Non-Lyapunov annealed decay for 1d Anderson eigenfunctions

Macera, Davide

doi:10.1063/5.0166218

In Exact dynamical decay rate for the almost Mathieu operator by Jitomirskaya et al. [Math. Res. Lett. 27(3), 789–808 (2020)], the authors analysed the dynamical decay in expectation for the supercritical almost-Mathieu operator in function of the coupling parameter, showing that it is equal to the Lyapunov exponent of its transfer matrix cocycle, and asked whether the same is true for the 1d Anderson model. We show that this is never true for bounded potentials when the disorder parameter is sufficiently large.

I. INTRODUCTION

Consider the one-dimensional Anderson model, i.e., the operator H acting on a dense subset of

ℓ^{2} (Z)

via

(H ψ) (x) = ψ (x + 1) + V_{x} ψ (x) + ψ (x - 1), x \in Z,

(1)

where V_x are i.i.d. random variables. We assume that the distribution of V₀ is bounded and not concentrated at one point (in most of the discussion below, the first assumption can be relaxed to the existence of a finite fractional moment

E | V_{0} |^{η} < \infty

⁠). Carmona–Klein–Martinelli⁵ showed that under these assumptions H exhibits Anderson localisation, i.e., almost surely H has pure point spectrum, and moreover

P \{\forall (λ, ψ) \in E \underset{x \to \pm \infty}{lim sup} \frac{1}{| x |} \log | ψ (x) | = - γ (λ)\} = 1,

(2)

where

E = E [H]

is the collection of eigenpairs of H (the spectrum is almost surely simple, so |ψ| is well-defined), and γ(λ) is the Lyapunov exponent of H at energy λ. Under more restrictive assumptions on H, the pure point nature of the spectrum was first proved by Goldsheid–Molchanov–Pastur⁸ and by Kunz–Souillard;¹² the exponential decay of the eigenfunctions was first established by Molchanov.¹⁵

While the proof of Ref. 5 employs multi-scale analysis, single-scale proofs have recently been found by Bucaj et al.,⁴ Gorodetski–Kleptsyn,⁹ and Jitomirskaya–Zhu.¹¹ Generalisations to models with off-diagonal disorder and to matrix-valued potentials are studied in Refs. 13 and 16.

A stronger notion of Anderson localisation involves the notion of eigenfunction correlator, introduced by Aizenman.¹ Denote

Q_{H} (x, y) = \sup \{| f (H) (x, y) | : f : R \to C, \sup | f | ⩽ 1\},

where the supremum is taken over Borel functions. If H has pure point spectrum, the correlator takes the form

Q_{H} (x, y) = \sum_{(λ, ψ) \in E} | ψ_{λ} (x) | | ψ_{λ} (y) | .

(3)

Then there exists γ > 0 such that for any x,

P \{\underset{y \to \pm \infty}{lim sup} \frac{1}{| y - x |} \log Q (x, y) ⩽ - γ\} = 1 .

(4)

In fact, in the current setting holds with γ = γ_inf, where

γ_{\inf} = \inf_{λ \in σ (H)} γ_{λ} (H)

(5)

and σ(H) is the spectrum of H (a deterministic set)—see Ref. 11. This strong form of implies , as well as dynamical localisation, decay of the Fermi projection as well as other properties of relevance in quantum dynamics.

Ge and Zhao built on the work¹¹ and proved the following:

Theorem 1.1

(Ge–Zhao). For the operator H of (1) with V₀ bounded and not concentrated at one point, one has, for any

x \in Z

,

γ^{E} = - \underset{y \to \pm \infty}{lim sup} \frac{1}{| y - x |} \log E Q (x, y) > 0 .

(6)

In Sec. II we give another, arguably, simpler, proof of this result, adopting an argument from Ref. 6.

Jitomirskaya et al. ¹⁰ studied the validity of (6) in the almost-periodic setting, namely, for the supercritical almost-Mathieu operator with Diophantine frequency, and showed that in that setting $γ^{E}$ can be taken to be equal to γ_inf. They asked whether the same is true for the Anderson model. We show that this is not the case. A first counterexample comes from the Anderson–Bernoulli model:

Theorem 1.2.

For a > 0, consider the operator H^a = H₀ + aV with V_x being a bounded random variable having an atom at 0. Then $γ^{E}$ is bounded from above uniformly in a.

