the second sum, the mean value theorem gives that and thus, Hence, since f ′ m (0) ≫ m 1/2−ε (cf. (27)), Thus, for any fixed spherical harmonic Y , and for every λ ∈ Λ l , Since these are the spherical harmonics coefficients of the measure 1 1+A l δ Ω(l) + A l 1+A l ν, the proof is concluded. (Recall that ν denotes the uniform measure.)

Proof of Theorem 2
We start by finding a sequence of new eigenvalues lying close to the set of old eigenvalues. To do so we will again use Lemma 5. Recall that (strong coupling) and in analogy with the three dimensional case we define where (for some fixed η > 131/146) Proposition 9. Let F (λ) be as above, and given γ ∈ (0, 1/10) let M γ be the set of integers given by Lemma 6. Then, for any m ∈ M γ , there exists a new eigenvalue λ such that |λ − m| ≤ γ and Proof. Given m ∈ M γ we start by finding (at least one) nearby new eigenvalue. To do so, rewrite the eigenvalue equation (i.e., (7) in the weak coupling limit, or (10) in the strong coupling limit) Thus, with λ = m + δ, and defining f (δ) = H m (m + δ) − r 2 (m) m m 2 +1 , we wish to find (small) solutions to f (δ) = r 2 (m) δ Now, by (32), f ′ (δ) is always a sum of positive terms, hence we may drop all terms but one, say the one corresponding to k = m + 3 (recall that m = n 2 + 1, hence k = n 2 + 4 is a sum of two squares), and find that 10 for |δ| ≤ 1/10. By Lemma 5, there exists δ 0 such that Using the above estimate on δ we next show that the lower bound on f ′ (δ) is essentially given by the size of H ′ m (m). Since m − ≤ m − 1 and m + ≥ m + 1, we find that holds for all n ∈ N 2 \ {m} and |δ| ≤ γ. Thus, for |δ| ≤ γ. Hence we may take A = r 2 (m) and B = H ′ m (m + δ 0 )(1 + O(γ)) in Lemma 5; on squaring the estimate δ 0 ≤ A/B we find that Remark 10. The above argument in fact gives the following: if r 2 (m+ h) ≥ R · r 2 (m) > 0 for some 0 < h < H, then (again for |δ| < 1/10), and thus there exists a nearby new eigenvalue λ = m + δ 0 with |δ 0 | ≪ H/ √ R, and For γ ∈ (0, 1/10) let M γ be the set given by Lemma 6, and Λ γ be the set of corresponding new eigenvalues given by Proposition 9. By restricting to a subsequence we may assume that for any n ∈ M γ , the sets Ξ(n) := ξ |ξ| : |ξ| 2 = n converge to a limit set Ξ(∞) of bounded cardinality.
It is now straightforward to exhibit scarring in momentum space.
By the definition of G λ and Op(a) we see that using the new notation (cf. (8), (11)) and therefore (38) Op(e w )g λ , g λ = As we aim to show that (38) is bounded from below in absolute value, we may assume that x 0 = 0. We will show that the sum in the numerator is essentially bounded from below by two terms in the sum, namely v such |v| = |v + w|.
In what follows, γ ∈ (0, 1/10) is small (and to be determined later), m = n 2 + 1 will always denote an element of M γ (recall that by construction, all elements of M γ are of this form), and given n we define a vector u ∈ Z 2 by u := (n, −1). For λ ∈ Λ γ let m ∈ M γ be the corresponding nearby integer (i.e., |m − λ| < γ by Proposition 9), set R = √ λ, and let C m := {v ∈ R 2 : |v| 2 = m} denote the circle of radius √ m centered at the origin. Define as the annulus of width 2|w| containing C m , and let A * R = A * R,w := {v ∈ R 2 : |v| ∈ [R−|w|, R+|w|], |v| 2 = m, |v+w| 2 = m}. The following Lemma will allow us to bound contribution of the negative terms in the sum in the numerator of the right hand side of (38).
Proof. Since C(v, w) < 0 if and only if the line segment joining v and v + w intersects C R , the first assertion follows from the triangle inequality.
We now write v = (x, y) for x, y ∈ Z.
and C(v, w) ≪ 1 δ √ R . Using a similar argument we get that C(v, w) ≪ 1 δ √ R if |v + w| 2 = m. Therefore, since r 2 (m) is bounded, and (39) follows. The contribution from the remaining, more subtle, terms are treated by "pairing off" negative summands with positive ones, and thereby getting some extra savings.
By corollaries 13 and 15 On the other hand, by Proposition 9 and, recalling that r 2 (m) is bounded, we can choose γ small enough (recall that |m − λ| = |δ| ≤ γ) so that Op(e w )g λ , g λ ≥ is uniformly bounded from below.

Proof of Theorem 3
Recall first the setting proved in Proposition 8: There exist an integer H ≥ 1 with the property that for all sufficiently large R there exist an integer h ∈ (0, H 2 ) such that |{n ∈ N 2 : n ≤ x, 0 < r 2 (n) ≪ R H , r 2 (n + h) ≥ R · r 2 (n)}| ≫ R x/(log x) H as x → ∞.
As noted in Remark 10, if r 2 (m + h) ≥ R · r 2 (m) > 0 for some integer h such that 0 < h < H, then there exists a new eigenvalue λ = m + δ 0 with The argument in Section 5.1 then shows that λ gives rise to a momentum scar provided r 2 (m) is also bounded. Proposition 8 then gives, upon choosing R sufficently large, that the number of such m ≤ x is ≫ x/(log x) H .