MODIFIED PRÜFER AND EFGP TRANSFORMS

AND THE SPECTRAL ANALYSIS OF

ONE-DIMENSIONAL SCHRÖDINGER OPERATORS

Alexander Kiselev^{1}, Yoram Last^{2}, and Barry Simon^{2,3}

April 3, 1997

Abstract. Using control of the growth of the transfer matrices, we discuss the spectral analysis of continuum and discrete half-line Schrödinger operators with slowly decaying potentials. Among our results we show if V (x) = ∑_{n=1}^{∞} a_{n}W (x−x_{n}) where W has compact support and x_{n}/x_{n+1} → 0, then H has purely a.c. (resp. purely s.c.) spectrum on (0, ∞) if ∑ a_{n}^{2} < ∞ (resp. ∑ a_{n}^{2} = ∞). For λn^{−1/2}a_{n} potentials where a_{n} are independent, identically distributed random variables with E(a_{n}) = 0, E(a_{n}^{2}) = 1, and λ < 2, we find singular continuous spectrum with explicitly computable fractional Hausdorff dimension.

§1. Introduction

In this paper, we will study continuum and discrete Schrödinger operators on the half-line (while we don’t always make them explicit, given theory in [10, 26, 32], many of our results extend to suitable whole-line problems). Explicitly, we are interested in the spectral analysis of operators H on L^{2}(0, ∞; dx) and on ℓ^{2}([1, ∞)) given by

(Hu)(x) = −\frac{d^{2}}{dx^{2}} + V (x)  (1.1)

in the continuum case and

(Hu)(n) = u(n + 1) + u(n − 1) + V (n)u(n)  (1.2)

1 Mathematical Sciences Research Institute, 1000 Centennial Drive, Berkeley, CA 94720. Research supported in part by NSF Grant No. DMS-9022140.

2 Division of Physics, Mathematics, and Astronomy, California Institute of Technology 253-37, Pasadena, CA 91125

3 This material is based upon work supported by the National Science Foundation under Grant No. DMS-9401491. The Government has certain rights in this material.

To be submitted to Commun. Math. Phys.

Typeset by AMS-TEX

in the discrete case.

Suitable boundary conditions are set at x (or n) = 0 so that H is self-adjoint (since in all our examples V is limit point at infinity, a boundary condition is not needed there). We are interested in the spectral properties of such operators in situations where |V (n)| → 0 as n → ∞, but so slowly that the usual scattering theory will not apply. Our main theme in this paper is that there are perturbation techniques of remarkable power for such operators based on two ideas.

The first is that one can use the transfer matrix to analyze spectral properties. The transfer or fundamental matrix is a 2 × 2 unimodular matrix defined in the continuum case for any E by

T_{E}(x, 0)\binom{a}{b} = \binom{u′(x)}{u(x)}  (1.3)

where −u′′ + V u = Eu, u′(0) = a, u(0) = b. In the discrete case

T_{E}(n, 0)\binom{a}{b} = \binom{u(n + 1)}{u(n)}  (1.4)

where u(1) = a, u(0) = b, and u(n + 1) + u(n − 1) + V (n)u(n) = Eu(n).

The second idea is that one can control the transfer matrix by controlling the norms of two solutions of −u′′ + V u = Eu and that the critical equations are ones that involve those norms.

Two of us [22] have recently found several new criteria for singular or absolutely continuous spectra in terms of transfer matrices, and these criteria will make some of our results here possible. The perturbation equations we will use have not been systematically used in this context except in the paper of Kotani-Ushiroya [21] whose techniques have some overlap with this paper. But they didn’t control the discrete case and their method is so entwined with certain Martingale inequalities that it is unclear how to use them in other contexts.

While we were writing up the work for this paper, we received a preprint from Remling [29] that uses this two-pronged approach and has considerable overlap with our Sections 5 and 6. We will discuss the connection shortly.

Here are some of the theorems that we will use that relate spectral properties to behavior of T (n). The first is from [22]:

Theorem 1.1. Suppose that there is a fixed sequence n_{i} → ∞ and S is a subset of ℝ so that for a.e. E ∈ S, lim_{i→∞} ∥T_{E}(n_{i}, 0)∥ = ∞. Then μ_{ac}(S) = 0, where μ_{ac} is the absolutely continuous part of the spectral measure for H.

Remarks. 1. The interesting aspect of this theorem is that n_{i} is arbitrary. The result actually allows a more general sequence ∥T_{E}(n_{i}, m_{i})∥.

2. In typical applications, S is an interval in the essential spectrum.

In the other direction, one has the following pair of results:

Theorem 1.2. Suppose S is a set so that for a.e. E ∈ S, \overline{lim}_{x→∞} ∥T_{E}(x, 0)∥ < ∞. Then μ_{ac}(Q) > 0 for any Q ⊂ S with |Q| > 0 where | · | = Lebesgue measure.

Theorem 1.3. Suppose there is a sequence n_{i} → ∞ so that ∫_{a}^{b} ∥T_{E}(n_{i}, 0)∥^{4} dE < ∞. Then (a, b) ⊂ spec(H) and the spectral measure is purely absolutely continuous on (a, b) and μ_{ac}(Q) > 0 for any Q with |Q ∩ (a, b)| > 0.

Remarks. 1. That Theorem 1.2 is implied by the Gilbert-Pearson [11] theory was noted by Stolz [36]. A simple proof can be found in [33]. Last-Simon [22] prove a stronger variant in which ∥T_{E}(x, 0)∥ is replaced by ∫_{−1}^{1} ∥T_{E}(x + y, 0)∥ dy and \overline{lim} by \underline{lim} (The discrete analog holds with \underline{lim} and without any local integration).

2. Theorem 1.3 is from Last-Simon [22] although the method used there is not much more than what is in Carmona [1].

As to distinguishing dense pure point from singular continuous spectrum, in one direction we have the following elementary result from Simon-Stolz [35].

Theorem 1.4. If ∑_{n} ∥T_{E}(n, 0)∥^{−2} = ∞ in the discrete case or ∫_{0}^{∞} ∥T_{E}(x, 0)∥^{−2} dx = ∞ in the continuum case, then Hu = Eu has no solution which is L^{2} at infinity.

The paradigm of results that guarantees a solution L^{2} at ∞ is Ruelle’s proof [30] of Osceledec’s theorem. His argument is abstracted in [22]. We will need the following in Section 8:

Theorem 1.5. If lim_{n→∞}[log ∥T_{E}(n, 0)∥/n^{α}] exists and lies in (0, ∞) for some α > 0, then Hu = Eu has an L^{2} solution.

[22] also has a general abstract result on power decay which, to get an L^{2} solution, requires

lim_{n→∞} \frac{log ∥T_{E}(n, 0)∥}{log n} > \frac{3}{2} .

[22] also has an example where the limit is \frac{3}{2} and there is no ℓ^{2} solution. But there are stronger results that hold a.e. in certain probabilistic situations, so we won’t discuss the power decay result here. In Section 8, we will discuss the probabilistic result.

As for the technique to control the growth of solutions, in the continuum case we will use modified Prüfer variables defined for E > 0 by

u′(x) = \sqrt{E} R(x) cos(θ(x))  (1.5a)
u(x) = R(x) sin(θ(x)).  (1.5b)

One finds these obey the differential equations (with k = \sqrt{E})

\frac{dθ}{dx} = k − \frac{V (x)}{k} sin^{2}(θ(x))  (1.6)
\frac{d log R(x)}{dx} = \frac{1}{2k} V (x) sin(2θ(x)).  (1.7)

Two features of these equations are immediately noteworthy:

(a) They separate in the sense that (1.6) does not involve R and after solving it, one obtains R by integration. That R drops out of (1.6) and the right side of (1.7) is an expression of the linearity of the initial equations.

(b) If V = 0 in some region (a, b), then in that region R is constant and θ(x) = θ(a)+ k(x−a). It is this fact that leads one to take the factor \sqrt{E} in (1.5a). The addition of this \sqrt{E} is what distinguishes this from ordinary Prüfer transformations.

There is a third significant feature which we will turn to momentarily.

Given how common these continuum equations are, we would have expected their discrete analogs would have been rediscovered and used many times, but even after some efforts at tracking various literature, we’ve found them in a single chain of four papers (and we are a fifth in this chain, since we learned of them from the fourth paper!).

The original discoverer of the correct equation was Thomas Eggarter [9] in 1971. He was not looking at an explicit difference equation but rather a continuum equation with V (x) = V_{0} ∑_{i=1}^{n} δ(x − x_{i}). By integrating modified Prüfer variables across the δ-functions, he was led to the transforms (E = 2 cos(k)),

u(n) − cos(k)u(n − 1) = R(n) cos(θ(n))  (1.8a)
sin(k)u(n − 1) = R(n) sin(θ(n))  (1.8b)

in which case we have, after some calculation (see Section 2),

cot(θ(n + 1)) = cot(k + θ(n)) − (sin(k))^{−1}V (n)  (1.9)
\frac{R(n + 1)^{2}}{R(n)^{2}} = 1 − \frac{V (n)}{sin(k)} sin(2θ(n) + 2k) + \frac{V (n)^{2}}{sin^{2}(k)} sin^{2}(θ(n) + k).  (1.10)

Actually, he had only an equation of the form (1.9). The definition of θ(n) and precise (1.9) is in a 1975 paper of Gredeskul-Pastur [13] who followed up on Eggarter’s work.

[9, 13] focus on (1.9) because they use the transform to study the integrated density of states. Pastur-Figotin [26] defined R and exploited (1.10) to study the Lyapunov exponent. In recognition of these seminal works, we call (1.8) the EFGP transform. Their approach was further exploited in Chulaevsky-Spencer [2]. It will often be useful to use an equivalent form of (1.9) that appears as (2.14).

Notice that (1.9), (1.10) have the two critical properties (a), (b) mentioned for (1.6), (1.7) in the continuum case. In particular, if V (n) = 0 for n in some interval [n_{0}, n_{1}], then in that region R(n) is constant and

θ(n) = θ(n_{0}) + k(n − n_{0}).

While the EFGP transform was obtained by integrating a continuum δ-function model, it could also be found by looking for a transform with property (b). We will explain this in Section 2.

[9, 13, 26, 2] all consider V ’s with no decay as n → ∞ but with a small coupling so that any calculations are only asymptotic in coupling constant. It turns out that the methods are especially well suited when V (n) → 0 at infinity and one obtains results that are exact for a fixed V. For example, in Section 8, we will find exact formulas for the local Hausdorff dimensions of certain singular continuous spectral measures.

The third critical factor of the modified Prüfer and EFGP transforms is a major theme of this paper, namely, that first-order terms in V are oscillatory while the second-order term has a strong tendency to be strictly positive. This idea is already seen in [26, 2] where γ(E) is O(g^{2}) with g a coupling constant because the first-order terms drop out.

Let us be explicit about this idea. In (1.6), one might think the positivity comes via the square in sin^{2}(θ(x)) but that is wrong! Indeed, in writing sin^{2}(θ) = \frac{1}{2} − \frac{1}{2} cos(2θ), it is the cos(2θ) that is critical! Formally, (1.6) says

θ(x) = kx + θ_{0} − \frac{V (x)}{k} sin^{2}(kx + θ_{0}) + O(V^{2}) ≡ kx + θ_{0} + δθ + O(V^{2})

and then using

sin(2θ) = sin(2kx + 2θ_{0}) + 2 cos(2kx + 2θ_{0})δθ + O(V^{2}),

we get

\frac{d log R}{dx} = t_{1} + t_{2} + O(V^{3}),

where

t_{1} = \frac{1}{2k} V (x) sin(2(kx + θ_{0})) − \frac{1}{2k^{2}} (V (x) ∫_{x_{0}}^{x} V (y) dy) cos(2(kx + θ_{0}))

is the oscillatory term that is often unimportant, while

t_{2} = \frac{1}{4k^{2}} \frac{d}{dx} [ ∫_{x_{0}}^{x} V (y) cos(2ky + 2θ_{0}) dy]^{2}

has a positive integral, second order in V.

In explicit cases, it is more subtle to prove the second order is strictly positive and, indeed, for examples like V (x) = x^{−α}, α < \frac{1}{2} where the spectrum is absolutely continuous (by Weidmann [37]), the second-order terms do not cause divergences. This means that results that depend on a finite second-order term should hold more generally than those that depend on an infinite second-order term. Indeed, we

Conjecture. If V is bounded and in L^{2}(ℝ, dx) (or ℓ^{2}(ℤ_{+})), then the essential support of the a.c. part of the spectrum is all of (0, ∞) (or (−2, 2) in the discrete case).

Our idea is that for almost all (but not all; see, e.g., [24, 25, 34]) k, the oscillations should kill the first-order term, and so the L^{2} condition should suffice to give a bounded transfer matrix for a.e. k and so the stated conclusion about the a.c. spectrum by Theorem 1.2.

After discussing the modified Prüfer and EFGP transforms and their relation to the growth of the transfer matrix in Section 2, we turn to two warm-up problems in Sections 3 and 4. In Section 3, we show these transforms can replace the Harris-Lutz [15] method in many cases where that method is applicable. In Section 4, we look at potentials V with \overline{lim} x|V (x)| finite and show that for such potentials their positive eigenvalues can only coalesce at E = 0. Since examples are known with countable many eigenvalues embedded in (0, ∞), this result is interesting.

In Sections 5–7, we study sparse potentials.

Definition. A Pearson potential is one of the form

V (x) = ∑_{n=1}^{∞} a_{n}W (x − x_{n}),  (1.11)

where W is a bounded, non-negative function of compact support, a_{n} → 0, and 1 ≤ x_{1} < x_{2} < x_{3} < ⋯,

\frac{x_{n}}{x_{n+1}} → 0.  (1.12)

The name is in honor of David Pearson who considered potentials of the form (1.11) where ∑_{n=1}^{∞} a_{n}^{2} = ∞ and x_{n} went to infinity sufficiently fast. To make things precise, think of the example x_{n} = n!.

Our major goal in Sections 5–6 is to prove the following:

Theorem 1.6. Let V be a Pearson potential. Then

(1) If ∑_{n=1}^{∞} a_{n}^{2} < ∞, the spectrum of −\frac{d^{2}}{dx^{2}} + V (x) is purely absolutely continuous on (0, ∞) for any boundary condition at 0.

(2) If ∑_{n=1}^{∞} a_{n}^{2} = ∞, the spectrum of −\frac{d^{2}}{dx^{2}} + V (x) is purely singular continuous on (0, ∞) for any boundary condition at 0.

In Section 5, we will actually prove a stronger version of (1):

Theorem 1.6′. Let V have the form (1.11) where

\overline{lim} \frac{x_{n}}{x_{n+1}} < 1.  (1.13)

Then (1) holds.

Pearson [27, 28] proved a weak version of (2) in that if ∑_{n=1}^{∞} a_{n}^{2} = ∞, there exists some set of x_{n}’s so that the spectrum is purely singular continuous. In [27], there are hints that a result of type (1) (again with x_{n} sufficiently large) should hold, but nothing explicit.

As noted at the end of Section 5, for (1) the bumps W (x − x_{n}) can be n-dependent.

At the end of Section 6, for [a, b] ≡ S ⊂ (0, ∞), we construct Pearson-like potentials (bumps whose width grows with n) so that there is purely a.c. spectrum on S and purely s.c. spectrum on (0, ∞)\S.