In particular, if V_x is a Bernoulli random variable with parameter p, by a result of Martinelli and Micheli,¹⁴ γ_inf ⩾ c log a for sufficiently large a. Therefore, by the above theorem, $γ^{E} (H^{a}) \neq γ_{I} (H^{a})$ for a large enough.

Furthermore, the above theorem remains true for any bounded random potential satisfying mild conditions, at sufficiently high disorder:

Theorem 1.3.

Let $V = {V_{i}}_{i \in Z}$ be a nondeterministic, bounded, i.i.d. random potential, and let H^a ≔ H₀ + aV.

Then, for any a large enough

\inf_{E \in σ (H^{a})} γ_{E} (H^{a}) > γ^{E} (H^{a}) .

II. PROOF OF THEOREM 1.1

For

a, b \in Z

⁠, denote by H_[a,b] the restriction of H to [a, b] (with Dirichlet boundary conditions), and let

G_{E} [H_{[a, b]}] = {(H_{[a, b]} - E)}^{- 1}

⁠. Let τ > 0,

E \in R

and N ⩾ 1. A site

x \in Z

is called (τ, E, N)-nonresonant [x ∉ Res(τ, E, N)] if

| G_{E} [H_{[x - N, x + N]}] (x, x \pm N) | ⩽ e^{- (γ (E) - τ) N} .

Otherwise, x is called (τ, E, N)-resonant [x ∈ Res(τ, E, N)]. The proof of the theorem uses the following.

Claim 1.

Assume that V₀ is bounded and not concentrated at one point. Then for any τ > 0 there exist C, c > 0 such that

P \{\forall E \in R d i a m (R e s (τ . E, N) \cap [- N^{2}, N^{2}]) > 2 N\} ⩽ C e^{- c N} .

See Ref. 13, Proposition 2.1 for this formulation (in the more general case of matrix potentials) and Ref. 11, Theorem 4.1, for a similar statement in the pure one-dimensional case.

Next, we need a representation for the eigenfunction correlator as a singular integral [see (7.4) at p. 102 of Ref. 2]:

Q (x, y) = \lim_{L \to \infty} Q^{L} (x, y), Q^{L} (x, y) = \lim_{ϵ \to 0^{+}} \frac{ϵ}{2} \int | G_{E} [H_{[- L, L]}] (x, y) |^{1 - ϵ} d E ⩽ 1 .

(7)

Having these two ingredients, we argue as follows. Without loss of generality we can assume that x = 0. Set

τ = \frac{1}{2} \min_{E \in σ (H)} γ (E)

and

N = ⌊ \frac{y}{10} ⌋

⁠, and consider the event

R = \{\exists E \in R : 0, y \in R e s (τ, E, N)\} .

According to Claim 1,

P (R) ⩽ C e^{- c N}

⁠. On the complement

Ω \ R

⁠, we have

R = A \cup B

⁠, where

A = \{E \in R : 0 \notin R e s (τ, E, N)\}, B = \{E \in R : y \notin R e s (τ, E, N)\} .

For E ∈ A,

\begin{aligned} G_{E} [H_{[- L, L]}] (0, y) = & G_{E} [H_{[- N, N]}] (0, N) G_{E} [H_{[- L, L]}] (N + 1, y) \\ + & G_{E} [H_{[- N, N]}] (0, - N) G_{E} [H_{[- L, L]}] (- N - 1, y), \end{aligned}

whence

| G_{E} [H_{[- L, L]}] (0, y) | ⩽ e^{- τ N} (| G_{E} [H_{[- L, L]}] (N + 1, y) | + | G_{E} [H_{[- L, L]}] (- N - 1, y) | .

and an analogous bound can be deduced for B.

Thus

\lim_{ϵ \to + 0} \frac{ϵ}{2} [\int_{A} + \int_{B}] | G_{E} [H_{[- L, L]}] (0, y) |^{1 - ϵ} d E ⩽ 4 e^{- τ N} .

Finally,

\begin{aligned} E Q (0, y) & = E (Q (0, y) | R) P (R) + E (Q (0, y) | Ω \ R) (1 - P (R)) \\ ⩽ P (R) + E (Q (0, y) | Ω \ R) ⩽ C e^{- c N} + 2 e^{- τ N} . \end{aligned}

Thus,

γ^{E} ⩾ \min (c, τ) > 0

⁠.

Remark.

