Must the Spectrum of a Random Schrödinger Operator Contain an Interval?

Abstract

We consider Schrödinger operators in $\ell ^2({\mathbb Z})$ whose potentials are given by independent (not necessarily identically distributed) random variables. We ask whether it is true that almost surely its spectrum contains an interval. We provide an affirmative answer in the case of random potentials given by a sum of a perturbatively small quasi-periodic potential with analytic sampling function and Diophantine frequency vector and a term of Anderson type, given by independent identically distributed random variables (with some small-gap assumption for the support of the single-site distribution). The proof proceeds by extending a result about the presence of ground states for atypical realizations of the classical Anderson model, which we prove here as well and which appears to be new.

Appendices

Appendix A: A Remark on Unbounded Background Potentials

In this section we expand on a comment made in Remark 1.1. There it was pointed out that if the boundedness of $V_\mathrm {bg}$ is not assumed, it is possible to find a counterexample to the problem posed in Problem 1.2, namely one can find an unbounded background potential $V_\mathrm {bg}$ and a compactly supported non-degenerate $\nu$ such that all the resulting operators $H_\omega$ have empty essential spectrum, and hence it is impossible for the spectrum to contain an interval. In fact, the underlying principle is both simple and purely deterministic:

Proposition A.1

Suppose $V_\mathrm {bg} : {\mathbb Z}\rightarrow {\mathbb R}$ is given by $V_\mathrm {bg}(n) = n$, $n \in {\mathbb Z}$, and $V_\mathrm {b} : {\mathbb Z}\rightarrow {\mathbb R}$ is bounded. Then the essential spectrum of the Schrödinger operator H in $\ell ^2({\mathbb Z})$ with potential $V_\mathrm {bg}+ V_\mathrm {b}$ is empty.

Proof

Consider the transfer matrices

$$\begin{aligned} \Pi _n=\begin{pmatrix} E-n-V_\mathrm {b}(n) &{} -1 \\ 1 &{} 0 \\ \end{pmatrix} \end{aligned}$$

and the cone in ${\mathbb {R}}^2$ given by

$$\begin{aligned} K=\left\{ {\bar{v}}=(v_1, v_2)\ |\ |v_1|>|v_2|\right\} . \end{aligned}$$

Set $M := \Vert V_\mathrm {b}\Vert _\infty$ and fix a bounded open interval $I\subset {\mathbb {R}}$. Then for any $E\in I$, all sufficiently large n, for any ${\bar{v}}\in K$ we have

$$\begin{aligned} |(E-n-V_\mathrm {b}(n))v_1-v_2 |&\ge |n-E-M| \cdot |v_1|-|v_2| \\&\ge |n-E-M| \cdot |v_1| - |v_1| \\&\ge (n-E-M-1)|v_1|>|v_1|, \end{aligned}$$

Thus, since

$$\begin{aligned} \Pi _n{\bar{v}} = \begin{bmatrix}(E-n-V_\mathrm {b}(n))v_1-v_2 \\ v_1 \end{bmatrix}, \end{aligned}$$

$\Pi _n$ sends the cone K to itself. Moreover, $\Pi _n$ expands the vectors in K. Indeed,

$$\begin{aligned} |(E-n-V_\mathrm {b}(n))v_1-v_2 | + |v_1|&\ge (n-E-M)|v_1| \\&> \frac{1}{2}(n-E-M)(|v_1|+|v_2|). \end{aligned}$$

It follows that, for sufficiently large N (with a largeness condition that depends on M and I), the restriction of H to the half line $[N,+\infty )$ with a Dirichlet boundary condition at N has no spectrum on the open interval I, as the solution obeying the boundary condition is not a generalized eigenfunction for any $E \in I$.³

This in turn shows that the restriction of the operator H to the half-line $[0, +\infty )$ (with any self-adjoint boundary condition at zero) has no essential spectrum on the interval I.⁴

Similar arguments show that the half-line operator obtained by restriction of H to $(-\infty , 0]$ (again with any self-adjoint boundary condition at zero) has no essential spectrum on the interval I.

Combining the two statements, it then follows that H itself has no essential spectrum on the interval I. Since the choice of the bounded open interval I was arbitrary, Proposition A.1 follows. $\square$

Remark A.1

As was pointed out by one of the referees, there is another way of proving this proposition, by appealing to Weyl’s theorem about the stability of the essential spectrum under relatively compact perturbations. This argument is more general and conceptual than the hands-on argument given above. We thank the referee for pointing this out.