In a recent paper, coincident with our work, Remling [29] obtained results related to Theorem 1.6(1) using similar methods. He only obtains the existence of absolutely continuous spectrum (his results are consistent with simultaneous singular continuous spectrum while we prove there is none), and he needs at least exp(\frac{3}{2}n log n) growth on the x_{n} (whereas, if f(n) is a monotone function with f(m) → ∞ no matter how slowly, then x_{n} = exp(nf(n)) obeys (1.12) and x_{n} = exp(an) obeys (1.13)).

After this manuscript was completed, we obtained a preliminary version of a preprint of Molchanov [23] with considerable overlap with our results in Sections 5 and 6.

In Section 7, we will prove

Theorem 1.7. Let x_{n} ∈ ℤ obey x_{n}/x_{n+1} → 0. Let V be the potential with

V (x_{n}) = a_{n}
V (x) = 0  x ≠ x_{n} for any n.

Then,

(1) If ∑ a_{n}^{2} < ∞, the discrete Schrödinger operator with potential V has purely a.c. spectrum for (−2, 2).

(2) If ∑ a_{n}^{2} = ∞, the operator has purely singular continuous spectrum on (−2, 2).

In Sections 8 and 9, we discuss models with randomness and decay, first studied by Simon [31] and then by Delyon, et al. [7], Delyon [6], and Kotani-Ushiroya [21]. Typical of the models discussed in these sections is (g is positive constant)

V (n) = gn^{−α}a_{n}

where the a_{n} are independent, identically distributed random variables, uniformly distributed in [−1, 1]. We prove

(i) If α > \frac{1}{2}, the spectrum is almost surely purely absolutely continuous in (−2, 2). (ii) If 0 < α < \frac{1}{2}, the spectrum is almost surely dense pure point in (−2, 2). (iii) If α = \frac{1}{2}, the spectrum is almost surely purely singular continuous in the region

|E| < (4 − \frac{1}{3}g^{2})^{1/2} and dense pure point in the region (4 − \frac{1}{3}g^{2})^{1/2} ≤ |E| < 2 (if g^{2} > 12, interpret (4 − \frac{1}{3}g^{2})^{1/2} as 0).

(iv) In case α = \frac{1}{2} and g^{2} < 12, in the region |E| < (4 − \frac{1}{3}g^{2})^{1/2}, the spectrum has

fractional Hausdorff dimension with local dimension (4 − E^{2} − \frac{g^{2}}{3} )/(4 − E^{2}). Section 8 handles the discrete case, and Section 9 the continuum case. For sparse potentials, we give the details in the continuum case and sketch the discrete

case; while for random decaying potentials, we give details in the discrete case and sketch the continuum case.

A.K. would like to thank the hospitality of I.H.E.S., and B.S. the hospitality of Hebrew University where some of this work was done.

§2. Modified Prüfer and EFGP Transforms

We will be interested in solutions of

−u′′(x) + V (x)u(x) = k^{2}u(x).  (2.1)

Change variables to

u′(x) = kR(x) cos(θ(x))  (2.2a)
u(x) = R(x) sin(θ(x)).  (2.2b)

These are called modified Prüfer variables. The 2π ambiguity in θ is fixed by choosing θ(0) ∈ [0, 2π) and demanding θ(x) be continuous in x.

Then a straightforward calculation shows (2.1) is equivalent to the pair of equations

\frac{dθ}{dx} = k − \frac{V (x)}{k} sin^{2}(θ(x))  (2.3)
\frac{d(log R)(x)}{dx} = \frac{1}{2k} V (x) sin(2θ(x)).  (2.4)

This change of variables is so very useful because if V = 0, then θ(x) = θ_{0} + kx, R(x) = R_{0}. We will be able to study V as a perturbation about this solution.

As explained in the introduction, one needs to study the asymptotic behavior of the norm of the transfer matrix T (x, 0). For any θ_{0} in [0, π), let θ(x, θ_{0}) solve (2.3) with initial condition θ(0) = θ_{0}. Then let R(x, θ_{0}) solve (2.4) with R(0, θ_{0}) = 1. Then

Theorem 2.1. For any α, β ∈ (0, ∞) and θ_{1} ≠ θ_{2}, there exists non-zero, finite constants C_{1} and C_{2} (independent of x and V) so that

C_{1} max(R(x, θ_{1}), R(x, θ_{2})) ≤ ∥T (x, 0)∥ ≤ C_{2} max(R(x, θ_{1}), R(x, θ_{2}))  (2.5)

for all k ∈ (α, β).

Proof. Define ∥(a, b)∥_{k}^{2} = (ka)^{2} + b^{2}. Then min(1, k)∥(a, b)∥ ≤ ∥(a, b)∥_{k} ≤ max(1, k) ∥(a, b)∥. So defining operator norms in terms of ∥ · ∥_{k}, we see min(k, k^{−1})∥T (x, 0)∥_{k} ≤ ∥T (x, 0)∥ ≤ max(k, k^{−1})∥T (x, 0)∥_{k}, so it suffices to prove (2.5) with ∥ · ∥_{k} rather than ∥ · ∥. But

∥T (x, 0)∥_{k} ≥ max(R(x, θ_{1}), R(x, θ_{2}))

is trivial and

∥T (x, 0)∥_{k} ≤ \{min[sin(\frac{1}{2} |θ_{1} − θ_{2}|), cos(\frac{1}{2} |θ_{1} − θ_{2}|)]\}^{−1} max(R(x, θ_{1}), R(x, θ_{2}))

by the lemma below. □

If |θ_{1} − θ_{2}| ≤ \frac{π}{2} (which can be done by replacing θ_{1} by π + θ, if need be), then this proof shows we can take

C_{1} = min(α, β^{−1})
C_{2} = max(β, α^{−1})[sin(\frac{1}{2} |θ_{1} − θ_{2}|)]^{−1}.

Lemma 2.2. Let A be a unimodular matrix. Let u_{θ} = (cos(θ), sin(θ)). Then if |θ_{1}−θ_{2}| ≤ \frac{π}{2} , then

∥A∥ ≤ sin(\frac{1}{2} |θ_{1} − θ_{2}|)^{−1} max(∥Au_{θ_{1}}∥, ∥Au_{θ_{2}}∥).

Proof. There exists θ_{0} so that

∥Au_{θ}∥^{2} ≥ ∥A∥^{2} sin^{2}(θ − θ_{0}).

If |θ_{1} − θ_{2}| < \frac{π}{2} , for any θ_{0} at least one of | sin(θ_{0} − θ_{i})| is larger than or equal to | sin(\frac{1}{2}(θ_{1} − θ_{2})|. □

Remark. One might worry that the lemma involves ∥ · ∥ and not ∥ · ∥_{k} but ∥A∥_{k} =

∥( \begin{matrix}k & 0 \\ 0 & 1 \end{matrix} ) A ( \begin{matrix}k & 0 \\ 0 & 1 \end{matrix} )^{−1}∥ and this product is also unimodular.

For the discrete case, we are interested in solutions of (0 ≤ k ≤ π)

u(n + 1) + u(n − 1) + V (n)u(n) = 2 cos(k)u(n).  (2.6)

EFGP variables R(n), θ(n) are defined by

R(n) cos(θ(n)) = u(n) − cos(k)u(n − 1)  (2.7a)
R(n) sin(θ(n)) = sin(k)u(n − 1).  (2.7b)

A priori θ(n) is only determined mod (2π). We will fix this ambiguity later. Noticing that

R(n) sin(k + θ(n)) = sin(k)u(n)  (2.8)
\frac{u(n)}{u(n − 1)} = \frac{sin(k + θ(n))}{sin(θ(n))}.  (2.9)

Similarly,

R(n) cos(k + θ(n)) = cos(k)u(n) − u(n − 1).  (2.10)

Thus,

cot(k + θ(n)) = \frac{cos(k)u(n) − u(n − 1)}{sin(k)u(n)}

where by definition,

− cot(θ(n + 1)) = \frac{cos(k)u(n) − u(n + 1)}{sin(k)u(n)} .

Thus, (2.6) is equivalent to

cot(θ(n + 1)) = cot(k + θ(n)) − \frac{V (n)}{sin(k)} .  (2.11)

Writing θ̄(n) ≡ θ(n) + k, we see, using first (2.7) and then (2.8)/(2.9):

R(n + 1)^{2} = sin^{2}(k)u(n)^{2} + (u(n + 1) − cos(k)u(n))^{2}

= sin^{2}(k)u(n)^{2} + (u(n − 1) − cos(k)u(n) + V (n)u(n))^{2}

= R(n)^{2} sin^{2}(θ̄(n)) + R(n)^{2}(cos(θ̄(n)) − \frac{V (n)}{sin(k)} sin(θ̄(n)))^{2}

= R(n)^{2}[1 − \frac{V (n)}{sin(k)} sin(2θ̄(n)) + \frac{V (n)^{2}}{sin^{2}(k)} sin^{2}(θ̄(n))].

We can summarize with the EFGP equations:

ν_{k}(n) ≡ −\frac{V (n)}{sin(k)} ; θ̄(n) = θ(n) + k  (2.12a)
cot(θ(n + 1)) = cot(θ̄(n)) + ν_{k}(n)  (2.12b)
\frac{R(n + 1)^{2}}{R(n)^{2}} = 1 + ν_{k}(n) sin(2θ̄(n)) + ν_{k}(n)^{2} sin^{2}(θ̄).  (2.12c)

We will fix the ambiguity in θ by demanding θ(n + 1) − θ̄(n) ∈ [−π, π). (2.12) can be regarded as analogs of modified Prüfer equations in that if V = 0, R(n) = constant, and θ(n) = θ(0) + kn.

As noted in the introduction, Eggarter arrived at the first version of the EFGP transform by looking at continuum models with δ-function potential ((2.12b) is especially transparent in this mode). But, one could have arrived at it by noting that when V (n) ≡ 0, the transfer matrix is powers of ( \begin{matrix}2 cos(k) & −1 \\ 1 & 0 \end{matrix} ). This matrix has eigenval-

ues e^{±ik} and so it must be similar to ( \begin{matrix}cos(k) & sin(k) \\ − sin(k) & cos(k) \end{matrix} ). That similarity transformation will make the powers simple. Indeed,

( \begin{matrix}0 & sin(k) \\ 1 & − cos(k) \end{matrix} ) ( \begin{matrix}2 cos(k) & −1 \\ 1 & 0 \end{matrix} ) = ( \begin{matrix}cos(k) & sin(k) \\ − sin(k) & cos(k) \end{matrix} ) ( \begin{matrix}0 & sin(k) \\ 1 & − cos(k) \end{matrix} )

so the transform (2.7) precisely realizes the similarity.

There is an analog of Theorem 2.1. Define R(n, θ) by requiring R(1) = 1, θ(1) = θ in [0, π). Then Theorem 2.3. For any α ∈ (0, \frac{π}{2} ) and θ_{1} ≠ θ_{2}, there exists non-zero, finite constants C_{1} and C_{2} (independent of x and V) so that for all k ∈ (α, π − α),

C_{1} max(R(n, θ_{1}), R(n, θ_{2})) ≤ ∥T (n − 1, 0)∥ ≤ C_{2} max(R(n, θ_{1}), R(n, θ_{2})).  (2.13)

Because of the arccot, (2.12b) is somewhat awkward to deal with. Pastur-Figotin [26] have noted an equivalent form of (2.12b) which is straightforward from

e^{2iφ} = 1 + \frac{1}{2} \frac{1}{1 + i cot(φ)}

viz.,

e^{2iθ(n+1)} = e^{2iθ̄(n)} + \frac{iν_{k}(n)}{2}(\frac{(e^{2iθ̄(n)} − 1)^{2}}{1 − \frac{iν_{k}(n)}{2}(e^{2iθ̄(n)} − 1)}).  (2.14)

As an application of (2.14) we have

Proposition 2.4. If |ν_{k}(n)| < \frac{1}{2}, then

|θ(n + 1) − θ̄(n)| ≤ π|ν_{k}(n)|  (2.15)

Proof. If |ν_{k}(n)| < \frac{1}{2}, then (2.14) implies that

|e^{2iθ(n+1)} − e^{2iθ̄(n)}| ≤ \frac{|ν_{k}(n)|}{2} \frac{4}{\frac{1}{2}} = 4|ν_{k}(n)|.

Since |e^{iη} − 1| ≥ \frac{2|η|}{π}, we get

|θ(n + 1) − θ̄(n)| ≤ \frac{π}{4} |e^{2iθ(n+1)} − e^{2iθ̄(n)}|

and so the claimed result. □

Note. Kiselev, Remling, and Simon [20] present a way of defining R, θ that makes the analogy to the continuum case transparent, makes (2.14) transparent, improves (2.15), and extends to more general h_{0}.

§3. Conditional Integrals and A.C. Spectrum

It follows from [11, 16, 17] that for both continuum and discrete Schrödinger operators on [0, ∞), we have (see also [33] for a quick proof):

Proposition 3.1. If S is a set of reals so that for each λ ∈ S, sup_{x} ∥T_{λ}(x, 0)∥ < ∞, then H has purely a.c. spectrum on S in the sense that

(i) For any boundary condition θ and any T ⊂ S with |T | > 0, we have ρ_{θ}^{ac}(T) > 0. (ii) For any boundary condition θ, ρ_{θ}^{sing}(S) = 0.

Thus, bounded transfer matrices have important spectral consequences. By Theorems 2.1 and 2.3, if we can show R( · , θ) remains bounded for two initial θ’s, we have boundedness of T. From this and (2.4), (2.12c), one easily obtains the well-known result that if ∫ |V (x)| dx < ∞, (resp. ∑ |V (n)| < ∞), then the spectrum is purely a.c. in (0, ∞) (resp. (−2, 2)). Here is a result allowing more general decay, first in the continuum case.

Theorem 3.2. Fix k ≠ 0. Suppose that lim_{β→∞} ∫_{x}^{β} V (y)e^{2iky} dy exists and that

W_{k}(x) = ∫_{x}^{∞} V (y)e^{2iky} dy  (3.1)

obeys

∫ |V (x)W_{k}(x)| dx < ∞.  (3.2)

Then

\overline{lim}_{x→∞} ∥T (x, 0)∥ < ∞.  (3.3)

Remarks. 1. This result is not new; it is essentially due to Harris-Lutz [15]. This is a new proof.

2. This result implies that if V (x) = ∑_{m=1}^{N} a_{m} sin(k_{m}x)/x^{β}, β > \frac{1}{2}, and k ≠ ±\frac{1}{2}k_{m} for any m, then (3.3) holds, and so by Proposition 3.1, the spectrum is purely a.c. except for possible positive eigenvalues of \{\frac{1}{4}k_{m}^{2}\}.