This proof can be extended to quasi-one-dimensional operator, such as the Anderson model on the strip of width W or the more general model studied in Ref. 13. A slightly weaker version of is still true in this case (see Ref. 6):

Q^{L} (x, y) ⩽ \lim_{ϵ \to 0^{+}} \frac{ϵ}{2} \int ‖ G_{E} [H_{[- L, L]}] (x, y) ‖^{1 - ϵ} d E ⩽ W,

and the argument above follows with minor modifications.

III. PROOF OF THEOREM 1.2

Let K > 0 be a large numerical constant (independent of any parameters), to be specified later. For x > 0, consider the event

Ω_{K, x} = \{\forall y \in [- K x, (K + 1) x], V_{y} = 0\} .

We shall prove the following: for any ϵ > 0, one has on Ω_K,x for sufficiently large x:

Q (0, x) ⩾ e^{- c | x |} .

(8)

Since

P (Ω_{K, x}) = {(1 - p)}^{2 K x + 1}

⁠, this this would imply that

γ^{E} ⩽ - \lim_{x \to \infty} \frac{1}{x} \log {(1 - p)}^{2 K x + 1} e^{- c x} ⩽ c - 2 K \log (1 - p)

(9)

as claimed.

We now turn to the proof of . Since the argument is uniform in a, we will use H for H^a. Observe that for any δ > 0,

Q (x, y) ⩾ δ | G_{δ} [H] (x, y) | .

(10)

In fact,

Q (x, y) = \sum_{(λ, ψ) \in E} | ψ_{λ} (0) | | ψ_{λ} (x) | ⩾ δ \sum_{(λ, ψ) \in E} \frac{| ψ_{λ} (0) | | ψ_{λ} (x) |}{| i δ - λ |} ⩾ δ | G_{i δ} [H] (0, x) | .

Let T = H₀ be the free Laplacian [obtained by setting V_x ≡ 0 in ], and let T_K,x be the restriction of T to the finite volume [−Kx, (K + 1)x]. Then, by applying the resolvent identity and the reverse triangle inequality twice, we get

\begin{aligned} | G_{i δ} [H] (0, x) | \\ ⩾ | G_{i δ} [T_{K, x}] (0, x) | \\ - | G_{i δ} [T_{K, x}] (0, - K x) | | G_{i δ} [H] (- K x - 1, x) | \\ - | G_{i δ} [T_{K, x}] (0, (K + 1) x) | | G_{i δ} [H] ((K + 1) x + 1, x) | \\ ⩾ | G_{i δ} [T] (0, x) | \\ - | G_{i δ} [T_{K, x}] (0, - K x) | | G_{i δ} [H] (- K x - 1, x) | \\ - | G_{i δ} [T_{K, x}] (0, (K + 1) x) | | G_{i δ} [H] ((K + 1) x + 1, x) | \\ - | G_{i δ} [T_{K, x}] (0, - K x) | | G_{i δ} [T] (- K x - 1, x) | \\ - | G_{i δ} [T_{K, x}] (0, (K + 1) x) | | G_{i δ} [T] ((K + 1) x + 1, x) | . \end{aligned}

By the Combes–Thomas estimate (Ref. 2, Theorem 10.5), we deduce that for δ ∈ (0, 1),

| G_{i δ} [H] (0, x) | ⩾ | G_{i δ} [T] (0, x) | - \frac{C_{1}}{δ} e^{- c_{1} (2 K + 1) δ | x |} .

Now,

| G_{i δ} [T] (0, x) | ⩾ C_{2} e^{- c_{2} δ | x |} .

(11)

Indeed, g(x) = G_iδ[T](0, x) is by definition the square-summable solution to the equation

g (x + 1) + g (x - 1) + i δ g (x) = δ_{x, 0} .

Plugging in the ansatz g(x) = αe^−ξ|x|, we find that this is indeed a solution provided that

ξ = a r c c o s h (\frac{i δ}{2}); α = {(2 + 2 e^{- ξ} + δ)}^{- 1},

hence

| g (x) | ⩾ | α | e^{- (R ξ) | x |}

⁠, where

R ξ ⩽ c_{2} δ

for δ > 0 small enough, as

R ξ = \log (\frac{δ}{2} + \sqrt{\frac{δ^{2}}{4} + 1}) = \log (1 + \frac{δ}{2} + o (δ^{2})) = \frac{δ}{2} + o (δ^{2})

in a neighbourhood of 0.

Having set c = c₂δ and 2K + 1 = ⌈100 c₂/c₁⌉, we obtain (8).

