On the spatial extent of localized eigenfunctions for random Schrödinger operators

Abstract

On $\mathbb {Z}^d$, consider $\varphi$, an $\ell ^2$-normalized function that decays exponentially at $\infty$ at a rate at least $\mu$. One can define the onset length (of the exponential decay) of $\varphi$ as the radius of the smallest ball, say, B, such that one has the following global bound $\displaystyle |\varphi (x)|\le \Vert \varphi \Vert _\infty e ^{-\mu \, \text {dist}(x,B)}$. The present paper is devoted to the study of the onset lengths of the localized eigenfunctions of random Schrödinger operators. Under suitable assumptions, we prove that, with probability one, the number of eigenfunctions in the localization regime having onset length larger than $\ell$ and localization center in a ball of radius L is smaller than $C L^d\exp (-c \ell )$, for $\ell >0$ large (for some constants $C,c>0$). Thus, most eigenfunctions localize on small size balls independent of the system size which is the physicists understanding of localization; to our knowledge, this did not result from existing mathematical estimates. For energies near the edge of the spectrum, we also provide a lower bound of the same type on the number of those eigenfunctions; in dimension 1, the upper and lower bounds only differ by a logarithmic correction. Finally, we give a number of numerical results that exemplify situations giving rise to large onset lengths, that corroborate the validity of our main result and that suggest that, up to lower order terms, the above defined cumulative distribution of onset lengths shows asymptotic exponential decay at some definite rate.

Appendices

SULE Bound from Eigenfunction Correlators

In the literature, spectral localization is frequently expressed via a bound

$$\begin{aligned} \sum _{x} e^{\nu |x-y|} {\mathbb E}\left( Q_\Omega (I,x,y) \right) \ \le \ A \end{aligned}$$

(A.1)

with constants A and $\nu$ independent of $\Omega$, where $Q_\Omega (I,x,y)$ is the eigenfunction correlator of $H_\Omega$ on I (see [2, Chapter 7]). For a finite region $\Omega$,

$$\begin{aligned} Q_\Omega (I,x,y) \ = \ \sum _{E\in I\cap \sigma ((H_\omega )_\Omega )} |\varphi _E(x)||\varphi _E(y)|, \end{aligned}$$

where $\varphi _E$ is the normalized eigenvector corresponding to eigenvalue E. For the operators considered here, the spectrum is known to be almost surely simple [16, 21]; for operators with degenerate spectrum the term $|\varphi _E(x)||\varphi _E(y)|$ should be replaced by $|\langle \delta _x, P_E \delta _y\rangle |$, with $P_E$ the corresponding eigen-projection. For an infinite region, one may replace this definition with

$$\begin{aligned} Q_\Omega (I,x,y) \ = \ \sup _{f} \left| f((H_\omega )_\Omega )(x,y) \right| , \end{aligned}$$

where the supremum is taken over Borel measurable functions f with support in I and $|f(x)|\le 1$ everywhere. A posteriori, one concludes from (A.1) that $(H_\omega )_\Omega$ has pure point spectrum in I (almost surely), and (since the spectrum is simple) that

$$\begin{aligned} Q_\Omega (I,x,y) \ = \ \sum _{E\in I\cap {\mathcal E}((H_\omega )_\Omega ))} |\varphi _E(x)||\varphi _E(y)| . \end{aligned}$$

(A.2)

We now recall the derivation of a SULE estimate of the form (A3) from spectral localization (A.1).

Proposition A.1

Let $(H_\omega )_\Omega$ be a random operator on a region $\Omega \subset {\mathbb Z}^d$ such that $(H_\omega )_\Omega$ has simple, pure-point spectrum in I almost surely and (A.1) holds and let $\epsilon >0$. If $S\subset \Omega$ is a finite set, then, with probability greater than $1-\epsilon$, every eigenvector $\varphi _E$ of $(H_\omega )_\Omega$ with eigenvalue $E\in I$ and $\mathcal {C}(\varphi _E)\cap S \ne \emptyset$ satisfies

$$\begin{aligned} \left( \sum _{x\in \Omega } e^{\nu |x-y|} |\varphi _E(x)|^2 \right) ^{\frac{1}{2}} \ \le \ A \left( \frac{\#S}{\epsilon } \right) ^{\frac{1}{2}} \end{aligned}$$

(A.3)

for any $y\in \mathcal {C}(\varphi _E)\cap S$. In particular, (2.5) holds with $A_{\mathrm {AL}}=A$ and $\mu =\frac{\nu }{2}$.