3. In [19], Kiselev proved that if V (x) = O(x^{− \frac{3}{4} −ϵ}), then (3.2) holds off a set of Lebesgue measure zero.

Proof. We will show for any θ_{0}, R(x, θ_{0}) is bounded, and then one can appeal to Theorem 2.1 to complete the proof of (3.3). Write θ(x) = kx + φ(x), so by (2.3), φ obeys

\frac{dφ}{dx} = −\frac{V (x)}{k} sin^{2}(kx + φ).  (3.4)

By (2.4) (and R(0) = 1),

log R(x) = ∫_{0}^{x} \frac{1}{2k} Im [(\frac{dW_{k}}{dx} )e^{2iφ}] dx

= Im [\frac{1}{2k} [W_{k}(x)e^{2iφ(x)} − W_{k}(0)e^{2iθ_{0}}] − \frac{2i}{2k} ∫_{0}^{x} W_{k} \frac{dφ}{dx} e^{2iφ}] dx

if we integrate by parts. By hypothesis, W_{k}(x) is bdd so using (3.4),

| log R(x)| ≤ bdd + \frac{1}{k} ∫_{0}^{x} |W_{k}(y)V (y)| dy

is bounded by (3.2). □ Remark. A similar argument proves that

lim_{x→∞} θ − kx − \frac{1}{2k} ∫_{0}^{x} V (y) dy

exists. This in turn lets one prove there are complex solutions η_{±}(k, x) with

η_{±}(k, x) exp((∓i(kx − \frac{1}{2k} ∫_{0}^{x} V (y) dy)) → 1
η′_{±}(k, x) exp(∓i(kx − \frac{1}{2k} ∫_{0}^{x} V (y) dy)) → ik.

Notice that if V ∈ L^{2},

kx − \frac{1}{2k} ∫_{0}^{x} V (y) dy = ∫_{0}^{x} \sqrt{k^{2} − V (y)} dy + Q(x)

where lim_{x→∞} Q(x) exists. So if V ∈ L^{2}, this says that WKB-type solutions exist. This is also what the Harris-Lutz method gives [19].

We are heading toward a proof of Theorem 3.3. Fix k ≠ 0, π. Suppose V (n) is a discrete potential with

lim_{B→∞} ∑_{m=n}^{B} V (m)e^{2ikm} = W_{k}(n)

exists and that

∑_{n=1}^{∞} |V (n)W_{k}(n)| + |V (n)W_{k}(n + 1)| < ∞.  (3.5)

Then

\overline{lim}_{n→∞} ∥T (n, 0)∥ < ∞.

Given a function f on {1, 2, … }, define (δf)(n) = f(n + 1) − f(n) and note that summation by parts takes the form

∑_{m=a}^{b} g(m)(δf)(m) = − ∑_{m=a}^{b} f(m + 1)(δg)(m) + (fg)(b + 1) − (fg)(a).

14 A. KISELEV, Y. LAST, AND B. SIMON

Lemma 3.4. If (3.5) holds for some k, then ∑_{n=1}^{∞} |V (n)|^{2} < ∞. Proof. Since W exists, V → 0 at ∞ and so V is bounded. Thus, writing V (n) = −e^{−2ikn}(δW_{k})(n), and summing by parts,

∑_{n=1}^{B} V (n)^{2} = bdd + ∑_{n=2}^{B+1} V (n)W_{k}(n)e^{−2ikn} − ∑_{n=1}^{B} V (n)W_{k}(n + 1)e^{−2ikn}

is bounded by (3.5). □ Lemma 3.5. Suppose that {a_{n}}_{n=1}^{∞} is a real sequence so that

a_{n} → 0  as n → ∞  (3.6)

and

∑_{n=1}^{N} a_{n} is bounded.  (3.7)

Then ∏_{n=1}^{N}(1 + a_{n}) is bounded. Proof. By (3.6), |a_{n}| → 0, so without loss we can suppose that |a_{n}| < 1. Then |1 + a_{n}| ≤ 1 + a_{n} ≤ e^{a_{n}} and (3.7) implies the result. □ Proof of Theorem 3.3. By (2.12c), Lemma 3.4, and Lemma 3.5, it suffices to prove that

∑_{n=1}^{N} ν_{k}(n)e^{2iθ̄(n)} ≡ G(N)  (3.8)

is bounded. Define

φ(n) = θ(n) − k(n − 1) = θ̄(n) − kn.

Proposition 2.4 and Lemma 3.4 imply that for n large

|(δφ)(n)| ≤ π|ν_{k}(n)|.  (3.9)

By the definition (3.8),

G(N) = − ∑_{n=1}^{N} δW_{k}(n)(sin k)^{−1}e^{2iφ(n)}

= bdd + (sin k)^{−1} ∑_{n=1}^{N} W_{k}(n + 1)δ(e^{2iφ})(n).

But |δ(e^{2iφ})| ≤ 2|δφ|, so by (3.9)

|G(N) − bdd| ≤ C_{1} ∑_{n=1}^{N} |W_{k}(n + 1)ν_{k}(n)|

≤ C_{1}[ ∑_{n=1}^{N} |W_{k}(n)ν_{k}(n)| + |ν_{k}(n)|^{2}] < ∞. □

Sometimes it is better to use slightly different Prüfer variables. For example, if R, θ are defined by

u′(x) = \sqrt{E − V (x)} R(x) cos(θ(x))
u(x) = R(x) sin(θ(x)),

then

\frac{d log(R)}{dx} = \frac{1}{2} \frac{∂V}{∂x} cos^{2}(θ(x)),

from which we see if V (x) → 0 at infinity and \frac{∂V}{∂x} ∈ L^{1}, then solutions are bounded. (This is essentially the proof of Weidmann’s theorem [37] in [33].) If one tries out an integration by parts argument, one needs both \frac{∂V}{∂x} ∈ L^{1} and V ∈ L^{2}.

§4. Bound States for O(x^{−1}) Potentials

If |V (x)| = o(x^{−1}), Eastham-Kalf [8] show that −\frac{d^{2}}{dx^{2}} + V (x) has no positive eigenvalues; more generally, if \overline{lim} x|V (x)| = C < ∞, they show any eigenvalue λ must obey λ ≤ C^{2}.

On the other hand, Naboko [24] and Simon [34] have constructed V (x) decaying arbitrarily slower than x^{−1} with eigenvalues dense in [0, ∞). In fact, Simon [34] constructed V (x) with V (x) = O(x^{−1}) so that there are infinitely many eigenvalues with λ_{i} → 0 as long as ∑ \sqrt{λ_{i}} < ∞. In this section, we will handle the borderline case and improve Eastham-Kalf [8] by showing:

Theorem 4.1. Let V (x) obey C = \overline{lim}_{x→∞} x|V (x)| < ∞. Then there are at most countably many positive eigenvalues λ_{n} for which there are solutions u_{n} of

−u′′_{n} + V (x)u_{n} = λ_{n}u_{n}

and u_{n} ∈ L^{2}. Moreover,

∑_{n} λ_{n} ≤ \frac{C^{2}}{2} .  (4.1)

Remarks. 1. We do not specify boundary conditions on V, that is, (4.1) is a bound on all possible boundary conditions at once.

2. There are λ_{n} so that (4.1)holds, but ∑ \sqrt{λ_{n}} = ∞ (e.g., λ_{n} = \frac{3C^{2}}{π^{2}n^{2}} ) so there is a gap between Simon’s examples and what our bounds allow. We believe the optimal result would be to prove that ∑_{n} \sqrt{λ_{n}} ≤ C.

Without loss of generality by slightly increasing C and looking at [x, ∞), we can suppose that

|V (x)| ≤ C(1 + |x|)^{−1}  (4.2)

which we henceforth do.

The following is standard (see, e.g., Eastham-Kalf [8]):

Lemma 4.2. If V is bounded and u solves −u′′ + V u = λu and u ∈ L^{2}, then u′ ∈ L^{2}. In particular, R(x, θ_{0}) ∈ L^{2} for that θ_{0} with (u(0), u′(0)) = (R_{0} sin(θ_{0}), kR_{0} cos(θ_{0})).

Proof.

∫_{0}^{N} |u′|^{2} dx = u′u|_{0}^{N}− ∫_{0}^{N} u′′u dx

= u′u|_{0}^{N}+ ∫_{0}^{N}(λ − V)u^{2} dx

so if lim_{N→∞} ∫_{0}^{N} |u′|^{2} dx = ∞, then lim_{N→∞} u′u = ∞, but that implies u^{2}(N) = u(0)^{2} + 2 ∫_{0}^{N} u′u dx → ∞, contradicting the fact that u ∈ L^{2}. □

Lemma 4.3. Let f and g be C^{1} functions on [1, ∞) so that

|g′f| + |f′| ∈ L^{1}.

Then ∫_{0}^{N} f(x)e^{i(kx+g(x))} dx is bounded as N → ∞ for any k ≠ 0.

Proof. Write e^{ikx} = \frac{1}{ik} \frac{d}{dx}e^{ikx} and integrate by parts to see that

|∫_{1}^{N} f(x)e^{i(kx+g(x))} dx| ≤ \frac{|f(N)|}{|k|} + \frac{|f(1)|}{|k|} + \frac{1}{|k|} ∫_{1}^{N}(|f′| + |fg′|) dx.

Noting that |f(N)| = |f(1)| + ∫_{1}^{N} |f′(y)| dy, we see that the integral is bounded. □

Remark. If f(x) → 0 at infinity, this argument shows that lim_{N→∞} ∫_{1}^{N} f(x)e^{i(kx+g(x))} dx exists.

Lemma 4.4. Let {e_{i}}_{i=1}^{N} be a set of unit vectors in a Hilbert space H so that

α ≡ N sup_{i≠k}⟨e_{i}, e_{j}⟩ < 1.  (4.3)

Then

∑_{i=1}^{N} |⟨g, e_{i}⟩|^{2} ≤ (1 + α)∥g∥^{2}  (4.4)

MODIFIED PRÜFER AND EFGP TRANSFORMS 17

for any g ∈ H. Proof. Let A be the n × n matrix with a_{ij} = ⟨e_{i}, e_{j}⟩. Note that the Hilbert-Schmidt norm of A − 1 is bounded by (∑_{i≠j}⟨e_{i}, e_{j}⟩^{2})^{1/2} ≤ α so (4.3) says that A is invertible. If B is its inverse, then

f_{i} = ∑ B_{ij}e_{j}  (4.5)

obeys ⟨f_{i}, e_{j}⟩ = δ_{ij}, and thus

∑⟨g, e_{i}⟩f_{j} ≡ Proj of g to the span of the e’s

and so

∥g∥^{2} ≥ ∥∑⟨g, e_{i}⟩f_{i}∥^{2}.

By (4.5), ⟨f_{i}, f_{j}⟩ = B_{ij} and since ⟨h, A^{−1}h⟩_{ℂ^{n}} ≥ ∥A∥^{−1}⟨h, h⟩_{ℂ^{n}}, we see that

∑_{i=1}^{n} |⟨g, e_{i}⟩|^{2} ≤ ∥A∥ ∑_{i,j} \overline{⟨g, e_{i}⟩} ⟨f_{i}, f_{j}⟩⟨g, e_{j}⟩
≤ ∥A∥ ∥g∥^{2}

which is (4.4). □ Proof of Theorem 4.1. It obviously suffices to show for each fixed N < ∞ that

∑_{n=1}^{N} λ_{n} ≤ \frac{C^{2}}{2} .

Define R_{n}(x) to be the R corresponding to the L^{2} solution u(x, λ_{n}). Normalize u so R_{n}(0) = 1. By Lemma 4.2,

∑_{n=1}^{N} |R_{n}(x)|^{2} ∈ L^{1}

so

\underline{lim} x ∑_{n=1}^{N} |R_{n}(x)|^{2} = 0

(for if not, eventually ∑_{n=1}^{N} |R_{n}(x)|^{2} ≥ Cx^{−1} is not L^{1}). Thus, we can find B_{j} → ∞ so that for n = 1, …, N,

R_{n}(B_{j}) ≤ B_{j}^{−1/2}

or

∫_{0}^{B_{j}} \frac{d}{dx} (log R_{n}(y)) dy ≤ −\frac{1}{2} ln B_{j}

so by (2.4),

∫_{0}^{B_{j}} V (x) sin(2θ_{n}(y)) dy ≤ −\sqrt{λ_{n}} log B_{j}.  (4.6)

Now consider the Hilbert spaces

H_{j} = L^{2}((0, B_{j}), (1 + x) dx).

In H_{j}, we have

∥V ∥_{H_{j}}^{2} ≤ ∫_{0}^{B_{j}} C^{2}(1 + |x|)^{−2}(1 + x) dx = C^{2} log(B_{j}) + O(1).  (4.7)

Let

e_{n}^{(j)}(y) = \frac{sin(2θ_{n}(y))}{(1 + |y|)} \frac{1}{\sqrt{N_{n}^{(j)}}} χ_{[0,B_{j}]}(y),

where

N_{n}^{(j)} = ∫_{0}^{B_{j}} \frac{sin^{2}(2θ_{n}(y))}{(1 + |y|)} dy.

Notice that 4θ_{n}(y) − 4\sqrt{λ_{n}} and 2(θ_{n} ± θ_{m}) − 2(\sqrt{λ_{n}} ± \sqrt{λ_{m}}) have derivatives that are O(x^{−1}) by (2.3). Thus by Lemma 4.3,

∫_{0}^{B_{j}} \frac{sin(2θ_{n}(y)) sin(2θ_{m}(y)) − \frac{1}{2}δ_{nm}}{(1 + |y|)} dy

are bounded. We conclude that

N_{i}^{(j)} = \frac{1}{2} log B_{j} + O(1)  (4.8)
⟨e_{i}^{(j)}, e_{k}^{(j)}⟩ = O((log B_{j})^{−1})  i ≠ k.  (4.9)

(4.6) and (4.8) imply that

⟨V, e_{n}^{(j)}⟩_{H_{j}} ≤ −\sqrt{2λ_{n}} (log B_{j})^{1/2} + O(1).  (4.10)

Since the number N of eigenfunctions is fixed, but B_{j} → ∞ for j large, Lemma 4.4 applies and

∑_{n=1}^{N} |⟨V, e_{n}^{(j)}⟩_{H_{j}}|^{2} ≤ (1 + O((log B_{j})^{−1})∥V ∥_{H_{j}}^{2}.  (4.11)

But (4.10) and (4.7) then say that

2( ∑_{n=1}^{N} λ_{n}) log(B_{j}) ≤ C^{2} log(B_{j}) + O(1),

so

∑_{n=1}^{N} λ_{n} ≤ \frac{C^{2}}{2} . □

§5. Sparse Potentials: The Continuum, Absolutely Continuous Case

Our goal in this section is to prove assertion (1) in Theorem 1.6 and Theorem 1.6′. The idea will be to control ∥T (x)∥^{4} and then use Theorem 1.3. As explained in Section 1, the key is oscillations in sin(2θ(x)) for θ(x) ∼ kx_{n+1} for x near x_{n+1}. We will realize this using an integration by parts so we need a priori control on objects like \frac{d∥T (x_{n})∥}{dk}.

Fix a Pearson potential; a_{n} is assumed to obey a_{n} → 0 and x_{n+1} > x_{n} + 2Δ. Fix θ_{0} and solve the modified Prüfer equations for each k ∈ (0, ∞) to get functions θ(x, k) and R(x, k) (with initial conditions θ(x = 0, k) = θ_{0}, R(x = 0, k) = 1). Fix Δ so supp(W) ⊂ [−Δ, Δ].