IV. PROOF OF THEOREM 1.3: LOGARITHMIC DIVERGENCE FOR GENERAL POTENTIALS

A version of the Martinelli–Micheli bound the 1d Anderson model with absolutely continuous, bounded potential has been proven in 1983 by Avron et al. in Ref. 3.

Theorem 4.1

(Avron et al.³). Let H^a = H₀ + aV be a random Schrödinger operator where V is a bounded random potential with absolutely continuous density. Then the Lyapunov exponent γ_a of H^a is such that

γ_{a} ⩾ \log (a) - K,

where K is a finite constant.

We will adapt the proof in Ref. 3 to any potential having finite first moment (not necessarily absolutely continuous).⁷ Avron et al.’s proof relies on the Thouless’ formula for the Lyapunov exponent of a Schrödinger operator H, stating that

γ_{H} (E) = \int \log | E - E^{'} | d κ_{H} (E^{'}),

(12)

where κ_H(E′) denotes the integrated density of states of the operator H. They proceed then to bound the negative part of the logarithm in 12 using the Wegner estimate: If H = H₀ + V is a random Schrödinger operator with i.i.d. potential, then

\frac{d κ_{H} (E)}{d E} ⩽ ‖ ρ_{V} ‖_{\infty},

which is unfortunately proven true only when the distribution ρ_V of the potential is absolutely continuous.

Fortunately, Shubin et al. proved in Ref. 17 a slightly weaker bound for the IDS of a random Schrödinger operator whose potential satisfies the conditions of Theorem 1.3: if

E [| V_{0} |] < \infty

(condition that we have automatically since the distribution of V₀ is bounded), then

| κ_{H} (E) - κ_{H} (E^{'}) | ⩽ C | E - E^{'} |^{α}

(13)

for some α ∈ (0, 1) and some constant C > 0. This bound is sufficient to let the Avron–Craig–Simon argument work in the present generality.

By applying the Thouless formula to H^a and splitting the logarithm into its positive and negative parts, we get that

\begin{aligned} \int \log | E - E^{'} | d κ_{H^{a}} (E^{'}) \\ = & \int \log_{+} | E - E^{'} | d κ_{H^{a}} (E^{'}) - \int \log_{-} | E - E^{'} | d κ_{H^{a}} (E^{'}) \\ ⩾ & \int_{a / 2}^{\infty} \log_{+} | E - E^{'} | d κ_{H^{a}} (E^{'}) - \int_{0}^{\infty} κ_{H^{a}} {E^{'} : | E - E^{'} | ⩽ e^{- t}} d t . \end{aligned}

We claim that

\int_{0}^{\infty} κ_{H^{a}} {E^{'} : | E - E^{'} | ⩽ e^{- t}} d t ⩽ C;

(14)

for some C > 0 uniform in a, and that

\int_{a / 2}^{\infty} \log_{+} | E - E^{'} | d κ_{H^{a}} (E^{'}) ⩾ c \log a

(15)

for some positive constant c.

The first bound is proven by using inequality :

\begin{aligned} \int_{0}^{\infty} κ_{H^{a}} {E^{'} : | E - E^{'} | ⩽ e^{- t}} d t ⩽ & ‖ V ‖_{\infty} \int_{0}^{\infty} e^{- t} d t^{α} \\ ⩽ & C \int_{0}^{\infty} α t^{α - 1} e^{- t} d t \\ ⩽ & C Γ (α) ⩽ C^{'} . \end{aligned}

Remark.

We strongly believe that the main result of this section (and thus Theorem 1.3 as a whole) can be extended to the most general setting for which 1-d localisation has been proven (nondeterministic potential with any finite fractional moment). However, generalising (13) to the case where one only has a generic fractional moment appears to be nasty.

An alternative to this approach could be extending the proof of the logarithmic divergence for the Anderson–Bernoulli model in Ref. 14 to the general case; however, even if this seems to be doable and should not present major technical difficulties, additional estimates would be needed to make Martinelli’s and Micheli’s already six page long proof work for generic potentials, and many formulas would get much longer and nastier.

In conclusion, the length of the present paper would likely get doubled by such an attempt, therefore we avoid it to keep the paper short and more readable while keeping the result reasonably general.

