Lower Bounds on Anderson-Localised Eigenfunctions on a Strip

It is known that the eigenfunctions of a random Schrödinger operator on a strip decay exponentially, and that the rate of decay is not slower than prescribed by the slowest Lyapunov exponent. A variery of heuristic arguments suggest that no eigenfunction can decay faster than at this rate. We make a step towards this conjecture (in the case when the distribution of the potential is regular enough) by showing that, for each eigenfunction, the rate of exponential decay along any subsequence is strictly slower than the fastest Lyapunov exponent, and that there exists a subsequence along which it is equal to the slowest Lyapunov exponent.

On an event of full probability, H exhibits Anderson localisation which manifests itself in the following spectral properties: the spectrum of H is pure point, and the eigenfunctions decay exponentially, meaning that there exists a deterministic γ > 0 such that for each eigenfunction ψ of H lim sup where · denotes the Euclidean norm in R W . For W = 1, the pure point nature of the spectrum was first established in [18], and exponential decay -by Molchanov in [32]; see further Kunz and Souillard [26]. In these works, it was assumed that the distribution of the potential is absolutely continuous with bounded density, The case of singular potentials was settled by Carmona, Klein, and Martinelli [6]. For W > 1 (Schrödinger case) with absolutely continuous distribution of the potential, the pure point nature of the spectrum was first proved in [15], and exponential decay -by Lacroix in [27,28]. The general Schrödinger case was settled by Klein, Lacroix and Speis in [25], building on [17]. The argument of [25] can be extended to the general situation (2), once the result of [16] (discussed below) is taken into account; an alternative argument avoiding multi-scale analysis and applicable to the general model (1) (and also to its further generalisation allowing for random hopping) is given in [31]. In this paper, we do not discuss Anderson localisation in dimension d > 1, and refer to the works of Fröhlich and Spencer [9] and Aizenman and Molchanov [2] and also to the monograph of Aizenman and Warzel [3].
Following precursory work by Tutubalin (see the survey [34]) and Virtser [37], Guivarc h and Raugi showed [20] that if the action of the semigroup generated by the support S λ of T n (λ) on R 2W and its wedge powers is strongly irreducible and contractive, (6) then the Lyapunov exponents are distinct: In the case (3) with absolutely continuous distribution of v n,α , the condition (6) was verified in [27], while in [15] (7) was directly established using the results of [34]. In [17], the following general theorem is proved: (6) (and consequently also (7)) holds if the group generated by S λ is Zariski-dense in Sp(2W, R).
It was also shown in [17] that in the Schrödinger case (3) one has (8) for any λ ∈ R.
In [16], a general method to compute the Zariski closure of the group generated by the support of T n (λ) was developed; one of its consequences is that (8) holds for any λ ∈ R also in the generality of (2). Now we can state the full result of Klein, Lacroix and Speis [25] (in the current setting, covered by [31]): there is an event of full probability on which each eigenpair H ψ = λψ satisfies lim sup A variety of heuristic arguments indicate that (9) should be sharp in the following strong sense: there is an event of full probability on which each eigenpair H ψ = λψ satisfies which, in conjuction with (9), implies the existence of a limit equal to −γ W (λ). For example, the Fermi Golden Rule leads one to believe that eigenfunctions violating (10) are unstable under perturbation. From the point of view of random matrix products, an eigenfunction decaying at a rate faster than γ W indicates a non-generic intersection between the W -dimensional space of initial conditions with the W -dimensional Oseledec subspace of decaying solutions in R 2W . The relation (10) was repeatedly conjectured at least since the 1980s; however, we are not aware of any rigorous results improving on the trivial bound lim inf (which follows from a general result of Craig and Simon [7], or from its quantitative version, stated as Lemma 2.2 below). The main difficulty comes from the fact that, although for each fixed energy λ the probability to have an eigenfunction which decays at a rate faster than γ W (λ) is zero, one can not use the union bound over the uncountable set of all real λ.
In this paper we make a step towards (10) by improving upon (11) (in the case when the distribution of potential is regular enough). To state the results precisely, we introduce some notation. Let E(H ) = {(λ, ψ)} be the collection of eigenpairs of H , with the normalisation ψ(0) = 1 (the choice of the sign is not important for us, and spectral multiplicity is known to be a null event). For γ > 0 and a bounded interval I ⊂ R, consider the two realisation-dependent sets: These sets consist of the eigenvalues for which the corresponding eigenvector decays at rate ≥ γ (along a subsequence, or uniformly). We note that there is no simple way to define the sets as random variables on the underlying probability space (see Kendall [24] and Tsirelson [36] for possible frameworks to address such questions); this does not cause problems since we only work with the measureable events {Fast ± (γ ; I ) = ∅} and {Fast ± (γ ; I ) = ∅} (which in fact lie in the tail σ -algebra).
For λ in the spectrum σ (H ) of H , define the deterministic quantities Roughly speaking, the functions γ ± * (λ) measure the fastest possible decay of an eigenfunction in the vicinity of λ (recall that γ j (λ) are continuous, cf. below, and that σ (H ) is almost surely equal to a deterministic set). In this notation, (9) and (11) imply that whereas the conjecture (10) stipulates that the last two inequalities are in fact equalities: The results below show that the first inequality in (14) is strict (for W ≥ 2), whereas the last one is an equality, at least, if one asumes Here x + = max(x, 0). We observe that for W ≥ 3 γ * ,1 (λ) < γ 1 (λ), hence (15) indeed improves on (11). For W = 2, γ * ,1 = γ 1 (λ); however, we prove for all λ ∈ σ (H ).
As for γ − * , our methods yield the optimal result: The following corollary summarises our main conclusions: for all λ ∈ σ (H ).
In the proofs, we repeatedly use the following argument, inspired by the work of Kakutani [23] and its ramifications by Spencer and Aizenman [1], to estimate the probability of exceptional events. Suppose we want to bound P {A = ∅}, where A is a random subset of, say, the interval [0, 1]. Suppose we find η ∈ (0, 1] and a random superset A + ⊃ A with the following properties: (a) for each λ ∈ [0, 1], P λ ∈ A + ≤ p ("single-energy bound"); (b) if λ ∈ A and |λ − λ| < η, then λ ∈ A + ("propagation estimate"). Then the Chebyshev inequality and the Fubini theorem yield: The paper is organised as follows. Some preliminary estimates are collected in Sect. 2. In Sects. 3 and 4 we prove Theorems 2 and 3, respectively. In Sect. 5 we discuss the prospects of improving the bounds in Theorems 1 and 2, and point out the connection to the problem, going back to [13,14] and recently studied by Gorodetski and Kleptsyn [19], of uniform convergence to the Lyapunov exponents, i.e. whether the quantifier ∀λ in (5) can be inserted inside the curly brackets. We also prove Proposition 5.1, which is an Sp(2W, R)-counterpart of one of the results of [19]. The proof of Theorem 3 in Sect. 6 makes use of this proposition.
We conclude this introduction with two remarks. First, we have chosen to present the arguments for the one-sided strip Z + × {1, · · · , W }; similar arguments can be applied to the two-sided strip Z × {1, · · · , W }. Second, it is possible that Assumption 1.1 can be somewhat relaxed, and that a refinement of the current methods could be applicable when the invariant measure (describing the limiting distribution of the unitary matrices in the singular value decomposition of the transfer matrices n ) is absolutely continuous with bounded density with respect to the Haar measure on the compact symplectic group, or at least enjoys the Frostman property (upper bound on the measure of every ball by a power of the radius) with a sufficiently large exponent. On the other hand, it is known (see [21] for the case W = 1) that for singular distributions of V (n) the invariant measure may be supported on lower-dimensional subsets of the symplectic group. Extending our results to such cases would require additional ideas.