Proof

From (A.1), it follows that

$$\begin{aligned} {\mathbb E}\left( \sum _{y\in S} \sum _{x\in \Omega } Q_\Omega (I,x,y)e^{\nu |x-y|} \right) \ \le \ A \# S . \end{aligned}$$

By Markov’s inequality, with probability $\ge 1- \epsilon$, we have

$$\begin{aligned} \sum _{y\in S} \sum _{x\in \Omega } Q_\Omega (I,x,y)e^{\nu |x-y|} \ \le \ A \frac{ \# S}{\epsilon } \end{aligned}$$

from which we conclude, using (A.2), that

$$\begin{aligned} \sum _{x\in \Omega } e^{\nu |x-y|} |\varphi _E(x)||\varphi _E(y)| \ \le \ A \frac{ \# S}{\epsilon } \end{aligned}$$

for every eigenvalue $E\in {\mathcal E}(H_\Omega )$ and each $y\in S$. If $\mathcal {C}(\varphi _E)\cap S\ne \emptyset$, then taking $y \in \mathcal {C}(\varphi _E)\cap S$, we have

$$\begin{aligned} \sum _{x\in \Omega } e^{\nu |x-y|} |\varphi _E(x)| \Vert \varphi _E\Vert _\infty \ \le \ A \frac{\# S}{\epsilon }. \end{aligned}$$

Since $|\varphi _E(x)|\le \Vert \varphi \Vert _\infty$ for every x, we conclude that

$$\begin{aligned} \sum _{x\in \Omega } e^{\nu |x-y|}|\varphi _E(x)|^2 \ \le \ A \frac{\# S}{\epsilon }. \end{aligned}$$

Taking the square root yields (A.3).

A Large Deviation Principle

Proposition B.1

Let $X_1,\ldots ,X_N$ be identically distributed random variables with

$$\begin{aligned} \Pr [X_j=1] = p \quad \text { and } \quad \Pr [X_j=0]=1-p. \end{aligned}$$

Suppose there is a partition of $\{1,\ldots ,N\}$ into K-disjoint subsets $S_1,\ldots ,S_K$ such that, for each $j=1,\ldots ,K$, the variables $(X_m)_{m\in S_j}$ are mutually independent. Then for any $\alpha \ge 1$,

$$\begin{aligned} \Pr \left[ \sum _{m=1}^N X_m > N (p+\delta ) \right] \ \le \ \exp \left( -\frac{\delta ^2 }{3K} N \right) . \end{aligned}$$

(B.1)

Proof

Let $Z(t)= {\mathbb E}[e^{t\sum _m X_m}].$ By Hölder’s inequality and the assumption that $(X_m)_{m\in S_j}$ are mutually independent,

It follows that

$$\begin{aligned} \Pr \left[ \sum _{m=1}^N X_m > N (p+\delta ) \right]\le & {} Z(t)e^{-N (p+\delta ) t} \ \le \ e^{N\left( p \frac{e^{Kt}-1}{K} -(p+\delta ) t\right) } \\\le & {} e^{N \left( \frac{e^{Kt}-1}{K} - (1+\delta ) t \right) }, \end{aligned}$$

where in the last step we have used that $e^{Kt}-1 -Kt\ge 0$. Optimizing over t yields

$$\begin{aligned} \Pr \left[ \sum _{m=1}^N X_m > N (p+\delta ) \right] \ \le e^{\frac{N}{K}\left( \delta -(1+\delta ) \log (1+\delta ) \right) } . \end{aligned}$$

Finally, eq. (B.1) follows since for $0\le \delta \le 1$.