V. PROOF OF THEOREM 1.3: GREEN FUNCTION ESTIMATES

Since the above argument uses crucially the fact that a Bernoulli random variable is 0 with positive probability, one might suspect that the presence of an atom at zero is required for the annealed dynamical decay to be non-Lyapunov. However, in this section we will use a simple trick to eliminate the atom at 0. The trick relies on the observation that it is possible to decompose any dilated random variable aX having an atom as a sum of two (not necessarily independent) random variables, one of which is bounded in a, and the other has basically the same distribution as aX with the difference that the atom has been subtracted some mass. If we subtract in this way mass from the atom at a sufficient rate, and control the error given by the bounded addend, then we can show that the growth of the annealed decay rate in a is much slower than the logarithmic lower bound prescribed by the results of Avron–Craig–Simon and Shubin–Vakilian–Wolff.

Proof.

In order to exploit the case of a potential with an atom at 0, we will make use of the following observation.

Observation 5.1.

Let X be a bounded random variable, and let

\bar{x} \in s u p p (X) \subseteq [- R, R]

. Suppose that X is absolutely continuous in a neighborhood of

\bar{x}

and denote by

{\tilde{X}}^{ϵ}

the random variable having the same density as X, except for the fact that it has an atom at

\bar{x}

of mass ϵ, suitably renormalised. Then there exists a bounded random variable

\tilde{η}

(not necessarily independent on X) such that

a X \overset{d}{=} a^{'} {\tilde{X}}^{ϵ} + \tilde{η}

for some a′ > 0.

This observation basically asserts that we can remove (or, by extension, subtract mass to) an atom from the distribution of a random variable at the cost of adding another (dependent) random variable uniformly bounded in the coupling.

It follows by simply observing that if $\tilde{η}$ has the same distribution of X and ${\tilde{X}}^{ϵ}$ is chosen to take the same values as $\tilde{η}$ (so that ${\tilde{X}}^{ϵ}$ would retain its usual law and its atom at $\bar{x}$ ⁠, but becoming totally dependent on $\tilde{η}$ ⁠), then $a {\tilde{X}}^{ϵ} + \tilde{η} \overset{d}{=} (a + 1) X$ ⁠.

Observation 5.2.

Without loss of generality, we can take R = δ/10.

In fact, δ is by construction always positive and we can always multiply the potential by any finite constant and incorporate such constant into the disorder parameter a.

We now use these two observations to prove the general result. Take V such that supp(V) ⊆ [−R, R] and set R = δ/10. Apply Observation 5.1 to the potential V with ϵ = ϵ(a) ≫ a^−β for all β > 0, and decompose $a V = a {\tilde{V}}^{ϵ (a)} + \tilde{η}$ ⁠, where $\tilde{η} \overset{d}{=} V$ ⁠, and ${\tilde{V}}^{ϵ (a)}$ is a random variable having the same distribution as V except for having an atom at 0 of mass ϵ(a) (with the necessary renormalisation). Then $H^{a} = T^{R} + a {\tilde{V}}^{ϵ (a)}$ ⁠, where $T^{R} = T_{0} + \tilde{η}$ ⁠.

Thus again, if Q(0, x) ⩾ e^−c|x| we get, as in ,

γ^{E} ⩽ c + 2 K \log (ϵ (a)) ≪ \log (a) .

(16)

Furthermore, we call

T_{K, x}^{R}

the restriction of T^R to the box [−Kx, (K + 1)x], analogously as before. We will shift T^R by −2 · 1 so that the spectrum of the resulting operator lies below δ/2. A double resolvent expansion analogue to the one performed in the Proof of Theorem 1.2 and the Combes-Thomas bound yield

\begin{aligned} Q (0, x) ⩾ & δ G_{δ} [H^{a} - 2 \cdot 1] (0, x) \\ ⩾ & δ G_{δ} [T^{R} - 2 \cdot 1] (0, x) - C_{3} e^{- c_{3} (2 K + 1) \sqrt{δ / 2} | x |} \end{aligned}

on the event

Ω_{x, K; a} ≔ {V_{y}^{ϵ (a)} \equiv 0 \forall y \in [- K x, (K + 1) x]}, P {Ω_{x, K; a}} = {[ϵ (a)]}^{(2 K + 1) x} .

Remark.

Notice that this time we chose δ instead of iδ as a spectral parameter. The reason for this choice is that we need to use the negativity of the shifted Laplacian to compare its Green’s function to that of the (negative) shifted operator.

Eventually, the only thing left to us to show is that

G_{δ} [T^{R} - 2 \cdot 1] (0, x) ⩾ \tilde{C} e^{- \tilde{c} \sqrt{δ} | x |} .