We need two propositions to prepare for bounds in an integration by parts:

Proposition 5.1. Suppose that \underline{lim} x_{n+1}/x_{n} > 1. For each a, b > 0, there exists a constant C so that for each k ∈ (a, b),

|\frac{∂θ}{∂k} (x_{n} + Δ)| ≤ Cx_{n}  (5.1)

and

|\frac{∂^{2}θ}{∂k^{2}} (x_{n} + Δ)| ≤ Cx_{n}^{2}.  (5.2)

Moreover, uniformly for k ∈ (a, b),

lim_{x→∞} \frac{1}{x} \frac{∂θ}{∂k} (x) = 1  (5.3)
lim_{x→∞} \frac{1}{x^{2}} \frac{∂^{2}θ}{∂k^{2}} (x) = 0.  (5.4)

Proof. Let

β = inf_{n} \frac{x_{n+1}}{x_{n}} > 1  (5.5)

by hypothesis.

As a preliminary, note that if h, g, f are functions on [a, b], h is C^{1} and

h′(x) = f(x) + g(x)h(x).  (5.6)

Then

|h(b)| ≤ (|h(a)| + (b − a)∥f∥_{∞})e^{(b−a)∥g∥_{∞}}  (5.7)

as follows from the exact solution of (5.6):

h(x) = h(x)e^{∫_{a}^{x} g(y) dy} + ∫_{a}^{x} f(y)e^{∫_{y}^{x} g(z) dz} dy.

Now let h(x) = \frac{∂θ}{∂k} (x). From (2.3),

\frac{∂h}{∂x} = 1 + \frac{V (x)}{k^{2}} sin^{2}(θ(x)) − \frac{V (x)}{k} sin(2θ(x))h.  (5.8)

This means for x ∈ (x_{n−1} + Δ, x_{n} − Δ), we have that

\frac{∂h}{∂x} = 1.  (5.9)

By (5.7) and (5.8),

|h(x_{n} + Δ)| ≤ e^{2C|a_{n}|Δ}[|h(x_{n} − Δ)| + 2Δ + 2C|a_{n}|Δ]  (5.10)
≤ e^{2C|a_{n}|Δ}[|h(x_{n−1} + Δ)| + (x_{n} − x_{n−1}) + 2C|a_{n}|Δ]  (5.11)

where we used (5.9) to go from (5.10) to (5.11). In these equations, C is a constant only depending on (a, b). Throughout this proof, C is such a constant whose value can vary from one equation to the next.

Let β > 1 be given by (5.5). Pick n_{0} so large that for n ≥ n_{0}:

β^{−1}e^{2|a_{n}|CΔ} ≤ \frac{1}{2} (1 + β^{−1})  (5.12)

and

(1 + \frac{2C|a_{n}|Δ}{x_{n}})e^{2a_{n}CΔ} ≤ 1 + (\frac{1 − β^{−1}}{2} ).  (5.13)

Since β > 1 and a_{n} → 0, such an n_{0} exists. Next, pick D ≥ 2 so

|h(x_{n_{0}−1} + Δ)| ≤ Dx_{n_{0}−1}.  (5.14)

We claim inductively that for n ≥ n_{0} − 1, we have that

|h(x_{n} + Δ)| ≤ Dx_{n}  (5.15)

for by (5.14), this holds for n = n_{0} − 1, and if it holds for n − 1, then by (5.11) and x_{n−1} ≤ β^{−1}x_{n},

|h(x_{n} + Δ)| ≤ [Dx_{n−1} + x_{n} − x_{n−1} + 2C|a_{n}|Δ]e^{2C|a_{n}|Δ}

≤ x_{n}[(D − 1)β^{−1} + 1 + \frac{2C|a_{n}|Δ}{x_{n}}]e^{2C|a_{n}|Δ}

≤ x_{n}[(D − 1)(\frac{1}{2})(1 + β^{−1}) + 1 + (\frac{1 − β^{−1}}{2} )] (by (5.12)/(5.13))

= x_{n}[D − (D − 2)(\frac{1 − β^{−1}}{2} )] ≤ Dx_{n}

since D ≥ 2. Thus, we’ve proven (5.15).

Next, let H(x) = h(x) − x, so (5.8) implies that

|\frac{∂H}{∂x} | ≤ C|a_{n}|(1 + |H|)  (5.16)

on (x_{n} − Δ, x_{n} + Δ). Using (5.7) and (5.15), we conclude (recall the constant C changes from one equation to the next!)

|H(x_{n} + Δ) − H(x_{n} − Δ)| ≤ C|a_{n}|x_{n}.

Since H(x_{n−1} + Δ) = H(x_{n} − Δ), we have that for n ≥ n_{0},

|\frac{H(x_{n} + Δ)}{x_{n} + Δ}| ≤ \frac{C}{x_{n} + Δ} + ∑_{m=n_{0}}^{n} a_{m} \frac{x_{m}}{(x_{n} + Δ)}

≤ \frac{C}{x_{n} + Δ} + ∑_{m=n_{0}}^{n} a_{m}β^{−(n−m)} → 0

as n → ∞ since β > 1 and a_{m} → 0. From this and (5.16), we see that |\frac{H(x)}{x}| → 0 as x → ∞, which proves (5.1).

To prove (5.2), let g = \frac{∂h}{∂k} = \frac{∂^{2}θ}{∂k^{2}} . Then differentiating (5.8) with respect to k, we see that

\frac{∂g}{∂x} = 0  on (x_{n−1} + Δ, x_{n} − Δ)(5.17a)
\frac{∂g}{∂x} = A(x) + B(x)h(x) + D(x)g(x) + E(x)h^{2}(x) on (x_{n} − Δ, x_{n} + Δ) (5.17b)

where A, B, D, E are uniformly bounded by Ca_{n} on this interval with C uniformly bounded as k runs through (a, b).

Now use (5.7) and (5.1) to see that

|g(x_{n} + Δ)| ≤ e^{2Ca_{n}Δ}[g(x_{n−1} + Δ) + Ca_{n}x_{n}^{2}Δ].

As above, if n is so large that

β^{−2}e^{2Ca_{n}Δ} ≤ \frac{1}{2} (1 + β^{−1})  and  (Ca_{n}Δ)e^{2Ca_{n}Δ} ≤ \frac{1}{2} (1 − β^{−1})

then inductively,

g(x_{n} + Δ) ≤ Cx_{n}^{2}

for n large. This is (5.2). Plugging this into (5.17b), we see that

g(x_{n} + Δ) ≤ C(1 + ∑_{m=1}^{n} a_{m}x_{m}^{2}),  (5.17c)

which yields lim_{n→∞} g(x_{n} + Δ)/x_{n}^{2} = 0 from which (5.4) is immediate. □

Proposition 5.2. For any a, b > 0, there is a C so that for all k ∈ (a, b),

log R(x_{n} + Δ) ≤ C ∑_{m=1}^{n} |a_{m}|  (5.18)

\frac{∂ log R}{∂k} (x_{n} + Δ) ≤ C ∑_{m=1}^{n} |a_{m}x_{m}|.  (5.19)

Proof. By (2.4), log R(x) is constant for x ∈ (x_{n−1} + Δ, x_{n} − Δ) and

| log R(x_{n} + Δ) − log R(x_{n} − Δ)| ≤ 2k^{−1}|a_{n}| ∫ W (y) dy,

so (5.18) holds with C = 2 min(k)^{−1} ∫ W (y) dy.

¿From (2.4), we have

\frac{∂}{∂x} \frac{∂}{∂k} (k log R) = V (x) cos(2θ(x)) \frac{∂θ}{∂k}

so that the bound (5.1) implies (5.9). □

As a final preliminary, we note that

Lemma 5.3. Suppose that \underline{lim} x_{n+1}/x_{n} > 1. Then for a constant C,

∑_{n=1}^{∞} ∑_{m≤n} |a_{n}a_{m}| \frac{x_{m}}{x_{n}} ≤ C ∑_{n=1}^{∞} a_{n}^{2}.

Proof. Let β = lim x_{n+1}/x_{n}. Pick 1 < γ < β. Then for m ≤ n, x_{m}/x_{n} ≤ Cγ^{−|m−n|}. Thus, the lemma follows from Young’s inequality that

T (a)_{n} ≡ ∑_{m} γ^{−|m−n|}a_{m}

is bounded from ℓ^{2} to ℓ^{2} for any γ > 1. □

Proof of Theorem 1.6′. Let g be a non-negative C^{∞}-function compactly supported on (0, ∞). We will prove that

sup_{n} ∫ g(k)R(k, x_{n} + Δ)^{4} dk < ∞.  (5.20)

Proving this for two values of θ_{0} and appealing to Theorem 2.1 gets a uniform bound on ∫ g(k)∥T (0, x_{n} + Δ)∥^{4} dk. Theorem 1.3 then proves pure absolute continuity of the spectrum on (0, ∞).

Let B_{n} = ∫ g(k)R(x_{n} + Δ)^{4} dk. Notice that by (2.4), R(x_{n−1} + Δ) = R(x_{n} − Δ) and

R(x_{n} + Δ)^{4} = R(x_{n} − Δ)^{4} exp(Q_{n}),  (5.21)

where

Q_{n} = \frac{2}{k} ∫_{−Δ}^{Δ} a_{n}W (y) sin(2θ(x_{n} + y)) dy.

Since k^{−1} and a_{n} are bounded, Q_{n} is uniformly bounded in n, and so

\begin{aligned}exp(Q_{n}) & ≤ 1 + Q_{n} + CQ_{n}^{2} \\ & ≤ 1 + Q_{n} + Ca_{n}^{2} \end{aligned}  (5.22)

(where again C is a constant that varies from formula to formula).

For y ∈ (−Δ, Δ), we have by (2.3)

|θ(x_{n} + y) − θ̃_{n}(y)| ≤ Ca_{n}

where

θ̃_{n}(y) = θ(x_{n−1} + Δ) + k(x_{n} + y − x_{n−1} − Δ),

so

|Q_{n} − \frac{2}{k} ∫_{−Δ}^{Δ} a_{n}W (y) sin(2θ̃_{n}(y)) dy| ≤ Ca_{n}^{2}.  (5.23)

By (5.21)–(5.23),

B_{n} ≤ B_{n−1}(1 + Ca_{n}^{2}) + E_{n}  (5.24)

where

E_{n} = a_{n} ∫_{−Δ}^{Δ} dy ∫ \frac{2g(k)}{k} R(x_{n−1} + Δ, k)^{4}W (y) sin(θ̃_{n}(y)) dk.

Notice that we’re implementing our basic strategy: We separate out the second-order terms (which will present no problem since ∏_{n=1}^{∞}(1 + Ca_{n}^{2}) < ∞) and need to control the first-order terms where we have an explicit highly oscillatory factor since θ_{n} ∼ kx_{n}.

Now

\frac{∂θ̃_{n}}{∂k} (y) = x_{n} + y − x_{n−1} − Δ + \frac{∂θ(x_{n−1} + Δ)}{∂k} > \frac{1}{2} x_{n}  (5.25)

for n large by the bound (5.3).

Thus, we can write

sin(θ̃_{n}(y)) = \frac{1}{\frac{∂θ̃_{n}}{∂k}} \frac{∂}{∂k} (− cos(θ̃_{n}(y)))

and integrate by parts.

24 A. KISELEV, Y. LAST, AND B. SIMON

After integration by parts, we have three terms

E_{n}^{(1)} coming from \frac{∂[k^{−1}g(k)]}{∂k}
E_{n}^{(2)} coming from \frac{∂R^{4}}{∂k}
E_{n}^{(3)} coming from \frac{∂}{∂k} ( \frac{1}{\frac{∂θ̃}{∂k}} ).

For the E_{n}^{(1)} term, we can bound R^{4} as follows using (5.18) and

x_{n} ≥ Cβ^{n}.  (5.26)

By (5.10), for n large,

R^{4} ≤ C exp(C ∑_{m=1}^{n} a_{m})

≤ C exp(\frac{n}{2} ln(β))

since a_{n} → 0. Thus, by (5.19) and (5.26),

E_{n}^{(1)} ≤ Cβ^{n/2}β^{−n} = Cβ^{−n/2}.  (5.27)

For the E_{n}^{(2)} term, we use \frac{∂R^{4}}{∂k} = R^{4} \frac{∂ log R}{∂k}, (5.19), and (5.25) to see that

E_{n}^{(2)} ≤ CB_{n−1}b_{n}

where

b_{n} = ∑_{m=1}^{n−1} a_{n}a_{m} \frac{x_{m}}{x_{n}} .

Note now that by ∑ a_{n}^{2} < ∞ and Lemma 5.3, we have

∑_{n=1}^{∞} b_{n} < ∞.  (5.28)

For the E_{n}^{(3)} term, we use (5.25) and (5.17c) to see that E_{n}^{(3)} ≤ B_{n−1}c_{n} where

c_{n} = Ca_{n} \frac{(1 + ∑_{m=1}^{n−1} a_{m}x_{m}^{2})}{x_{n}^{2}} .

As in the proof of Lemma 5.3,

∑ c_{n} < ∞.  (5.29)

By (5.24) and the above estimates on E_{n}^{(i)},

max(B_{n}, 1) ≤ (1 + Ca_{n}^{2} + Cb_{n} + Cc_{n} + Cβ^{−n/2}) max(B_{n−1}, 1).  (5.30)

By hypothesis, ∑ a_{n}^{2} < ∞, and by (5.28–5.29), ∑ b_{n} + c_{n} < ∞. Thus

∏_{n=1}^{N}(1 + Ca_{n}^{2} + Cb_{n} + Cc_{n} + Cβ^{−n/2})

is bounded and consequently, so is B_{n}. □

It is easy to see that the methods of this section extend to prove:

Theorem 5.4. Suppose V (x) = ∑ W_{n}(x − x_{n}) where (i) \overline{lim} x_{n}/x_{n+1} < 1 (ii) supp W_{n} ⊂ [−Δ, Δ] for some fixed Δ (iii) ∑_{n} ∫ |W_{n}(y)|^{2} dy < ∞.

Then −\frac{d^{2}}{dx^{2}} + V (x) has purely a.c. spectrum on (0, ∞).

§6. Sparse Potentials: The Continuum, Singular Continuous Case

In this section, we will prove assertion (2) in Theorem 1.6. The idea will be to force ∥T (k^{2}, x_{n})∥ to infinity for almost all k and suitable x_{n}. To do this, we will need to isolate a strictly positive second-order term and show that these second-order terms then dominate the first-order terms because of oscillations.

Here is a warm-up problem to show this cancellation mechanism. Let X_{n} be independent, identically distributed random variables taking the values ±1 with probability \frac{1}{2}. Let ϵ > 0 and let a_{n} be a sequence going to zero as n → ∞. Finally, let

Y_{n} = ∑_{m=1}^{n}(ϵa_{m}^{2} + a_{m}X_{m}).

Suppose that ∑ a_{n}^{2} = ∞. We claim there exists a subsequence n(i) → ∞, so with probability 1,

lim_{i→∞} Y_{n(i)} = ∞.  (6.1)

The reason (6.1) holds is that by the central limit theorem ∑_{m=1}^{n} a_{n}X_{n} is typically not more negative than O( − \sqrt{∑ a_{n}^{2}} ) and, because of the square root, this is smaller than ϵ ∑_{m=1}^{n} a_{n}^{2}.