Preliminaries
Convergence to the Lyapunov exponent Assume that (6) holds at some λ. Then (6) also holds in a neighbourhood of λ, and then (see e.g. [25, Corollary 2.5]) the Lyapunov exponents γ j (λ) are continuous at λ. For each > 0, let r (λ) ∈ (0, 1/2] be such that The following large deviation estimate goes back to the work of Le Page [30].
Lemma 2.1 (see [8], [5, Section V.6]) Assume (1). Let I ⊂ R be a finite interval such that (6) holds for all λ ∈ I . Then there exist C > 0 and c > 0 such that for each λ ∈ I , 1 ≤ j ≤ W , ∈ (0, 1], and n ≥ 1 The arguments leading to the following corollary of Lemma 2.1 are also well known (for W = 1, see e.g. Jitomirskaya and Zhu [22,Section 5]; we also mention a result of Craig-Simon [7, Theorem 2.3], which is not quantitative, but on the other hand holds in more general setting).
Proof. If n ≤ 100W 2 or 2 ≤ 100 log n/n, we can ensure the desired inequality by adjusting the constants, therefore we assume that n > 100W 2 and 2 ≥ 100 log n/n. Consider the j-th exterior power n (λ ) ∧ j of n (λ ), so that Each matrix element p(λ ) of n (λ ) ∧ j (where p runs in a finite set P enumerating the matrix elements) is a polynomial of degree ≤ jn ≤ W n in λ. Now we use the following result of Bernstein [4], although we require much less than its full strength (in place of the logarithmic dependence on the degree with a precise constant, we could do with any prefactor growing slower than exponentially): for any polynomial q of degree n Returning to our setting, let then we have for any p ∈ P: By Lemma 2.1 and the choice of r , Thus by (19) P and this completes the proof.
The probability density of transfer matrices The following lemma builds on the arguments going back to the work of Ricci and Stein [33]. In the context of random Schrödinger operators, similar reasoning appears in the work Shubin, Vakilian and Wolff [35]. Recently, a general argument in the setting of motivic morphisms has been developed by Glazer and Hendel [11]; further arguments are discussed in [12]. For completeness, we sketch a proof (restricted to the generality of the current discussion) below. Then there exists C such that for any n ≥ n 0 the distributions of V n (λ), U n (λ) and V * n (λ)U (λ) are absolutely continuous with density ≤ C with respect to the Haar measure on the compact symplectic group Sp(2W, R) ∩ SO(2W, R).