Appendix B: The Almost Sure (Essential) Spectrum

In this section we discuss non-randomness aspects of spectra that are well known in the ergodic setting. However, since we are interested in non-stationary random potentials in this paper, we need extensions of these results. It will turn out that Kolmogorov’s zero-one law can serve as a substitute, leading to the non-randomness of the essential spectrum. This will be explained in Subsection B.2 below. In Subsection B.3 we then derive an extension of Kotani’s support theorem from the ergodic setting to the non-stationary case. Along the way we also explain why under suitable additional assumptions, one does have a non-random spectrum, as formulated in Theorem 1.1 in the Introduction. Before presenting the proofs of these results in Subsections B.2 and B.3 we recall a useful characterization of the essential spectrum of a deterministic Schrödinger operator in terms of transfer matrix behavior in Subsection B.1, as this tool will be used in those proofs.

1.1 B.1. A characterization of the essential spectrum

The following characterization of the essential spectrum in terms of transfer matrices can be extracted from the denseness of generalized eigenvalues (energies for which there are polynomially bounded solutions of the Schrödinger equation) and the classical Weyl criterion; see [24, Proposition B.2].

Proposition B.1

Let $V:{\mathbb {Z}}\rightarrow {\mathbb {R}}$ be a bounded potential of the discrete Schrödinger operator H acting on $\ell ^2({\mathbb Z})$ via

$$\begin{aligned}{}[H u](n) = u(n+1) + u(n-1) + V(n) u(n). \end{aligned}$$

(B.1)

Then energy $E\in {\mathbb {R}}$ belongs to the essential spectrum of the operator H if and only if there exists $K>0$ such that for any $N\in {\mathbb {N}}$ there is a sequence $\{m_j\}_{j\in {\mathbb {N}}}, m_j\in {\mathbb {Z}},$ with $|m_j-m_{j'}|>2N$ if $j\ne j'$, and unit vectors ${\bar{u}}_j$, $|{\bar{u}}_j|=1$, such that $|T_{[m_j, m_j+i], E}\,{\bar{u}}_j|\le K$ for all $|i|\le N$ and all $j\in {\mathbb {N}}$, where $T_{[m, m+i], E}$ is the product of transfer matrices given by

$$\begin{aligned} T_{[m, m+i], E}=\left\{ \begin{array}{lll} \Pi _{m+i-1, E}\ldots \Pi _{m, E}, &{} \hbox { if }i> 0; \\ {\mathrm{Id}}, &{} \hbox { if }i= 0; \\ \Pi _{m+i, E}^{-1}\ldots \Pi _{m-1, E}^{-1}, &{} \hbox { if }i<0, \end{array} \right. \end{aligned}$$

and $\Pi _{n, E}=\left( \begin{array}{cc} E-V(n) &{} -1 \\ 1 &{} 0 \\ \end{array} \right)$.

1.2 B.2. Existence of the almost sure (Essential) spectrum

Recall that Theorem 1.1 asserts the existence of a non-random spectrum under the assumption that the background potential is generated by continuous sampling along the orbit of a minimal homeomorphism.

Proof of Theorem 1.1

Denote by $G:(\text {supp}\,\nu )^{{\mathbb {Z}}}\rightarrow (\text {supp}\,\nu )^{{\mathbb {Z}}}$ the left shift on the space of sequences $\omega \in (\text {supp}\,\nu )^{{\mathbb {Z}}}$, and consider the map

$$\begin{aligned} T\times G:X\times (\text {supp}\,\nu )^{{\mathbb {Z}}}\rightarrow X\times (\text {supp}\,\nu )^{{\mathbb {Z}}}. \end{aligned}$$

We have the following statement:

Lemma B.1

For every $x\in X$ and $\nu ^{\mathbb {Z}}$-almost every $\omega$, the $(T\times G)$-orbit of $(x, \omega )$ is dense in $X\times (\text {supp}\,\nu )^{{\mathbb {Z}}}$.

Proof

Fix any small $\varepsilon >0$, and any point $(y, \omega ')\in X\times (\text {supp}\, \nu )^{\mathbb {Z}}$. Due to the minimality of $T:X\rightarrow X$, there exists a sequence $m_j\in {\mathbb {N}}$, $m_j\rightarrow \infty$ as $j\rightarrow \infty$, such that $\text {dist}_X(T^{m_j}(x), y)<\varepsilon$. For any $j\in {\mathbb {N}}$, the probability of the event

$$\begin{aligned} \left\{ |{{\tilde{\omega }}}_{m_j+i}-\omega '_{m_j+i}|<\varepsilon \ \text {for any}\ |i|<\frac{1}{\varepsilon }, \ \text {where}\ (T\times G)^{m_j}(x, \omega )=(T^{m_j}(x), {{\tilde{\omega }}})\right\} \end{aligned}$$

is bounded away from zero, and if $|m_j-m_{j'}|>\frac{2}{\varepsilon }$, then those events are independent. Hence with probability one infinitely many of them must happen. Since $\varepsilon >0$ could be chosen arbitrarily small, Lemma B.1 follows. $\square$