To make a proof, notice that since ∑_{m=1}^{n} a_{m}^{2} → ∞, we can choose n(i) so that ∑_{m=1}^{n(i)} a_{m}^{2} ≥ i^{2}. By a Tschbechev inequality,

Prob( ∑_{1}^{n(i)} a_{m}X_{m} ≥ \frac{ϵ}{2} ∑_{1}^{n(i)} a_{m}^{2}) ≤ \frac{∥ ∑_{1}^{n(i)} a_{m}X_{m}∥^{2}}{(\frac{ϵ}{2} ∑_{1}^{n(i)} a_{m}^{2})^{2}} = \frac{4}{ϵ^{2}} \frac{1}{∑_{1}^{n(i)} a_{m}^{2}} ≤ \frac{4}{ϵ^{2}i^{2}} .

∑ \frac{1}{i^{2}} < ∞, so by the Borel-Cantelli lemma, with probability 1, eventually

∑_{1}^{n(i)} a_{m}X_{m} ≤ \frac{ϵ}{2} ∑_{1}^{n(i)} a_{n}^{2}

and thus eventually,

Y_{n(i)} ≥ \frac{ϵ}{2} ∑_{1}^{n(i)} a_{m}^{2}

diverges.

The usual Kolmogorov stopping argument that lets one prove things without subsequences isn’t obviously applicable here in a situation where we assume no regularity on the a_{m}’s (see Section 8 for the case a_{m} = m^{−α}). Since a subsequence suffices for our application, we have not tried to push the argument through to get lim Y_{n} = ∞, even in the toy problem.

Notice that independence of the X_{n}’s was not needed; rather, it suffices to have enough control of E(X_{n}X_{m}) to show that the first-order term is small compared to the secondorder term. In the case at hand, we will use integration by parts in k as we did in the last section to get this control.

We summarize the key to the above argument with

Lemma 6.1. Let P_{n}, Q_{n} be random variables so that (i) P_{n}(x) ≥ α_{n} > 0 for a.e. x and positive reals α_{n} (ii) ∑ α_{n}^{−1}Exp(|Q_{n}|) < ∞ (iii) lim_{n→∞} α_{n} = ∞.

Then P_{n}(x) + Q_{n}(x) → ∞ for a.e. x. If (ii) is replaced with

(ii′) \underline{lim}_{n→∞} α_{n}^{−1}Exp(|Q_{n}|) = 0, then there exists a subsequence n(i) so that P_{n(i)}(x) + Q_{n(i)}(x) → ∞ for a.e. x.

Proof. If (ii′) holds, we can find a subsequence so that (ii) holds. Thus, it suffices to prove the result assuming (ii).

By (ii), ∑ α_{n}^{−1}|Q_{n}(x)| < ∞ for a.e. x. In particular, α_{n}^{−1}Q_{n}(x) → 0 so P_{n} + Q_{n} ≥ α_{n}[1 − α_{n}^{−1}|Q_{n}(x)|] → ∞. □

We will also need the following lemma:

Lemma 6.2. Suppose that B_{n}, α_{n}, β_{n} ≥ 0 are real numbers and that

B_{n} ≤ B_{n−1} + 2α_{n}\sqrt{B_{n−1}} + β_{n}  (n ≥ 1).  (6.2)

Then,

\sqrt{B_{n}} ≤ \sqrt{B_{0}} + ∑_{k=1}^{n} α_{k} + \sqrt{∑_{k=1}^{n} β_{k}} .  (6.3)

Proof. We give a proof by induction. (6.2) holds for n = 0. Let a_{n} = ∑_{k=1}^{n} α_{k}, b_{n} = ∑_{k=1}^{n} β_{k}. By the induction hypothesis,

\sqrt{B_{n−1}} ≤ \sqrt{B_{0}} + a_{n−1} + \sqrt{b_{n−1}} .  (6.4)

(6.2) implies that

B_{n} ≤ (\sqrt{B_{n−1}} + α_{n})^{2} + β_{n}.

So by (6.4),

B_{n} ≤ (\sqrt{B_{0}} + a_{n} + \sqrt{b_{n−1}} )^{2} + β_{n}
≤ (\sqrt{B_{0}} + a_{n})^{2} + b_{n} + 2\sqrt{b_{n−1}} (\sqrt{B_{0}} + a_{n})
≤ (\sqrt{B_{0}} + a_{n} + \sqrt{b_{n}} )^{2}

proving (6.3) inductively. □

So fix a Pearson potential with ∑ a_{n}^{2} = ∞. Fix θ_{0} and let R(x, k) be the solution of (2.3/2.4). Let

Y_{n}(k) = log R(x_{n} + Δ, k)

and

δY_{n}(k) = Y_{n}(k) − Y_{n−1}(k).

By (2.4),

δY_{n}(k) = \frac{a_{n}}{2k} ∫_{−Δ}^{Δ} W (y) sin 2θ(x_{n} + y) dy.  (6.5)

As in Section 5, we write

θ̃_{n}(y) = θ(x_{n−1} + Δ) + k(x_{n} + y − x_{n−1} − Δ).

But we expand θ to the next order by letting

θ_{n}^{(1)}(y) = −\frac{a_{n}}{k} ∫_{−Δ}^{y} W (y) sin^{2}(θ̃_{n}(y)) dy.  (6.6)

Then by (2.3),

θ(x_{n} + y) = θ̃_{n}(y) + θ_{n}^{(1)}(y) + O(a_{n}^{2}),

so by (6.5),

δY_{n}(k) = a_{n}X_{n}^{(1)} + a_{n}^{2}S_{n} + O(a_{n}^{3})  (6.7a)
X_{n}^{(1)} = \frac{1}{2k} ∫_{−Δ}^{Δ} W (y) sin(2θ̃_{n}(y)) dy  (6.7b)

S_{n} = \frac{1}{k} ∫_{−Δ}^{Δ} W (y) cos(2θ̃_{n}(y))[\frac{θ_{n}^{(1)}(y)}{a_{n}}].  (6.7c)

In the formula for θ_{n}^{(1)}, use

sin^{2}(θ̃_{n}(y)) = \frac{1}{2}(1 − cos(2θ̃_{n}(y))).

The cos term from this formula when plugged into (6.7c) gives

\frac{1}{2} k^{2} ∫_{−Δ}^{Δ} W (y) cos(2θ̃_{n}(y))( ∫_{−Δ}^{y} W (s) cos(2θ̃_{n}(s))) dy

= \frac{1}{4k^{2}} ( ∫_{−Δ}^{Δ} W (y) cos(2θ̃_{n}(y)) dy)^{2} (6.8)

We lump the contribution of the \frac{1}{2} term with the first-order term. Defining X(y) = ∫_{−Δ}^{y} W (s) ds, we find

δY_{n}(k) = [a_{n}^{2}Z_{n}(k) + a_{n}X_{n}(k)] + O(a_{n}^{3}),  (6.9)

where

Z_{n}(k) = \frac{1}{4k^{2}} ( ∫_{−Δ}^{Δ} W (y) cos(2θ̃_{n}(y)) dy)^{2}
X_{n}(k) = \frac{1}{2k} ∫_{−Δ}^{Δ}[W (y) sin(2θ̃_{n}(y)) − \frac{a_{n}W (y)X(y)}{2k} cos(2θ̃_{n}(y))] dy.

In (6.9), the O(a_{n}^{3}) means an error bounded by Ca_{n}^{3} where C is a finite constant for k ∈ [a, b] any compact subinterval of (0, ∞).

Define

W̃(k) = ∫_{−Δ}^{Δ} W (y)e^{2iky} dy.

Then,

Z_{n}(k) = \frac{1}{8k^{2}} |W̃(k)|^{2} + X̃_{n}(k),  (6.10)

where

X̃_{n}(k) = \frac{1}{8k^{2}} |W̃(k)|^{2} cos(4(θ̃_{n}(0, k) + φ(k)),  (6.11)

where φ(k) = \frac{1}{2}Arg(W̃(k)).

For let θ̃_{n}(y) = θ̃_{n}(0) + ky. If W̃(k) = |W̃(k)|e^{2iφ(k)}, then

Z_{n}(k) = \frac{1}{4k^{2}} (Re ∫_{−Δ}^{Δ} W (y)e^{2i(θ̃_{n}(0)+ky)} dy)^{2}

= \frac{1}{4k^{2}} |W̃(k)|^{2} cos^{2}(2(θ̃_{n}(0, k) + φ(k))).

Proof of Theorem 1.6, Part (2). Let

P_{n}(k) = \frac{1}{8k^{2}} |W̃(k)|^{2} ∑_{m=1}^{n} a_{m}^{2}
Q_{n}(k) = Y_{n}(k) − P_{n}(k); δQ_{n}(k) = Q_{n}(k) − Q_{n−1}(k),

so

δQ_{n}(k) = a_{n}^{2}X̃_{n}(k) + a_{n}X_{n}(k) + O(a_{n}^{3}).

Let g be a C^{∞}-function compactly supported in {k ∈ (0, ∞) | W̃(k) ≠ 0}. Let

B_{n} = ∫ g(k)| ∑_{m=1}^{n} a_{m}X_{m}(k)|^{2} dk

B̃_{n} = ∫ g(k)| ∑_{m=1}^{n} a_{m}^{2}X̃_{m}(k)|^{2} dk.

We will prove that

\sqrt{B_{n}} / ∑_{m=1}^{n} a_{m}^{2} → 0  (6.12)

as n → ∞, and similarly for B̃_{n}. Since ∑_{m=1}^{n} a_{m}^{3}/ ∑_{m=1}^{n} a_{m}^{2} → 0 (on account of a_{n} → 0 and ∑_{m=1}^{n} a_{m}^{2} → ∞), (6.12) and the Schwartz inequality imply that

∫ g(k)|Q_{n}(k)| dk/ ∑_{m=1}^{n} a_{m}^{2} → 0

so by Lemma 6.1 and inf_{k∈supp g} \frac{|W̃(k)|}{8k^{2}} > 0 implies that there is a subsequence n(i) so that Y_{n(i)}(k) → ∞ for a.e. k in supp g. By doing this for two values of θ_{0} and using Theorem 2.1 and Theorem 1.1, we conclude there is no a.c. spectrum on supp g.

Since W̃ is an entire function, it has isolated zeros and thus, this argument shows σ_{ac} is empty. By Theorem 1.4, σ_{pp} ∩ (0, ∞) is empty, and an elementary argument proves that σ(H) ⊃ [0, ∞). So the spectrum on (0, ∞) is purely singular continuous. It thus suffices to prove (6.12) (the proof for B̃_{n} is essentially identical).

Let M_{n−1}(k) = ∑_{m=1}^{n−1} a_{m}X_{m}(k). Then

B_{n} ≤ B_{n−1} + |∫ g(k)M_{n−1}(k)a_{n}X_{n}(k)| + Ca_{n}^{2}

for a suitable constant C. Now X_{n} has cos(2θ̃_{n}(y)) and sin(2θ̃_{n}(y)) terms. As in the last section, we write those as a suitable [dθ̃_{n}(y, k)/dk]^{−1} \frac{d}{dk} […] and integrate by parts and get three terms:

One coming from \frac{∂[k^{−1}g(k)]}{∂k} ∂k. Noting that |M_{n}(k)| ≤ Cn, we have that this is bounded by \frac{Cn}{x_{n}}.

One coming from \frac{∂M_{n}(k)}{∂k}. Using (5.1), this term is bounded by

C ∑_{m=1}^{n−1} a_{n}a_{m} \frac{x_{m}}{x_{n}} .

One coming from L_{n} = [\frac{∂^{2}θ_{n}}{∂k^{2}} ]/[\frac{∂θ_{n}}{∂k}]^{2}. As in the last section, this L_{n} is bounded by C(∑_{m=1}^{n−1} a_{m}x_{m}^{2})/x_{n}^{2}. We can use the Schwartz inequality to control ∫ g(k)|M_{n}(k)| dk, and so bound this term by C\sqrt{B_{n−1}} a_{n} ∑_{m=1}^{n−1} a_{m}x_{m}^{2}/x_{n}^{2}. The net result is the bound

B_{n} ≤ B_{n−1} + 2α_{n}\sqrt{B_{n−1}} + β_{n},  (6.13)

where

α_{n} = C ∑_{m=1}^{n−1} |a_{n}a_{m}| \frac{x_{m}^{2}}{x_{n}^{2}}

and

β_{n} = C[a_{n}^{2} + \frac{n}{x_{n}} + ∑_{m=1}^{n−1} a_{n}a_{m} \frac{x_{m}}{x_{n}} ].

By the argument in Lemma 5.3 with x_{n−1}/x_{n} → 0 and ∑_{m=1}^{∞} a_{n}^{2} → ∞, we see that

∑_{m=1}^{n} α_{m}/ ∑_{m=1}^{n} a_{m}^{2} → 0  (6.14)

and that

∑_{m=1}^{n} β_{m} ≤ C(1 + ∑_{m=1}^{n} a_{m}^{2}),

so

\sqrt{∑_{m=1}^{n} β_{m}} / ∑_{m=1}^{n} a_{m}^{2} → 0.  (6.15)

Lemma 6.2 and (6.13–6.15) imply (6.12). □

One can modify this construction to make examples of decaying potentials for which the associated Schrödinger operator has regions of a.c. spectrum and regions of s.c. spectrum. The idea is to arrange that W̃(k) vanishes in a whole interval so that even though a_{n} ∉ ℓ_{2}, we have a.c. spectrum for those k. Of course, W̃(k) cannot vanish if W has compact support, so we will take the bump functions of increasing support converging toward a function whose Fourier transform vanishes in an interval. So, let S = [a, b] ⊂ (0, ∞). Let f be an even Schwartz class function that vanishes if k^{2} ∈ S and is strictly positive on [0, ∞)\S.

Let a_{n} = n^{−1/2}, x_{n} = (n!)^{2}, Δ_{n} = n^{−1/12}. Notice that ∑ a_{n}^{2} = ∞. Define

f̃(x) = \frac{1}{4π} ∫ exp(−2ikx)f(k) dk

W_{n}(x) = f̃(x)χ_{(−Δ_{n},Δ_{n})}(x)

and

V (x) = ∑_{n} a_{n}W_{n}(x − x_{n}).

We are heading toward:

Theorem 6.3. The half-axis Schrödinger operator −\frac{d^{2}}{dx^{2}} + V (x) has purely singular spectrum on (0, ∞)\S and purely a.c. spectrum on S.

Lemma 6.4. For any m > 0, there exists a constant C_{m} with

|f(k) − ∫ e^{2ikx}W_{n}(x) dx| ≤ C_{m}n^{−m}.  (6.16)

Proof. Let f_{n}(k) = ∫ e^{−2ikx}W_{n}(x) dx. Then

f_{n}(k) = \frac{1}{2π} ∫ \frac{sin(Δ_{n}(k − k′))}{(k − k′)} f(k′) dk′

so the left side of (6.16) is

\frac{1}{2π} |∫ \frac{sin Δ_{n}(k − k′)}{k − k′} [f(k) − f(k′)] dk′|,

which has the form

|∫ g(y, k) sin Δ_{n}y dy|,

where g(y, k) is Schwartz space in y with bounds (including bounds on derivatives) uniform in k. If we integrate by parts 12m times, we will get (6.16). □

Proposition 5.1 extends with no change. In the region where f(k) ≠ 0, the analysis earlier in this section shows that log R(x_{n(i)} + Δ_{n(i)}) → ∞ for a.e. k and a suitable subsequence x_{n(i)}, so we know the spectrum in (0, ∞)\S is purely singular continuous.