Moreover, the bounds in (a)-(b) are locally uniform in λ.
Remark 2.4. For concreteness, we may assume that the singular value decomposition is constructed so that n is diagonal with strictly decreasing positive entries on the diagonal, and the first non-zero entry of eich column of U n and V n is positive.
Proof. Consider the product map According to [ . Hence det[(D F n 1 ) * (D F n 1 )] is not identically zero; by continuity, the maximum of its absolute value is bounded away from zero locally uniformly in λ.
The map (20) is real analytic, therefore the probability density of n 1 (λ) lies in L p for some p > 1 (this can be proved directly as in [12] or deduced from [33, Proposition 2.1] using an appropriate embedding theorem), and, again, both p and the bound are locally uniform in λ. Applying the inequality (which is a simple special case of the Young convolution inequality on Sp(2W, R)), we obtain that for n 0 = n 1 ( (1 − 1/ p) −1 + 1) the density of n 0 (λ) is bounded. This proves the first item, from which the second one follows.
A geometric lemma Denote by S(F) the unit sphere of an Euclidean vector space F. For future reference, we record the following fact (attributed to Archimedes): if u is a random vector uniformly distributed on S(R ), then the probability density of the random vector P F u, where P F : R → F be the orthogonal projection onto a fixed k-dimensional subspace F ⊂ R , 1 ≤ k ≤ − 1, is given by

Lemma 2.5. Let U be a random matrix taking values in
where a 1 ≥ a 2 ≥ · · · ≥ a , and let F ⊂ R be a k-dimensional subspace. Then for any Proof. It is sufficient to prove the estimate for the ∞ norm · ∞ in place of the Euclidean norm, as this will only affect the value of the numerical constant C . We first observe that for a fixed u ∈ S(R ) Indeed, let j 0 be such that a j 0 ≥ a > a j 0 +1 . The random vector ((U u) j ) j 0 j=1 has bounded density in a neighbourhood of zero (according to (21), for j 0 ≤ − 2 the density is uniformly bounded, whereas for j 0 = − 1 it explodes only on the boundary of the unit ball). Therefore thus concluding the proof of (22).
For any k-dimensional subspace F 1 ⊂ R and v ∈ S(F 1 ), the k −1 dimensional measure of the intersection of Q v with S(F 1 ) admits the lower bound whence by the Chebyshev inequality, the Fubini theorem and (22)
Denote by u j (λ ) ( j = 1, 2, 3, 4) the right singular vectors of n (λ ) (i.e. the eigenvectors of n (λ ) * n (λ ); the choice of the direction of the vectors will be specified later), and by P F 0 -the orthogonal projection onto F 0 . Let where C > 0 will be specified shortly. The required estimate (27) follows from (24) and the following two ingredients: a propagation estimate which replaces Claim 3.1, and the single-energy bound which replaces Claim 3.2.
To prove (37), we use once again that if U is uniformly distributed on the compact symplectic group Sp(2W, R) ∩ SO(2W, R), then each column of U is uniformly distributed on the unit sphere. Thus, according to Lemma 2.3, the probability density of u 1 (λ ) with respect to the Haar measure on S(R 2W ) is bounded uniformly in n ≥ n 0 . Hence by (21) This concludes the proof of (37) and of the theorem.