Since for any two potentials defined by initial conditions that have dense orbits, the spectra of the corresponding operators coincide, Theorem 1.1 follows. $\square$

In cases where the background potential is not of the form considered in Theorem 1.1, we have the following substitute result. We are grateful to Victor Kleptsyn for communicating it to us.

Theorem B.1

Suppose $\{\nu _n\}_{n\in {\mathbb {Z}}}$ is a family of probability distributions on ${\mathbb {R}}$ with uniformly bounded supports. Let $V:{\mathbb {Z}}\rightarrow {\mathbb {R}}$ be a random potential chosen (independently at each site) with respect to the measure $\mu =\prod _{n \in {\mathbb Z}} \nu _n$. Then there exists a (non-random) compact set $\Sigma \subset {\mathbb {R}}$ such that $\mu$-almost surely, the essential spectrum of the discrete Schrödinger operator with potential V is equal to $\Sigma$.

Proof

Notice that Proposition B.1 implies that for a given point $E_0\in {\mathbb {R}}$, the event “the energy $E_0$ belongs to the essential spectrum of the Schrödinger operator with the random potential V” is a “tail event”, and hence due to Kolmogorov’s zero-one law must have probability either zero or one. Similarly, for any given closed interval I, the event “I has non-empty intersection with the essential spectrum” is a “tail event”, and has probability either zero or one. Since there is a countable base of the topology of ${\mathbb {R}}$ consisting of intervals, and a countable intersection of tail events is a tail event, Theorem B.1 follows. $\square$

1.3 B.3. Monotonicity: the support theorem

Here we formulate and prove a generalization of Kotani’s support theorem, originally proved in the continuum ergodic setting in [35]:

Theorem B.2

Let $\nu _1$ and $\nu _2$ be two probability distributions on ${\mathbb {R}}$ with bounded support. Let $V_{\mathrm {bg}}:{\mathbb {Z}}\rightarrow {\mathbb {R}}$ be a bounded background potential. Denote by $\Sigma _1$ the almost sure essential spectrum of the discrete Schrödinger operator given by the random potential $V_{\mathrm {bg}}+V^\mathrm {AM}_{\omega }$, where $V^\mathrm {AM}_\omega$ is a random sequence generated with respect to the distribution $\nu _1$ at each site. Define $\Sigma _2$ similarly, using the distribution $\nu _2$. If $\mathrm {supp} \, \nu _1 \subseteq \mathrm {supp}\, \nu _2$, then $\Sigma _1 \subseteq \Sigma _2$.

Proof

Suppose that $E_0\in \Sigma _1$. Then, due to Proposition B.1 there exists $K>0$ such that for any $N\in {\mathbb {N}}$, there are a sequence $\{m_j\}_{j\in {\mathbb {N}}}, m_j\in {\mathbb {Z}},$ with $|m_j-m_{j'}|>2N$ if $j\ne j'$, unit vectors ${\bar{u}}_j$, $|\bar{u}_j|=1$, and $(2N+1)$-tuples $\{t^1_{-N, j}, \ldots , t^1_{0, j}, \ldots , t^1_{N, j}\}$ of real numbers, $t^1_{i,j}\in \text {supp}\, \nu _1$, such that $|T_{[m_j, m_j+i], E}\,{\bar{u}}_j|\le K$ for all $|i|\le N$ and all $j\in {\mathbb {N}}$, where $T_{[m_j, m_j+i]}$ are the products of transfer matrices

$$\begin{aligned} \Pi _{m_j+i, E_0}=\left( \begin{array}{cc} E_0-(V_{\mathrm {bg}}(m_j+i)+ t^1_{i,j}) &{} -1 \\ 1 &{} 0 \\ \end{array} \right) , \ \ |i|\le N. \end{aligned}$$