On the other hand, if g is C^{∞} supported in S, we claim that

sup_{n} ∫ g(k)R(k, x_{n} + Δ_{n})^{4} dk < ∞.  (6.17)

The proof is similar to that in the last section. In place of (5.22), we need to use

exp(Q_{n}) ≤ 1 + Q_{n} + \frac{1}{2} Q_{n}^{2} + O(a_{n}^{3}).

As in this section, Q_{n}^{2} has a term a_{n}^{2}|W̃_{n}(k)|^{2}/8k^{2} and oscillatory terms that we can integrate by parts. Noting that

∑_{m=1}^{n−1} a_{n}a_{m} \frac{x_{m}}{x_{n}} ≤ n^{−2}n^{−1/2} ∑_{m=1}^{n−1} m^{−1/2} ≤ Cn^{−2}

is still summable and that ∑ a_{n}^{2}|W̃_{n}(k)|^{2} is summable by Lemma 6.4, we obtain (6.17). □

§7. Sparse Potentials: The Discrete Case

In this section, we will sketch the proof of Theorem 1.7. The proof follows closely that in the last two sections with (2.12) replacing (2.3/2.4). We will make use of (2.14), the Pastur-Figotin form of (2.12b).

Fix α > 0 and pick k ∈ (α, π − α) and then N so large that for all such k, |ν_{k}(n)| < \frac{1}{2} for n ≥ N_{0}. (2.14) can then be effectively used to prove the analogs of (5.1/5.2), that is,

|\frac{∂θ}{∂k} (x_{n})| ≤ Cx_{n}  |\frac{∂^{2}θ}{∂k^{2}} (x_{n})| ≤ Cx_{n}^{2}.  (7.1)

(2.12c) can be rewritten

log R(n + 1) − log R(n) = \frac{1}{2} log(1 + ν_{k}(n) sin(2θ̄) + ν_{k}(n)^{2} sin^{2}(θ̄)).  (7.2)

This implies the bound

log R(x_{n}) ≤ C ∑_{m=1}^{n} |a_{m}|.  (7.3)

Next notice that

1 + α sin(2θ) + α^{2} sin^{2}(θ) = (1 + \frac{1}{2} α sin(2θ))^{2} + α^{2} sin^{4}(θ).

This provides a uniform bound on the argument of the log(·) in (7.2), and so allows one to prove

|\frac{∂}{∂k} log R(x_{n})| ≤ C ∑_{m=1}^{n} a_{n}x_{m}.  (7.4)

With these tools, the proof of assertion (1) of Theorem 1.6 is similar to that in Section 5, only a little simpler since (2.12c) implies

R(n + 1)^{4} ≤ R(n)^{4}(1 + ν_{k}(n) sin(2θ̄(n)) + C_{n}a_{n}^{2}).

The same integration by parts used in Sections 5 and 6 shows that

∫ g(k)R(n, k)^{4}ν_{k}(n) sin(2θ(n)) dk = C(b_{n} + c_{n} + B^{−n/2})(1 + ∫ g(k)R(n, k)^{4} dk)

with b_{n} = ∑_{m−1}^{n−1} a_{n}a_{m} x_{m}/x_{n} and c_{n} is like b_{n} with x_{m}^{2}/x_{n}^{2} replacing x^{m}/x^{n}. As in Section 5, this proves assertion (1) in Theorem 1.7.

To prove assertion (2), we must identify a strictly positive second-order term. We write

log(1 + α sin(2θ) + α^{2} sin^{2}(θ)) = α sin(2θ) + α^{2}(sin^{2}(θ) − \frac{1}{2} sin^{2}(2θ)) + O(α^{3})  (7.5)
= α sin(2θ) + \frac{1}{4} α^{2} cos(4θ) − \frac{1}{2} α^{2} cos(2θ) + \frac{1}{4} α^{2} + O(α^{3}).

(7.6)

This lets us write

log R(n + 1) − log R(n) = \frac{1}{2} a_{n}^{2} + a_{n}X_{n}

and, as in Section 6, use the integration by parts machine to prove

(∫ (∑_{n=1}^{N} a_{n}X_{n})^{2}g(k) dk)^{1/2}/ ∑ a_{n}^{2} → 0

and complete the proof as there.

In this case, we don’t need to worry about zeros of W̃(k) since the analog of W here is δ_{n0} and so W̃(k) = 1.

§8. Random Decaying Potentials: The Discrete Case

In this section, we consider discrete situations where the V (n) are independent random variables of zero mean and decaying variance. The results that imply a.c. spectrum require no regularity in E(V (n)^{2}), while those for singular spectrum require some kind of regular decay, as we will explain.

The results for a.c. spectrum are so general yet so simple to prove that they are a paradigm of the usefulness of the EFGP transform.

Theorem 8.1. Suppose V_{ω}(n) are independent random variables with E(V_{ω}(n)) = 0 and

∑_{n} E(V_{ω}(n)^{2}) + E(V_{ω}(n)^{4}) < ∞.  (8.1)

Then for a.e. ω, h_{ω} has purely a.c. spectrum on (−2, 2).

Remarks. 1. For E(V_{ω}^{2})^{1/2} ≤ Cn^{−α} with V bounded and α > \frac{1}{2}, we get a.c. spectrum recovering results of Delyon, et al. [7].

2. If the V_{ω}(n) are uniformly bounded, then E(V_{ω}(n)^{4}) ≤ CE(V_{ω}(n)^{2}) and so (8.1) becomes ∑_{n} E(V_{ω}(n)^{2}) < ∞; we state the general bound because unbounded V ’s are so easy to accommodate.

3. The case E(V_{ω}(n)^{2})^{1/2} = n^{−1/2} log(n)^{−1} is of some interest. This sequence is ℓ^{2} so if V is bounded, the theorem proves a.c. spectrum. Kotani-Ushiroya [21] cannot handle such borderline cases.

Proof. Fix θ_{0}. Then R_{ω}(n) and θ_{ω}(n) become random variables which are measurable functions of {V_{ω}(j)}_{j≤n−1} and so independent of {V_{ω}(j)}_{j≥n}.

By (2.12c),

R(n + 1)^{4} = R(n)^{4}(1 + \frac{V_{ω}(n)}{sin k} sin(2θ̄_{ω}(n)) + O(V_{ω}^{2} + V_{ω}^{4})).

Since V_{ω}(n) is independent of θ̄(n) and R(n), we have

E(R_{ω}(n)^{4}V_{ω}(n) sin(2\overline{θ_{ω}(n)} )) = E(V_{ω}(n))E(R_{ω}^{4}(n) sin(2θ̄_{ω}(n))) = 0.

Using independence to bound E(R(n)^{4}V_{ω}^{j}) by E(R(n)^{4})E(V_{ω}^{j}), we see that

E(R_{ω}(n + 1)^{4}) ≤ [1 + CE(V_{ω}^{2}(n) + V_{ω}^{4}(n))]E(R_{ω}^{4}(n)),

where C is uniformly bounded for k in any (α, π − α) with α > 0. It follows that

E( ∫_{α}^{π−α} R_{ω}(n, k)^{4} dk) < ∞.

By Fatou’s lemma, for a.e. ω,

\underline{lim} ∫_{α}^{π−α} R_{ω}(n, k)^{4} dk < ∞,

and by Theorem 1.3, the spectrum is purely a.c. on (−2 cos(α), 2 cos(α)). □

For the case where ∑_{n=1}^{∞} E(V (n)^{2}) = ∞, we need some regularity of the fall-off. Rather than try to find complicated general conditions, we consider the case where E(V (n)^{2}) ∼ n^{−2α} with α ≤ \frac{1}{2}. The same method can handle a case like E(V (n)^{2}) = [n log(n + 1)]^{−1} (which always has singular continuous spectrum of Hausdorff dimension 1) by the kind of arguments we will discuss in the case α = \frac{1}{2}; in this case for typical energies ∥T (0, n)∥ grows like log(n).

Explicitly, we suppose

(i) E(V_{ω}(n)^{2})^{1/2} = λn^{−α} 0 < α ≤ \frac{1}{2}; λ > 0 (ii) E(V_{ω}(n)) = 0 (iii) For some ϵ > 0, sup_{ω} |V_{ω}(n)| ≤ Cn^{−(2α/3)−ϵ} (iv) V_{ω}(n) is independent of {V_{ω}(j)}_{j=1}^{n−1}.

Remarks. 1. Think of the case discussed in [26, 7] where V_{ω}(n) = n^{−α}X_{n}(ω) with X_{n} identically distributed bounded, independent random variables. If E(X) = 0 and X is bounded, then (i)–(iv) hold.

2. With some extra effort, we could allow unbounded distributions, and only require that lim_{n→∞} n^{+α}E(V_{ω}(n)^{2})^{1/2} exists and be non-zero.

Theorem 8.2. Suppose (i)–(iv) hold. Fix k in (0, π) with k ≠ \frac{π}{4} , \frac{2π}{4} \frac{3π}{4} . Then for a.e. ω,

lim_{n→∞} \frac{log ∥T_{2 cos(k)}(n, 0)∥}{(∑_{j=1}^{n} j^{−2α})} = \frac{λ^{2}}{8 sin^{2}(k)} .

Remark. In case α < \frac{1}{2}, this says ∥T (n, 0)∥ ∼ exp(Cn^{1−α}) with C = \frac{λ^{2}}{8(1−2α) sin^{2}(k)}. If

α = \frac{1}{2}, this says ∥T ∥ ∼ n^{C} with C = \frac{λ^{2}}{8 sin^{2}(k)}.

Proof. By Theorem 2.3, we need only prove this result with R(n) replacing T for each θ_{0}. So fix k and θ_{0}, and let θ_{ω}(n), R_{ω}(n) solve (2.12). By (2.12c),

log R(n + 1) − log R(n) = \frac{1}{2} log(1 + ν_{k}(n) sin(2θ̄(n)) + ν_{k}(n)^{2} sin^{2}(θ̄(n))).  (8.2)

Since sup_{ω} ν_{k}(n) → 0 as n → ∞, we can use

log(1 + x) = x − \frac{x^{2}}{2} + O(x^{3}).  (8.3)

We also use

sin^{2} θ − \frac{1}{2} sin^{2}(2θ) = \frac{1}{4} − \frac{1}{2} cos(2θ) + \frac{1}{4} cos(4θ).

The net result is

log R(n) = \frac{1}{8} ∑_{j=1}^{n} \frac{E(V_{ω}(n)^{2})}{sin^{2}(k)} + C_{1} + C_{2} + C_{3} + C_{4}

where the corrections have the form

C_{1} = −\frac{1}{2 sin(k)} ∑_{j=1}^{n} V_{ω}(j) sin(2θ̄_{ω}(j))

C_{2} = \frac{1}{2 sin^{2}(k)} ∑_{j=1}^{n}[V_{ω}(j)^{2} − E(V_{ω}(j)^{2})][sin^{2}(θ̄_{ω}(j)) − \frac{1}{2} sin^{2}(2θ̄_{ω}(j))]

C_{3} = \frac{1}{2 sin^{2}(k)} ∑_{j=1}^{n} E(V_{ω}(j)^{2})[\frac{1}{2} cos(2θ̄_{ω}(j)) − \frac{1}{4} cos(4θ̄_{ω}(j))]

C_{4} = ∑_{j=1}^{n} O(V_{ω}(j)^{3} + V_{ω}(j)^{4}).

The theorem follows if we prove that for each q = 1, 2, 3, 4 and a.e. ω,

\overline{lim}_{n→∞} \frac{|C_{q}(ω)|}{∑_{j=1}^{n} j^{−2α}} = 0.  (8.4)

(8.4) for q = 4 is an immediate consequence of hypothesis (iii).

C_{1}, C_{2} clearly have zero expectation values and variances that decay properly for us to hope (8.4) holds; the key to the proof will be a Martingale inequality. C_{3} will depend on the fact that cos(θ) has zero average and the slow variation of E(V_{ω}(n)^{2}).

We break the proof to present some needed lemmas. For the first two of these lemmas, let X_{0}, X_{1}, …, X_{N} be independent random variables where X_{0} can be vector valued. Suppose that for j = 1, …, N,

Z_{j} = X_{j}f_{j}(X_{1}, …, X_{j−1}; X_{0})  (8.5)

with f_{j} a measurable function, and that

E(X_{j}) = 0.  (8.6)

The following is a variant of a standard Martingale inequality; we provide a proof for the reader’s convenience:

Lemma 8.3.

E( sup_{n=1,2,...,N} |Z_{1} + ⋯ + Z_{n}| ≥ r) ≤ \frac{1}{r^{2}} E( ∑_{j=1}^{N} Z_{j}^{2})  (8.7)

Proof. Define

Y_{n} = ∑_{j=1}^{n} Z_{j}, Q_{n} = ∑_{j=n+1}^{N} Z_{j}

MODIFIED PRÜFER AND EFGP TRANSFORMS 37

and let

A_{j} = {ω | |Y_{1}| ≤ r, |Y_{2}| ≤ r, …, |Y_{j}| > r}.

Then χ_{n}, the characteristic function of A_{n}, is a function only of X_{0}, X_{1}, …, X_{n} and thus, if k > n,

E(Z_{k}Y_{n}χ_{A_{n}}) = E(X_{k})E(f_{k}(X_{1}, …, X_{k−1}, X_{0})Y_{n}χ_{A_{n}}) = 0.

Thus,

E(χ_{n}Y_{n}^{2}) ≤ E(χ_{n}(Y_{n} + Q_{n})^{2})

since the cross term has zero expectation when we expand the square. Thus,

r^{2} ∑_{j=1}^{n} E(χ_{j}) ≤ ∑_{j=1}^{N} E(χ_{j}Y_{j}^{2}) ≤ ∑_{j=1}^{N} E(χ_{j}Y_{N}^{2}) ≤ E(Y_{N}^{2})

which is (8.7). □ Lemma 8.4. Suppose E(Z_{n}^{2}) ≤ Cn^{−2α}. Then for a.e. ω:

(1) If α < \frac{1}{2} and β > \frac{1}{2}(1 − 2α), then

lim_{n→∞} |∑_{j=1}^{n} Z_{j}| n^{−β} = 0.

(2) If α = \frac{1}{2} and β > \frac{1}{2}, then

lim_{n→∞} |∑_{j=1}^{n} Z_{j}|(log n)^{−β} = 0.

(3) If α > \frac{1}{2},

lim_{n→∞} ∑_{j=1}^{n} Z_{j} = Y_{∞}

exists, and for any β < α − \frac{1}{2},

lim_{n→∞} n^{+β} |∑_{j=n}^{∞} Z_{j}| = 0.

Remark. Naively, fluctuations should behave as (∑_{j=1}^{n} j^{−2α})^{1/2}. This lemma shows they are not much worse. Since we only need that they are small compared to ∑_{j=1}^{n} j^{−2α}, the lemma suffices.