On the Uniform Convergence to the Lyapunov Exponents
A better understanding of the deviations of 1 n log s j ( n (λ)) from their limiting values γ j would allow us to strengthen the conclusion of Theorems 1 and 2, possibly up to the conjectured γ + * conj = γ 1 , as we now discuss. Recall the following result from [13,14] pertaining to W = 1: with probability one, the set 1 is dense in σ (H ). Subsequently, it was found that also the (possibly) smaller set 0 is almost surely dense in σ (H ). Recently, a general framework encompassing and generalising these results was developed by Gorodetski and Kleptsyn [19], who also provided detailed information on the structure of the exceptional sets τ , and showed that P ∀λ ∈ R : lim sup We are not aware of a published reference discussing the extension of this problem for W > 1. However, it is plausible that the arguments developed in the aforementioned works could yield that if it were true that P ∀λ ∈ σ (H ) lim n→∞ 1 n log s j ( n (λ)) = γ j (λ) one could significantly improve the results of the current paper: the argument of Theorem 1 would yield γ + * ≤ γ * ,2 , where γ * ,2 is the solution of whereas the argument of Theorem 2 would establish the optimal bound γ + * = γ W (for arbitrary W , cf. the proof of Theorem 3 below). If (42) is false, it would be helpful to understand is it true that P ∀λ ∈ σ (H ) lim sup Following Craig and Simon [7] (cf. Lemma 2.2), note that (43) holds (unconditionally) for j = 1. Also (according to the same lemma) (42) would imply (43).
In this section, we prove the following extension of (41) to 2W -dimensional cocycles. We confine ourselves to the setting of transfer matrices, which is used in the proof of Theorem 3. Denote In particular, P sup λ∈ [a,b] lim inf n→∞ Dev n (λ) = 0 = 1.

Remark 5.2.
Here n 2 can be replaced with any function tending to infinity faster than linearly.
Hered n 2 (·) is independent of d n (·). Also recall from Lemma 2.2 that P( Now, Lemma 2.1 implies that for each λ ∈ (λ − r (λ), λ + r (λ)) where |||X ||| = max α,β |X α,β | (the 1 to ∞ norm). Note that each matrix entry of n (λ ) is a polynomial in λ of degree n, therefore the set A (1) n of λ ∈ (λ − r (λ), λ + r (λ)) for which | 1 n log ||| n (λ )||| − γ 1 (λ)| ≥ 2 is a union of at most W 2 n intervals. Applying the same argument to the wedge powers n (λ) ∧ j , we construct the sets A (2) n is a union of at most C j (W )n intervals (where C j (W ) may depend only on j and W ), and We also construct similar sets A ( j) n 2 ,n corresponding to n 2 ,n (λ ), and let The set A n is a union of ≤ C(W )n intervals, whereas A n 2 ,n is a union of ≤ C(W )n 2 intervals. If these two sets intersect, than either one of the edges of the intervals comprising A n lies in A n 2 ,n , or vice versa. Invoking (48), we see that Observe that on (1) n ∩ (2) n ∩ (3) n , for each λ , either 1 n log s j ( n (λ )) is close to γ j (λ) for all j, or this holds true for 1 n 2 log s 1 ( n 2 (λ )). This concludes the proof of the proposition.
On the other hand, (21) implies that for each θ ∈ S(R W −1 ) Therefore the probability that there exists θ satisfying (56) is at most as claimed. This concludes the proof of (53) and of the theorem.