By continuity it follows that there exists $\varepsilon _{N}>0$ such that for any $2N+1$-tuple $\{{\tilde{t}}_{-N, j}, \ldots , {\tilde{t}}_{0, j}, \ldots , {\tilde{t}}_{N, j}\}$ with $|t^1_{i,j}-\tilde{t}_{i,j}|<\varepsilon _N$ we have $|{\tilde{T}}_{[m_j, m_j+i], E}\,\bar{u}_j|\le 2K$, where ${\tilde{T}}_{[m_j, m_j+i]}$ are the products of transfer matrices

$$\begin{aligned} {{\tilde{\Pi }}}_{m_j+i, E_0}=\left( \begin{array}{cc} E_0-(V_{\mathrm {bg}}(m_j+i)+ {\tilde{t}}_{i,j}) &{} -1 \\ 1 &{} 0 \\ \end{array} \right) , \ \ |i|\le N. \end{aligned}$$

Since $\text {supp}\, \nu _1\subseteq \text {supp}\, \nu _2$, this implies that with positive (and, by compactness arguments, uniformly in $j\in {\mathbb {N}}$ bounded away from zero) probability, a random sequence generated by i.i.d. random variables distributed with respect to $\nu _2$ will coincide (up to an error not greater than $\varepsilon _N>0$) with the $(2N+1)$-tuple $\{t^1_{-N, j}, \ldots , t^1_{0, j}, \ldots , t^1_{N, j}\}$ over the interval of indices $[m_j-N, m_j+N]$. Due to the second Borel-Cantelli Lemma, if the sum of probabilities of a sequence of independent events is infinite, with probability one infinitely many of those events happens. Therefore, another application of Proposition B.1 implies that $\Sigma _1 \subseteq \Sigma _2$. $\square$

Proof of Theorem 1.2

Since in the ergodic case the almost sure spectrum coincides with the almost sure essential spectrum, Theorem 1.2 follows from Theorem B.2. $\square$

Appendix C: Generalized Ground States

1.1 C.1. The d eterministic setting

Let us consider a Schrödinger operator

$$\begin{aligned}{}[H \psi ](n) = \psi (n+1) + \psi (n-1) + V(n) \psi (n) \end{aligned}$$

(C.1)

in $\ell ^2({\mathbb Z})$ with a bounded potential $V : {\mathbb Z}\rightarrow {\mathbb R}$. Thus, H is a bounded self-adjoint operator and its spectrum $\sigma (H)$ is a compact subset of ${\mathbb R}$. We refer to

$$\begin{aligned} E_\mathrm {max} := \max \sigma (H) \end{aligned}$$

(C.2)

as the ground state energy. Traditionally, one considers $-\Delta + V$ and refers to the bottom of the spectrum as the ground state energy. Since it is customary to drop the minus sign when considering discrete Schrödinger operators, one then switches focus from the bottom to the top of the spectrum.

Our goal is to discuss the ground state. In informal terms, this is “the smallest” solution of the difference equation

$$\begin{aligned} u(n+1) + u(n-1) + V(n) u(n) = E u(n) \end{aligned}$$

(C.3)

for $E = E_\mathrm {max}$. In the traditional setting, when considering atomic models $-\Delta +V$ with V having a non-trivial negative part and $|V(x)| \rightarrow 0$ as $|x| \rightarrow \infty$, the bottom of the spectrum (usually) is a discrete eigenvalue and one is often able to show that it is simple. A normalized associated eigenfunction, which could typically be shown to be strictly positive, was then referred to as the ground state. In our general setting, there may not be a square-summable eigenfunction, and hence we will need a more general concept to identify a ground state.

Let us recall the famous definition of subordinacy due to Gilbert and Pearson [23].

Definition

A solution u of (C.3) is called subordinate at $+ \infty$ if it does not vanish identically and we have

$$\begin{aligned} \lim _{N \rightarrow \infty } \frac{\sum _{n=1}^N |u(n)|^2}{\sum _{n=1}^N |{\tilde{u}}(n)|^2} = 0 \end{aligned}$$

for every solution ${\tilde{u}}$ of (C.3) that is linearly independent from u (i.e., ${\tilde{u}}$ is not a multiple of u).

Remark C.1

• (a) Subordinacy at $- \infty$ is defined analogously.

• (b) By the constancy of the Wronskian (or the equivalent fact that the transfer matrices are unimodular), we cannot have two linearly independent square-summable solutions. Thus, every solution that is square-summable at $+ \infty$ is subordinate at $+ \infty$, and hence subordinacy generalizes square-summability.

• (c) If there is a solution of (C.3) that is subordinate at $\pm \infty$, then it is unique up to a multiplicative constant. In this sense one can say that it is “the smallest solution.”