Proof. (1) Pick β_{1} so β > β_{1} ≥ \frac{1}{2}(1 − 2α). By Lemma 8.3,

\begin{aligned}E( sup_{j=1,...,2^{n−1}} | ∑_{k=2^{n−1}+1}^{2^{n−1}+j} Z_{k}| ≥ 2^{nβ_{1}}) & ≤ C 2^{−2nβ_{1}}2^{(n−1)}2^{−2(n−1)α} \\ & ≤ C 2^{−(n−1)(2α+2β_{1}−1)} \end{aligned}  (8.8)

is summable in n by the choice of β_{1}. Therefore, by the Borel-Cantelli lemma, for a.e. ω, there is an n_{0}(ω_{0}) so that the sup inside (8.8) is less than 2^{nβ_{1}} if n ≥ n_{0}. Let j be larger than 2^{n_{0}−1} and pick n so that 2^{n−1} + 1 ≤ j ≤ 2^{n}. Then

|Z_{1} + ⋯ + Z_{j}| ≤ |Z_{1} + ⋯ + Z_{2^{n_{0}}} | + ∑_{k=1}^{n} 2^{kβ_{1}}

≤ |Z_{1} + ⋯ + Z_{2^{n_{0}}} | + \frac{2^{nβ_{1}}}{2^{β_{1}} − 1}

≤ |Z_{1} + ⋯ + Z_{2^{n_{0}}} | + \frac{2^{β_{1}}}{2^{β_{1}} − 1} j^{β_{1}}.

Thus, \overline{lim} j^{−β_{1}}|Z_{1} + ⋯ + Z_{j}| < ∞. Since β > β_{1}, the limit for β is 0.

(2) Pick β_{1} with β > β_{1} > \frac{1}{2} and define

K_{n} = {ω | sup_{j=1,...,2^{n}} | ∑_{m=1}^{j} Z_{j}| ≥ n^{β_{1}}}.

Then by Lemma 8.3,

E(K_{n}) ≤ n^{−2β_{1}} ∑_{1}^{2^{n}} \frac{1}{j} ≤ n^{−2β_{1}}(1 + n log 2) ≤ Cn^{1−2β_{1}}

since ∑_{1}^{k} \frac{1}{j} ≤ 1 + log k.

Pick an integer m so m(2β_{1} − 1) > 1. Then

∑_{n=1}^{∞} E(K_{n^{m}}) < ∞.

So by the Borel-Cantelli lemma, for a.e. ω, there is n_{0}(ω) so if n ≥ n_{0}, then ω ∉ K_{n^{m}}. If j > 2^{n_{0}^{m}}, pick n so that

2^{(n−1)^{m}} < j ≤ 2^{n^{m}}.

Then

|Z_{1} + ⋯ + Z_{j}| ≤ (n^{m})^{β_{1}} ≤ 2^{mβ_{1}}(n − 1)^{mβ_{1}} ≤ 2^{mβ_{1}}(log 2)^{−β_{1}}(log j)^{β_{1}}.

(3) Pick β_{1} so β < β_{1} < α − \frac{1}{2}. Then

E( sup_{j=1,...,2^{n−1}} | ∑_{k=2^{n−1}+1}^{2^{n−1}+j} Z_{k}| ≥ 2^{−nβ_{1}}) ≤ C 2^{−2nβ_{1}}2^{n−1}2^{−2(n−1)α}
≤ C 2^{2β_{1}}2^{−2(n−1)[α−1/2−β_{1}]}

is summable. Thus, for a.e. ω, there is an n_{0}(ω) so that for n ≥ n_{0}(ω), the sup is bounded by 2^{−nβ_{1}}. Thus, if j_{1} ≥ j_{2} ≥ 2^{n_{2}−1} ≥ 2^{n_{0}−1},

| ∑_{k=j_{1}}^{j_{2}} Z_{k}| ≤ ∑_{n=n_{2}}^{∞} 2^{−nβ_{1}} → 0

as n_{2} → ∞. So the sum is convergent (i.e., the partial sums are Cauchy). Moreover, if j ≥ 2^{n_{0}−1} and n is picked so 2^{n−1} ≤ j ≤ 2^{n}, then

|∑_{k=j}^{∞} Z_{k}| ≤ ∑_{m=n}^{∞} 2^{−mβ_{1}} = \frac{2^{−nβ_{1}}}{1 − 2^{−β_{1}}} ≤ \frac{j^{−β_{1}}}{1 − 2^{−β_{1}}}

and thus, if we multiply by j^{β}, the limit is 0. □

Lemma 8.5. Suppose that k ∈ ℝ is not in ℤπ. Then there exist integers q_{ℓ} → ∞ so that for any θ_{0}, …, θ_{q_{ℓ}},

|∑_{j=1}^{q_{ℓ}} cos(θ_{j})| ≤ 1 + ∑_{j=1}^{q_{ℓ}} |θ_{j} − θ_{0} − kj|.

Remark. In essence, we show | ∑_{j=1}^{q} cos(θ_{0} + kj)| ≤ 1 a stronger result than the ergodic theory result that |\frac{1}{q} ∑_{j=1}^{q} cos(θ_{0} + kj)| → 0. The weaker ergodic theory result suffices for our application, but the proof of this lemma is easy so we give it.

Proof. By general number theory considerations [14], we can find p_{ℓ}, q_{ℓ} so that

|k − \frac{πp_{ℓ}}{q_{ℓ}} | ≤ \frac{1}{q_{ℓ}^{2}}  (8.8)

and p_{ℓ}/q_{ℓ} ∉ ℤ if k ∉ ℤπ. For any p/q ∉ ℤ and any θ_{0},

∑_{j=1}^{q} cos(θ_{0} + \frac{jpπ}{q} ) = 0.  (8.9)

Thus

|∑_{j=1}^{q_{ℓ}} cos(θ_{j})| = |∑_{j=1}^{q_{ℓ}} cos(θ_{j}) − cos(θ_{0} + \frac{jp_{ℓ}π}{q_{ℓ}} )| ≤ ∑_{j=1}^{q_{ℓ}}|(θ_{j} − θ_{0} − \frac{jp_{ℓ}π}{q_{ℓ}} )|

≤ ∑_{j=1}^{q_{ℓ}} |θ_{j} − θ_{0} − kj| + ∑_{j=1}^{q_{ℓ}} j |k − \frac{πp_{ℓ}}{q_{ℓ}} |

≤ \frac{q_{ℓ}(q_{ℓ} + 1)}{2q_{ℓ}^{2}} + ∑_{j=1}^{q_{ℓ}} |θ_{j} − θ_{0} − kj|. □

Conclusion of the Proof of Theorem 8.2. We need to verify (8.4) for q = 1, 2, 3. V_{ω}(n) sin(2θ̄_{ω}(n)) ≡ Z_{n} has the form (8.5) and E(Z_{n}^{2}) ≤ Cn^{−2α}, so by Lemma 8.4, for a.e. ω,

|C_{1}(ω)| = o( ∑_{j=1}^{n} j^{−2α}).

[V_{ω}(n)^{2}−E(V_{ω}(n)^{2})][sin^{2}(θ̄)− \frac{1}{2} sin^{2}(2θ)] also has the form of (8.4) since E(V_{ω}^{2}−E(V_{ω}^{2})) = 0. Since V is bounded,

E((V^{2} − E(V^{2}))^{2}) ≤ CE(V^{2}).

Thus, for a.e. ω,

|C_{2}(ω)| = o( ∑_{j=1}^{n} j^{−2α})

also.

Finally, we will show

∑_{j=1}^{n} j^{−2α} cos(4θ̄_{ω}(j)) = o( ∑_{j=1}^{n} j^{−2α}),

which proves (8.4) for q = 3. By hypothesis on k, 4k ∉ ℤπ so Lemma 8.5 applies. Let q_{ℓ} be as in that lemma. Note next that by hypothesis (iii) and Proposition 2.4 for j large,

|θ_{ω}(j + 1) − θ_{ω}(j) − k| ≤ C_{0}j^{−2α/3}.  (8.10)

Pick n_{0} so

n_{0} ≥ q_{ℓ}^{2}  (8.11)

and

4C_{0}n_{0}^{−2α/3} ≤ q_{ℓ}^{−2}.  (8.12)

Suppose N = n_{0} + Kq_{ℓ}. Then

| ∑_{j=n_{0}+1}^{N} j^{−2α} cos(4θ_{ω}(j))| = | ∑_{m=0}^{K} ∑_{j=1}^{q_{ℓ}}(n_{0} + mq_{ℓ} + j)^{−2α} cos(4θ_{ω}(mq_{ℓ} + j))| = A_{1} + A_{2}

where A_{1} is what we get by replacing (n_{0} + kq_{ℓ} + j)^{−2α} by (n_{0} + kq_{ℓ})^{−2α} and A_{2} is the difference. By Lemma 8.5, (8.10), and (8.12),

A_{1} ≤ ∑_{k=0}^{K}(n_{0} + kq_{ℓ})^{−2α}[1 + 1]

while using

|(n_{0} + kq_{ℓ} + j)^{−2α} − (n_{0} + kq_{ℓ})^{−2α}| ≤ (n_{0} + kq_{ℓ})^{−2α}jn_{0}^{−1}

and (8.11),

A_{2} ≤ ∑_{k=0}^{K}(n_{0} + kq_{ℓ})^{−2α}q_{ℓ}^{2}n_{0}^{−1} ≤ ∑_{k=0}^{K}(n_{0} + kq_{ℓ})^{−2α}.

Thus for any N,

|∑_{1}^{N} j^{−2α} cos(4θ_{ω}(j))| ≤ C_{ℓ} + 3q_{ℓ}^{−α} ∑_{j=1}^{N} j^{−2α}

and so

\overline{lim}_{N→∞}( ∑_{j=1}^{N} j^{−2α})^{−1}|∑_{1}^{N} j^{−2α} cos(4θ_{ω}(j))| ≤ 3q_{ℓ}^{−α}

uniformly in ω. Since we can take q_{ℓ} → ∞ by Lemma 8.5, the \overline{lim} is 0. □

Theorem 8.6. Suppose that (i)–(iv) hold with α < \frac{1}{2} but we consider V (1) as a continuous parameter. Then for a.e. ω:

(1) For a dense G_{δ} of values of V (1), H_{ω} has purely singular continuous spectrum in (−2, 2).

(2) For Lebesgue a.e. value of V (1), H_{ω} has dense pure point spectrum in (−2, 2) and the eigenfunctions obey H_{ω}u = 2 cos(k_{m})u with

lim_{n→∞} \frac{log(|u(n)^{2} + u(n + 1)^{2}|^{1/2})}{|n|^{1−2α}} = −\frac{(1 − 2α)λ^{2}}{8 sin^{2}(k_{m})} .  (8..13)

If we consider a whole-line problem with independent V_{ω}(n) where both {V_{ω}(n)}_{n=1}^{∞} and Ṽ_{ω}(n) ≡ V_{ω}(−n), n = 1, 2, … obey hypotheses (i)–(iv) and V_{ω}(0) has a purely a.c. density, then for a.e. ω, H_{ω} has dense pure point spectrum in (−2, 2) and (8.13) holds as |n| → ∞.

Remark. This strengthens the result originally proven in [31] and improved in [7] in two ways. First, we get the explicit constant in (8.13). Second, we only require one V_{ω}( ·) to have an a.e. distribution.

Proof. By Theorem 8.2 and Fubini’s theorem for a.e. ω, we have for a.e. k ∈ (0, π),

lim_{n→∞} \frac{log ∥T (n)∥}{n^{1−2α}} = \frac{(1 − 2α)λ^{2}}{8 sin^{2}(k)} .

Thus by Theorem 8.3 of [22], there is an L^{2}-solution obeying (8.13). The theorem follows from general principles on rank one perturbations [12, 4, 5, 28]. □

The case α = \frac{1}{2} has an extra subtlety we will need to deal with, using an argument modeled on Kotani-Ushiroya [21]. The following replaces an explicit but complex formula they use for the projection onto a decaying solution (and fills in a gap in their argument):

Lemma 8.7. Let u_{θ} = (cos θ, sin θ) in ℝ^{2}. For any unimodular matrix A with ∥A∥ > 1, let θ(A) be the unique θ ∈ (−\frac{π}{2} , \frac{π}{2} ] with ∥Au_{θ}∥ = ∥A∥^{−1}. Define ρ(A) = ∥Au_{0}∥/∥Au_{π/2}∥. Let A_{n} be a sequence of unimodular matrices with ∥A_{n}∥ → ∞ and ∥A_{n+1}A_{n}^{−1}∥/∥A_{n}∥∥A_{n+1}∥ → 0 as n → ∞. Let ρ_{n} = ρ(A_{n}), θ_{n} = θ(A_{n}). Then:

(i) θ_{n} has a limit θ_{∞} if and only if lim_{n→∞} ρ_{n} ≡ ρ_{∞} exists (ρ_{∞} = ∞ is allowed, but then we only have |θ_{n}| → \frac{π}{2} ).

(ii) Suppose θ_{n} has a limit θ_{∞} ≠ 0, \frac{π}{2} (equivalently, ρ_{∞} ≠ 0, ∞). Then

lim_{n→∞} \frac{log ∥A_{n}u_{∞}∥}{log ∥A_{n}∥} = −1  (8.14)

if and only if

\overline{lim}_{n→∞} \frac{log |ρ_{n} − ρ_{∞}|}{log ∥A_{n}∥} ≤ −2.  (8.15)

Remark. Consider

A_{n} = ( \begin{matrix}cosh(n) & (−1)^{n} sinh(n) \\ (−1)^{n} sinh(n) & cosh(n) \end{matrix} ) .

Then ρ(A_{n}) ≡ 1 and ∥A_{n}∥ → ∞ but θ_{n} = (−1)^{n+1}(\frac{π}{4} ) does not have a limit. This shows that the condition ∥A_{n+1}A_{n}^{−1}∥/∥A_{n}∥∥A_{n+1}∥ → 0 is required. Indeed, in this case that limit is 1. Kotani-Ushiroya miss this issue.

Proof. (i) Note first that

∥A_{n}u_{θ}∥^{2} = ∥A_{n}∥^{2} sin^{2}(θ − θ_{n}) + ∥A_{n}∥^{−2} cos^{2}(θ − θ_{n}).  (8.16)

Thus,

ρ_{n} = \frac{tan^{2}(θ_{n}) + ∥A_{n}∥^{−4}}{1 + ∥A_{n}∥^{−4} tan^{2}(θ_{n})} .  (8.17)

It follows that ρ_{n} has a finite limit ρ_{∞} if tan^{2}(θ_{n}) has a finite limit. By writing

ρ_{n}^{−1} = \frac{cot^{2}(θ_{n}) + ∥A_{n}∥^{−4}}{1 + ∥A_{n}∥^{−4} cot^{2}(θ_{n})} ,

this is true also for ρ_{n} → ∞ and tan^{2}(θ_{n}) → ∞.

Pick η ∈ [0, \frac{π}{2} ] so tan^{2}(θ_{n}) → tan^{2}(η). If η = 0, then θ_{n} → 0, and if η → \frac{π}{2} , then |θ_{n}| → \frac{π}{2} because of the continuity of tan(θ) on [−\frac{π}{2} , \frac{π}{2} ]. If 0 < η < \frac{π}{2} , we only have |θ_{n}| → η and have to worry about the sign (see the remark above).