By writing down the Neumann series for G_δ[T − 2 · 1], we get the following inequalities:

\begin{aligned} | G_{δ} [T^{R} - 2 \cdot 1] (x, y) | = & {[- T_{0} {(2 + δ - \tilde{η})}^{- 1}]}^{- 1} (x, y) \cdot \frac{1}{2 + δ - \tilde{η}} \\ = & |(\sum_{n = 0}^{\infty} {(2 + δ - \tilde{η})}^{- n} {(T_{0})}^{n} (x, y))| \cdot \frac{1}{2 + δ - \tilde{η}} \\ ⩾ & \frac{1}{2 + \frac{δ}{2}} |\sum_{n = 0}^{\infty} T_{0}^{n} (x, y) {(2 + \frac{δ}{2})}^{- n}| \\ = & | G_{δ / 2} [T_{0} - 2 \cdot 1] (x, y) | . \end{aligned}

We can compute G_δ/2[T₀ − 2 · 1](0, x) explicitly via the same method used in the proof of 1.2. In this case, we get that

| G_{δ / 2} [T_{0} - 2 \cdot 1] (x, y) | ⩾ | α | e^{- ξ | x |}, with α = {(2 e^{- ξ} + \frac{δ}{2} - 2)}^{- 1}, ξ = a r c c o s h (1 + \frac{δ}{4}) .

In particular, when δ is small, G_δ/2[T₀ − 2 · 1](x, y) decays exponentially with rate of order

\sqrt{δ}

and

G_{δ} [T^{R} - 2 \cdot 1] (0, x) ⩾ G_{δ / 2} (T_{0} - 2 \cdot 1)] (0, x) ⩾ \tilde{C} e^{- \tilde{c} \sqrt{δ / 2} | x |} .

Setting, again, K large enough so that

2 K + 1 = ⌈ 100 \tilde{c} / c_{3} ⌉

⁠, and setting

c = \tilde{c} \sqrt{\frac{δ}{2}}

⁠, we finally conclude that

Q (0, x) ⩾ \frac{δ}{2} G_{δ} [T^{R} - 2 \cdot 1] (0, x) ⩾ e^{- c | x |} .

This, combined with and the Avron–Craig–Simon bound for general bounded potentials proven in Paragraph 4, implies the thesis.□

ACKNOWLEDGMENTS

I am deeply indebted to my former Ph.D. advisor, Sasha Sodin, for suggesting the topic of this paper and for giving major contributions to its development. I believe he should have been a co-author of this paper, but for reasons I do not understand he asked me to erase his name from it.

I am also grateful to Alexander Elgart for many useful comments on a preliminary version of this paper, and to the anonymous referee for useful suggestions and for pointing out a flaw in the original argument.

This work has been supported by the Grant No. EPSRC EP∖T004290∖1.

This work was started when the author was a Ph.D. student at Roma Tre University.

AUTHOR DECLARATIONS

Conflict of Interest

The author has no conflicts to disclose.

Author Contributions

Davide Macera: Conceptualization (equal); Investigation (equal); Validation (equal); Writing – original draft (equal); Writing – review & editing (equal).

DATA AVAILABILITY

Data sharing is not applicable to this article as no new data were created or analyzed in this study.

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Published open access through an agreement with JISC Collections

Non-Lyapunov annealed decay for 1d Anderson eigenfunctions

I. INTRODUCTION

II. PROOF OF THEOREM 1.1

III. PROOF OF THEOREM 1.2

IV. PROOF OF THEOREM 1.3: LOGARITHMIC DIVERGENCE FOR GENERAL POTENTIALS

V. PROOF OF THEOREM 1.3: GREEN FUNCTION ESTIMATES

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Non-Lyapunov annealed decay for 1d Anderson eigenfunctions Open Access

I. INTRODUCTION

II. PROOF OF THEOREM 1.1

III. PROOF OF THEOREM 1.2

IV. PROOF OF THEOREM 1.3: LOGARITHMIC DIVERGENCE FOR GENERAL POTENTIALS

V. PROOF OF THEOREM 1.3: GREEN FUNCTION ESTIMATES

ACKNOWLEDGMENTS

AUTHOR DECLARATIONS

Conflict of Interest

Author Contributions

DATA AVAILABILITY

REFERENCES

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Non-Lyapunov annealed decay for 1d Anderson eigenfunctions