• (d) We will eventually be interested in dynamically defined potentials, where the associated Schrödinger cocycle at energy $E_\mathrm {max}$ is reducible to a constant parabolic matrix. Thus, for each element of the hull, there will then be a unique (up to a constant multiple) solution $u_\mathrm {bdd}^\pm$ that is bounded at $\pm \infty$, while each linearly independent solution grows linearly. It is easy to check that in this scenario, $u_\mathrm {bdd}^\pm$ is subordinate at $\pm \infty$.

Theorem C.1

Suppose that $E = E_\mathrm {max}$ and $u^\pm$ is a solution of (C.3) that is subordinate at $\pm \infty$. Then, up to a multiplicative constant, we have $u^\pm (n) > 0$ for every $n \in {\mathbb Z}$.

Proof

We only consider the $+$ case, as the proof for the − case is completely analogous.

Let us consider the ratios

$$\begin{aligned} d(n) = \frac{u^+(n)}{u^+(n-1)}. \end{aligned}$$

As $u^+$ is subordinate, $u^+$ does not vanish identically and hence cannot have two consecutive zeros. Thus, while it can happen that $u^+(n-1) = 0$, we then must have $u^+(n) \not = 0$, and we can unambiguously set $d(n) := \infty$ in this case. In all other cases, d(n) is a finite (complex) number.

In fact, we have $d(n) \in {\mathbb R}\cup \{\infty \}$ for every $n \in {\mathbb Z}$. To see this, assume this fails. Then a conjugate solution ${\bar{u}}^+$ is linearly independent from $u^+$, solves (C.3) as well (because E is real), and is subordinate at $+ \infty$ as well (for obvious reasons); a contradiction.

Note next that the claim of the theorem is equivalent to

$$\begin{aligned} d(n) \in (0,\infty ) \quad \text {for every } n \in {\mathbb Z}. \end{aligned}$$

Moreover, if $d(n) = \infty$ for some $n \in {\mathbb Z}$, then $d(n-1) = 0$. Thus, the failure of the claim of the theorem is equivalent to

$$\begin{aligned} \exists k \in {\mathbb Z}: d(k) \in (-\infty ,0]. \end{aligned}$$

(C.4)

Assume that (C.4) holds. We consider the restriction $H_k^+$ of H to $\ell ^2(\{ k, k+1, k+2, \ldots \})$ with Dirichlet boundary condition. We consider the Weyl-Titchmarsh function

$$\begin{aligned} m_k^+(z) := \langle \delta _k, (H^+_k - z)^{-1} \delta _k \rangle = \int \frac{d\mu _k^+(E)}{E - z}, \end{aligned}$$

(C.5)

where $\mu _k^+$ is the spectral measure corresponding to the pair $(H_k^+,\delta _k)$ and $z \in {\mathbb C}\setminus \sigma (H^+_k)$. It follows from general results (see, e.g., [15]) that we can also write

$$\begin{aligned} m_k^+(z) = - \frac{u^+_z(k)}{u^+_z(k-1)}, \end{aligned}$$

(C.6)

where $u^+_z$ is subordinate at $+\infty$ and solves (C.3) with $E = z$.

By the min-max principle, we have that $\sigma (H^+_k) \subset (-\infty ,E_\mathrm {max}]$ and hence $\mu ^+_k$ gives no weight to $(E_\mathrm {max},\infty )$. It therefore follows from (C.5) that

$$\begin{aligned} z \in (E_\mathrm {max},\infty ) \quad \Rightarrow \quad m_k^+(z) < 0. \end{aligned}$$

(C.7)

Moreover, it also follows from (C.5) that

$$\begin{aligned} z \in (E_\mathrm {max},\infty ) \quad \Rightarrow \quad (m_k^+)'(z) = \int \frac{d\mu _k^+(E)}{(E - z)^2} > 0. \end{aligned}$$

(C.8)

Obviously, (C.7) and (C.8) show that the following limit exists:

$$\begin{aligned} m_k^+(E_\mathrm {max}) = \lim _{z \downarrow E_\mathrm {max}} m^+_k(z) \in [-\infty , 0). \end{aligned}$$

(C.9)

On the other hand, the relation (C.6) then extends to $z = E_\mathrm {max}$ by subordinacy theory (see, e.g., [15]), and we obtain

$$\begin{aligned} m_k^+(E_\mathrm {max}) = - d(k) \in [0,\infty ). \end{aligned}$$

(C.10)

Since (C.9) and (C.10) are incompatible, it follows that (C.4) is impossible, and the proof is finished. $\square$

It is of interest to find sufficient conditions for the assumption of Theorem C.1. Let us discuss the presence of a subordinate solution on the right half line at the top of the spectrum. The case of the left half line is of course similar. In applications we will need the coincidence of the energy, that is, the top of the spectrum will have to be the same for the whole line, the left half line, and the right half line. This will be the case, for example, in the Anderson model.

So, we consider an operator H in $\ell ^2({\mathbb N})$, ${\mathbb N}= \{ 1, 2, 3, \ldots \}$, acting as (1.1) together with a Dirichlet boundary condition at the origin,

$$\begin{aligned} \psi (0) = 0. \end{aligned}$$

(C.11)

(Note that in the proof above, this operator would have been denoted by $H_1^+$.)

We have the following result.

Proposition C.1

Consider the setting just described and define the ground state energy $E_\mathrm {max}$ as in (C.2). Then the difference equation (C.3) admits a subordinate solution at $\infty$ for $E = E_\mathrm {max}$.

Proof

Denote the spectral measure of H and $\delta _1$ by $\mu$. Thus, the associated Weyl-Titchmarsh function $m : {\mathbb C}_+ \rightarrow {\mathbb C}_+$ is given by

$$\begin{aligned} m(z) := \langle \delta _1, (H - z)^{-1} \delta _1 \rangle = \int \frac{d\mu _(E)}{E - z}. \end{aligned}$$

(C.12)

Obviously, the right-hand side makes sense for every $z \not \in \mathrm {supp} \, \mu = \sigma (H)$, and we will make use of this fact.

It follows from general results (see, e.g., [15]) that we can also write

$$\begin{aligned} m(z) = - \frac{u^+_z(1)}{u^+_z(0)}, \end{aligned}$$

(C.13)

where $u^+_z$ is subordinate at $+\infty$ and solves (C.3) with $E = z$. The existence of such a solution for every $z \not \in \mathrm {supp} \, \mu = \sigma (H)$ is clear (simply apply $(H-z)^{-1}$ to $\delta _1$, obtain a solution away from one and modify around one to turn it into a genuine solution).

By the same reasoning as in the previous proof we again have the negativity statement

$$\begin{aligned} z \in (E_\mathrm {max},\infty ) \quad \Rightarrow \quad m(z) < 0 \end{aligned}$$

(C.14)

and the monotonicity statement

$$\begin{aligned} z \in (E_\mathrm {max},\infty ) \quad \Rightarrow \quad m'(z) = \int \frac{d\mu (E)}{(E - z)^2} > 0, \end{aligned}$$

(C.15)

which combined imply the limiting statement

$$\begin{aligned} m(E_\mathrm {max}) := \lim _{\varepsilon \downarrow 0} m(E_\mathrm {max} + \varepsilon ) \in [-\infty , 0). \end{aligned}$$

(C.16)

Observe that we have

$$\begin{aligned} \lim _{\varepsilon \downarrow 0} m(E_\mathrm {max} + i \varepsilon ) = \lim _{\varepsilon \downarrow 0} m(E_\mathrm {max} + \varepsilon ), \end{aligned}$$

(C.17)

that is, we claim that the limit on the left-hand side exists and equals the limit on the right-hand side. To see this, one can separate the two cases $m(E_\mathrm {max}) = - \infty$ and $m(E_\mathrm {max}) \in (-\infty ,0)$ and use monotone convergence in the first case and dominated convergence in the second case to verify existence of the limit on the left-hand side, as well as equality with $m(E_\mathrm {max})$.

Thus, (C.16) and (C.17) yield

$$\begin{aligned} \lim _{\varepsilon \downarrow 0} m(E_\mathrm {max} + i \varepsilon ) = m(E_\mathrm {max}) \in [-\infty , 0), \end{aligned}$$

which in turn implies the desired statement via subordinacy theory; compare [15, 28]. $\square$

Remark C.2

An alternative approach is presented in [48, Section 2.3]. The idea is to first show that for $E \ge E_\mathrm {max}$, every solution of (C.3) can change sign at most once, and to then consider the Dirichlet solution, which has a zero (and hence a sign change) at $n_0$, normalize it, and send $n_0$ to infinity. This recovers a positive solution $u_+(\cdot ,E)$, which can be shown to be minimal among all positive solutions in a suitable sense (namely, it has values 1 and $\phi _+(E)$ at the points 0 and 1, and a solution $u(\cdot ,E)$ with values 1 and $\phi (E)$ at the points 0 and 1 will be positive if and only if $\phi (E) \ge \phi _+(E)$). Moreover, it turns out to be strongly (i.e., no need for an average) subordinate:

$$\begin{aligned} \lim _{n \rightarrow \infty } \frac{u_+(n,E)}{u(n,E)} = 0. \end{aligned}$$

A similar treatment can be performed near $-\infty$, and hence one obtains positive solutions $u_\pm (\cdot , E)$ that may or may not be linearly dependent, but which are minimal on their respective half line. This leads to the two cases where $u_\pm$ are linearly dependent (called critical) or not (called subcritical).

Conversely, it can be shown that the presence of a positive solution at energy E implies that $\sigma (H) \subset (-\infty ,E]$, so that combining the two statements, one finds that $\sigma (H) \subset (-\infty ,E]$ if and only if there is a positive solution at E.

1.2 C.2. The dynamically defined setting

Let us take Remark C.1.(d) further and deduce some consequences of Theorem C.1 in the case of dynamically defined potentials. Before stating them, let us describe the framework. Suppose $(\Omega ,T)$ is a topological dynamical system given by a homeomorphism $T : \Omega \rightarrow \Omega$ of a compact metric space. Fix an ergodic Borel probability measure $\mu$ and a continuous sampling function $f : \Omega \rightarrow {\mathbb R}$. We obtain a family of potentials $\{ V_\omega \}_{\omega \in \Omega }$ given by

$$\begin{aligned} V_\omega (n) = f(T^n \omega ), \quad \omega \in \Omega , \; n \in {\mathbb Z}, \end{aligned}$$

and a family of Schrödinger operators $\{ H_\omega \}_{\omega \in \Omega }$ in $\ell ^2({\mathbb Z})$, acting via

$$\begin{aligned}{}[H_\omega \psi ](n) = \psi (n+1) + \psi (n-1) + V_\omega (n) \psi (n), \quad \omega \in \Omega , \; n \in {\mathbb Z}. \end{aligned}$$

It is a fundamental result (see, e.g., [15]) that there exists a compact set $\Sigma \subset {\mathbb R}$ such that $\sigma (H_\omega ) = \Sigma$ for $\mu$-almost every $\omega \in \Omega$. Moreover, in case T is minimal (i.e., all of its orbits are dense), then we even have $\sigma (H_\omega ) = \Sigma$ for every $\omega \in \Omega$ (and the choice of the ergodic measure plays no role). Let us set

$$\begin{aligned} E_\mathrm {max} := \max \Sigma . \end{aligned}$$

The difference equation associated with $H_\omega$,

$$\begin{aligned} u(n+1) + u(n-1) + V_\omega (n) u(n) = E u(n), \end{aligned}$$

(C.18)

can be recast in matrix-vector form as

$$\begin{aligned} \begin{pmatrix} u(n+1) \\ u(n) \end{pmatrix} = \begin{pmatrix} E - V_\omega (n) &{} - 1 \\ 1 &{} 0 \end{pmatrix} \begin{pmatrix} u(n) \\ u(n-1) \end{pmatrix}. \end{aligned}$$

Iterating this, one sees that the resulting matrix product is generated by considering the second component of the iterates of the following skew-product

$$\begin{aligned} (T,A_E) : \Omega \times {\mathbb R}^2 \rightarrow \Omega \times {\mathbb R}^2, \quad (\omega , v) \mapsto (T \omega , A_E(\omega ) v), \end{aligned}$$

where

$$\begin{aligned} A_E : \Omega \rightarrow \mathrm {SL}(2,{\mathbb R}), \quad \omega \mapsto \begin{pmatrix} E - f(\omega ) &{} -1 \\ 1 &{} 0 \end{pmatrix}. \end{aligned}$$

For $n \in {\mathbb Z}$ we define $A^n_E : \Omega \rightarrow \mathrm {SL}(2,{\mathbb R})$ by $(T,A_E)^n = (T^n,A^n_E)$ and then note that this is precisely the matrix product that sends $(u(0), u(-1))^t$ to $(u(n), u(n-1))^t$ for solutions of (C.18).

Corollary C.1

Suppose there are $c \in {\mathbb R}$ and a continuous map $W: \Omega \rightarrow \mathrm {SL}(2,{\mathbb R})$ such that

$$\begin{aligned} W(T \omega )^{-1} A_{E_\mathrm {max}}(\omega ) W(\omega ) = A_* := \begin{pmatrix} 1 &{} c \\ 0 &{} 1 \end{pmatrix}. \end{aligned}$$

(C.19)

Then $c\ne 0$, and there are a compact cone C in the open first quadrant of ${\mathbb R}^2$ and a continuous section $b : \Omega \rightarrow C \setminus \{ 0 \}$ so that

• (i) b is projectively invariant under the dynamics: $[b(T\omega )] = [A_{E_\mathrm {max}}(\omega ) b(\omega )]$, where $[ \cdot ] : {\mathbb R}^2 \setminus \{0\} \rightarrow {\mathbb R}{\mathbb {P}}^1$ denotes the canonical projection,

• (ii) for every $\omega \in \Omega$, the sequence $(A^n_{E_\mathrm {max}}(\omega ) b(\omega ))_{n \in {\mathbb Z}}$ is bounded.

Proof

Recall that the discussion preceding Proposition C.1 points out the relevance of the coincidence of the top of the spectrum for the line and half line operators. Let us explain that this will always hold in the dynamically defined situation.

We use the natural notation $H_\omega , H^+_\omega , H^-_\omega$ for the operators in question. First of all, it is well known and not hard to see (cf., e.g., [15]) that the essential spectra coincide almost surely, that is,

$$\begin{aligned} \sigma _\mathrm {ess}(H_\omega ) = \sigma _\mathrm {ess}(H^+_\omega ) = \sigma _\mathrm {ess}(H^-_\omega ) \quad \text { for } \mu -\text { almost every } \omega \in \Omega . \end{aligned}$$

(C.20)

Secondly, the spectrum of the whole-line operator is purely essential, that is,

$$\begin{aligned} \sigma (H_\omega ) = \sigma _\mathrm {ess}(H_\omega ) \quad \text { for } \mu -\text { almost every } \omega \in \Omega . \end{aligned}$$

(C.21)

Finally, the min-max theorem implies as before that

$$\begin{aligned} \max \sigma (H^\pm _\omega ) \le \max \sigma (H_\omega ) \quad \text {for every } \omega \in \Omega . \end{aligned}$$

(C.22)

Combining (C.20)–(C.22), we find that

$$\begin{aligned} \max \sigma (H^\pm _\omega ) = \max \sigma (H_\omega ) \quad \text { for } \mu -\text { almost every } \omega \in \Omega . \end{aligned}$$

(C.23)

In cases where the spectrum is $\omega$-independent, this identity then trivially extends to all $\omega$’s.

It follows from Proposition C.1 and (C.23) that whenever we have a conjugacy of the form (C.19) with a continuous map $W : \Omega \rightarrow \mathrm {SL}(2,{\mathbb R})$ and some $c \in {\mathbb R}$, then we must have $c \not = 0$.

Now let us set

$$\begin{aligned} {\tilde{b}}(\omega ) := W(\omega ) \begin{pmatrix} 1 \\ 0 \end{pmatrix} = \begin{pmatrix} W_{11}(\omega ) \\ W_{21}(\omega ) \end{pmatrix}. \end{aligned}$$

The conjugacy (C.19) shows that ${\tilde{b}}$ has property (i). Since

$$\begin{aligned} A_*^n \begin{pmatrix} 1 \\ 0\end{pmatrix} \end{aligned}$$

is bounded as n ranges over ${\mathbb Z}$, the conjugacy (C.19) also shows that

$$\begin{aligned} A^n_{E_\mathrm {max}}(\omega ) {\tilde{b}}(\omega ) \end{aligned}$$

is bounded as n ranges over ${\mathbb Z}$. Thus, ${\tilde{b}}$ has property (ii) as well. Moreover, it takes values in ${\mathbb R}^2 \setminus \{ 0 \}$. It remains to show that it can be modified so that (the two properties are preserved and) it takes values in a compact cone in the open first quadrant.

As discussed above, $(A^n_{E_\mathrm {max}}(\omega ) \tilde{b}(\omega ))_{n \in {\mathbb Z}}$ corresponds to a solution $u_\omega$ of (C.18) with $E = E_\mathrm {max}$, which must then also be bounded. Similarly, since $(A_*^n v)_{n \in {\mathbb Z}}$ is linearly growing in both directions (i.e., for $n \rightarrow \infty$ and for $n \rightarrow - \infty$) for any v that is linearly independent from $(1,0)^t$, we see that all solutions of this difference equation that are linearly independent from $u_\omega$ must be linearly growing in both directions.

It follows that the solution $u_{\omega }$ is subordinate at both $+ \infty$ and $- \infty$. By Theorem C.1 it is therefore strictly positive up to a multiplicative constant. This means that for a suitable $\mathrm {const} \not = 0$, $b := \mathrm {const} \cdot {\tilde{b}}$ takes values in the open first quadrant. By compactness of $\Omega$ and continuity of b, we can find a compact cone C in the open first quadrant such that $b : \Omega \rightarrow C \setminus \{ 0 \}$, completing the proof. $\square$