In (8.16), take θ = θ_{n+1} and see that

sin^{2}(θ_{n+1} − θ_{n}) ≤ ∥A_{n}∥^{−2}∥A_{n}A_{n+1}^{−1}∥^{2}∥A_{n+1}u_{θ_{n+1}}∥^{2} = ∥A_{n}∥^{−2}∥A_{n+1}∥^{−2}∥A_{n+1}A_{n}^{−1}∥^{2}

since A_{n}A_{n+1}^{−1} is unimodular, and thus ∥A_{n}A_{n+1}^{−1}∥ = ∥A_{n+1}A_{n}^{−1}∥. Thus by hypothesis,

sin^{2}(θ_{n+1} − θ_{n}) → 0.

This, together with |θ_{n}| → η ∈ (0, \frac{π}{2} ), implies that θ_{n} has a limit.

(ii) By (8.16), we have that (8.14) holds if and only if

lim_{n→∞} \frac{log |θ_{n} − θ_{∞}|}{log ∥A_{n}∥} ≤ −2.  (8.18)

Since θ_{∞} ≠ 0, π, this is true if and only if

lim_{n→∞} \frac{log | tan^{2}(θ_{n}) − tan^{2}(θ_{∞})|}{log ∥A_{n}∥} ≤ −2.

By (8.17) and θ_{∞} ≠ π,

|| tan^{2}(θ_{n}) − tan^{2}(θ_{∞})| − |ρ_{n} − ρ_{∞}|| ≤ C∥A_{n}∥^{−4}.

Thus, (8.18) holds if and only if (8.15) holds. □

Lemma 8.8. Suppose the hypotheses of Theorem 8.2 hold with α = \frac{1}{2} and k ≠ \frac{π}{4} , \frac{2π}{4} , \frac{3π}{4} is fixed. Then for a.e. ω, there exists an initial condition u_{θ(ω)} so that

lim_{n→∞} \frac{log ∥T_{2 cos(k)}(n, 0)u_{θ(ω)}∥}{log(n)} = −\frac{λ^{2}}{8 sin^{2}(k)} .

Remark. As noted in [22] (and gotten incorrectly in [21]), Ruelle’s deterministic argument doesn’t ever suffice in this ∥T ∥ ∼ n^{γ} case. If A_{n} is a sequence of unimodular matrices with lim_{n→∞} log ∥A_{n}∥/ log(n) = γ, then [22] has explicit examples (even coming from deterministic Schrödinger operators) for each γ > \frac{1}{2} where the decaying solution only obeys lim_{n→∞} log ∥A_{n}u_{∞}∥/ log(n) = −γ + 1. It also appears one needs γ > \frac{3}{2} to be sure of the existence of decaying solutions. But following [21], the probabilistic argument here can replace Ruelle’s argument.

Proof. Let β = \frac{λ^{2}}{8 sin^{2}(k)}. Let R_{1}(n) and R_{2}(n) be the R’s associated to θ = 0 and θ = \frac{1}{2}. By the proof of Theorem 8.2 for a.e. ω,

lim_{n→∞} \frac{log ∥R_{i}(n)∥}{log(n)} = β.  (8.19)

Let θ_{i}(n) be the corresponding EFGP angles. By (2.7),

R_{1}(n)R_{2}(n) sin(θ_{1}(n) − θ_{2}(n)) = sin(k)[u_{1}(n)u_{2}(n − 1) − u_{1}(n − 1)u_{2}(n)] = −1

(by the initial conditions R_{1}(1) = R_{2}(1) = 0, θ_{1}(1) = 0, θ_{2}(1) = \frac{π}{2} ) and constancy of the Wronskian. Thus by (8.19) for a.e. ω,

lim_{n→∞} \frac{log |θ_{1}(n) − θ_{2}(n)|}{log(n)} = −2β.  (8.20)

Let ρ_{n} = \frac{R_{1}(n)}{R_{2}(n)}. Then by (2.12c),

L_{ω}(n) ≡ [log ρ(n + 1) − log ρ(n)] = log(1 + A_{1}(n)) − log(1 + A_{2}(n)),

where

A_{i}(n) = −\frac{V_{ω}(n)}{sin(k)} sin(2θ_{i,ω}(n)) + \frac{V_{ω}(n)^{2}}{sin^{2}(k)} sin^{2}(θ_{i,ω}(n)).

Define

F (a, θ) = log(1 − a sin(2θ) + a^{2} sin^{2}(θ)).

By a finite Taylor expansion,

F (a, θ) = ∑_{j=1}^{J−1} a^{j}P_{j}(θ) + O(a^{J})

with P_{1}(θ) = sin(2θ) and the P ’s, C^{∞} in θ. Fix ϵ > 0 so for n large, use (8.20) to see that |θ_{1} − θ_{2}| = o(n^{−2β+ϵ}). Choosing J so n^{−J/3} = o(n^{−2β−1}), we see that

L_{ω}(n) = −\frac{V_{ω}(n)}{sin(k)} [sin(2θ_{1}(n)) − sin(2θ_{2}(n))] + O(n^{−2β−1+ϵ}).

Since θ_{j}(n) depend only on {V_{ω}(k)}_{k≤n−1}, we can apply part (3) of Lemma 8.4 (with 2α = 1 + 2β − ϵ) to see that for a.e. ω,

lim_{N→∞} ∑_{1}^{N} L_{ω}(n)  (8.21)

exists and

|∑_{N}^{∞} L_{ω}(n)| ≤ C_{ω}N^{−2β+ϵ}.  (8.22)

By (8.21), lim \frac{R_{1}(n)}{R_{2}(n)} ≡ ρ_{∞} exists and is different from 0 and ∞. Moreover, by (8.22),

\overline{lim} \frac{log |ρ(n) − ρ(∞)|}{log(n)} ≤ −2β.

Lemma 8.7 completes the proof. □

Theorem 8.9. Suppose (i)–(iv) hold with α = \frac{1}{2}. Then,

(1) For a.e. ω, the essential spectrum of H_{ω} is [−2, 2] and the absolutely continuous spectrum of H_{ω} is empty.

(2) If |λ| ≥ 2 and V_{ω}(1) has an absolutely continuous distribution, then for a.e. ω, H_{ω} has dense point spectrum and only dense point spectrum in (−2, 2).

(3) If |λ| < 2 and V_{ω}(1) has an absolutely continuous distribution, then for a.e. ω, H_{ω} has purely singular continuous spectrum in {E | |E| < (4 − λ^{2})^{1/2}} and only dense pure point spectrum in {E | (4 − λ^{2})^{1/2} < |E| < 2}.

In either case (2) or (3), in the region of point spectrum, there are almost surely eigenvectors of power decay n^{−β} with

β = \frac{λ^{2}}{8 − 2E^{2}} .  (8.23)

Remark. This theorem extends results of Delyon, et al. [7], Delyon [6], and Kotani-Ushiroya [21]. In particular, [7] conjectured that there is a region of point spectrum near E = ±2 no matter how small λ is.

Proof. By Theorem 8.2, lim_{n→∞} ∥T_{ω}(0, n)∥ = ∞ for a.e. E for a.e ω, so by Theorem 1.1, we conclude (3). By Lemma 8.8, for a.e. pairs (ω, E), there is a unique decaying solution with rate of decay n^{−β} with β = \frac{λ^{2}}{8−2E^{2}} . If β > \frac{1}{2}, this is ℓ^{2} and we have potential point spectrum. If β < \frac{1}{2}, there is no ℓ^{2} solution. The general theory of rank one perturbations ([32, 5]) then yields (2) and (3). □

We can compute the precise Hausdorff dimension of the singular continuous spectral measures in this case:

Theorem 8.10. Fix λ < 2 and a model obeying (i)–(iv) with α = \frac{1}{2}. In the region |E| ≤ (4 − λ^{2})^{1/2}, define

d(E, λ) = \frac{4 − E^{2} − λ^{2}}{4 − E^{2}} .

Suppose V_{ω}(1) has an absolutely continuous density. Then for a.e. ω, the spectral measure, μ, has dimension d(E, λ) at E in the sense that for any ϵ, there is a δ so that μ(A) = 0 if A is a subset of (E − δ, E + δ) of Hausdorff dimenion less than (d − ϵ), and there is a subset B of Hausdorff dimension less than (d + ϵ), so μ((E − δ, E + δ)\B) = 0.

Proof. Let ∥u∥_{L} = (∑_{j=1}^{L} u(j)^{2})^{1/2}. By the general theory of rank one perturbations, Theorem 8.2, Lemma 8.8, and the assumption of V_{ω}(1) for a.e. ω, μ is supported on the set of energies where most solutions grow as n^{β} and one decays as n^{−β} where β(E, λ) is given by (8.23). The hypothesis for singular spectrum is precisely β < \frac{1}{2}.

Since β < \frac{1}{2}, ∥u_{1}∥_{L} ∼ L^{−β}L^{1/2} while ∥u_{2}∥_{L} ∼ L^{β}L^{1/2} where a ∼ b is shorthand for lim \frac{log(a)}{log(b)} = 1. The Jitomirskaya-Last version [16, 17] of the Gilbert-Pearson [11] theory says that the Borel transform of the spectral measure is supported on the set of E’s where

|m(E + iϵ)| ∼ \frac{∥u_{2}∥_{L}}{∥u_{1}∥_{L}}  (8.26)

and E is given by

∥u_{1}∥_{L}∥u_{2}∥_{L} = \frac{1}{2ϵ}  (8.27)

(the ∼ in (8.26) holds in the strong sense that the ratio lies in the interval (5 − \sqrt{24} , 5 + \sqrt{24} )). Thus, ϵ ∼ L^{−1} and (8.26) says that

|m(E + iϵ)| ∼ ϵ^{−2β}.

Since β is continuous, the theory in [3] then says that the local dimension is given by 1 − 2β as claimed. □

§9. Random Decaying Potentials: The Continuum Case

Having done the discrete random case, we will only sketch the continuum case. We will specialize to a situation where {V (x)}_{n≤x<n+1} are independent for different n’s. Using ideas from [21], one can presumably use Martingale methods to control asymptotically independent situations.

Theorem 9.1. Let {V_{ω}(x)}_{0≤x<∞} be a family of random variables and let

a_{n}(ω) = ∫_{n}^{n+1} |V_{ω}(y)| dy.  (9.1)

Suppose

(i) E(V_{ω}(x)) = 0 for each x (ii) ∑_{n} E(a_{n}^{2}e^{Ca_{n}}) < ∞ for all C > 0 (iii) {V (x)}_{n≤x<n+1} are independent for different n’s.

Then for a.e. ω, −\frac{d^{2}}{dx^{2}} + V_{ω}(x) on L^{2}(0, ∞) has purely absolutely continuous spectrum on (0, ∞) for any boundary condition.

Remarks. 1. Our methods imply for all E > 0 and a.e. ω, T_{E}(n) is bounded, and that implies −\frac{d^{2}}{dx^{2}} +V_{ω}(x) is limit point at infinity, so we need not worry about self-adjointness issues.

2. A simple example where (ii) holds is if sup_{ω,x} |V_{ω}(x)| < ∞ and E(∫_{0}^{∞} V (y)^{2} dy) < ∞.

Proof. By (2.4),

R^{4}(n + 1) = R^{4}(n) exp(B_{n}(ω)),  (9.2)

where

B_{n}(ω) = \frac{2}{k} ∫_{n}^{n+1} V_{ω}(x) sin(2θ_{ω}(x)) dx.  (9.3)

By (2.3),

|θ_{ω}(x) − θ_{ω}(n) − k(x − n)| ≤ \frac{2}{k} a_{n}(ω).  (9.4)

Using

|e^{x} − 1 − x| ≤ \frac{1}{2}x^{2}e^{x},

we obtain from (9.2)–(9.4),

R^{4}(n + 1) ≤ R^{4}(n)(1 + Ca_{n}^{2}e^{Ca_{n}}) + Q_{n}  (9.5a)
Q_{n} = \frac{2}{k} R^{4}(n) ∫_{n}^{n+1} V_{ω}(x) sin(2θ_{ω}(n) + 2k(x − n)) dx  (9.5b)

for some constant C uniformly bounded for k in any compact of (0, ∞).

Since V_{ω}(x) is independent of {V_{ω}(y)}_{y≤n}, it is independent of R(n) and θ_{ω}(n), and so E(Q_{n}) = 0.

Moreover, a_{n} is independent of R(n), so (9.5) implies that

E(R^{4}(n + 1)) ≤ E(R^{4}(n))E(1 + Ca_{n}^{2}e^{Ca_{n}}).

By condition (ii), we see that

\overline{lim}_{n→∞} E(R^{4}(n)) < ∞

with bounds uniform in k on compacts of (0, ∞). Thus by Fatou’s lemma, for a.e. ω, \underline{lim} ∫_{a}^{b} R_{ω}^{4}(n, k) dk < ∞ and so the spectrum is purely absolutely continuous by Theorem 1.3. □

Theorem 9.2. Let f be supported on (0, 1) and let

V_{ω}(x) = ∑_{n=0}^{∞}(n + 1)^{−α}X_{n}(ω)f(x − n),

where {X_{n}(ω)} are independent, identically distributed bounded variables of mean zero and 0 < α ≤ \frac{1}{2}. Then for 4k ∉ ℤπ,

lim_{n→∞} \frac{log ∥T (n, 0)∥}{∑_{j=1}^{n} j^{−α}} = \frac{E(X_{n}^{2})}{8k^{2}} |∫_{0}^{1} f(y)e^{iky} dy|^{2}.  (9.6)

Remarks. 1. This implies pure point spectrum for a.e. ω if α < \frac{1}{2}.

2. If α = \frac{1}{2}, we get singular continuous spectrum for large E and pure point spectrum for small E (assuming ∫_{0}^{1} f(y) dy ≠ 0 or ∫_{0}^{1} yf(y) dy ≠ 0) and no a.c. spectrum.

Sketch. Define θ_{n}(y) = θ(n) + ky and

δθ_{n}(y) = −(n + 1)^{−α}X_{n} ∫_{0}^{y} f(y) sin^{2}(θ_{n}(y)) dy.  (9.7)

By (2.3),

|θ_{n}(n + y) − θ_{n}(y) − δθ_{n}(y)| = O(n^{−2α})

for y ∈ (0, 1).

Plugging this into (2.4), we find that

log R(n + 1) − log R(n) = Y_{n}^{(1)} + Y_{n}^{(2)} + O(n^{−2α}),

where

Y_{n}^{(1)} = \frac{(n + 1)^{−α}X_{n}(ω)}{2k} ∫_{0}^{1} f(y) sin(2θ_{n}(y)) dy

and

Y_{n}^{(2)} = \frac{(n + 1)^{−α}X_{n}(ω)}{2k} ∫_{0}^{1} 2f(y) cos(2θ_{n}(y))(δθ_{n})(y) dy.

By using Lemmas 8.2 and 8.3, one sees that

( ∑_{j=0}^{n}(j + 1)^{−2α})|∑_{j=0}^{n} Y_{j}^{(1)}| → 0

for a.e. ω. The same lemmas let us replace X_{n}(ω)^{2} by E(X_{n}^{2}(ω)) in Y_{n}^{(2)}. So if we let ≗ indicate equal up to o(∑_{j=0}^{n}(j + 1)^{−2α}) terms, we see that

log R(n) ≗ ∑_{j=0}^{n−1}(Y_{n}^{(3)} + Y_{n}^{(4)}),

where we use sin^{2}(θ_{n}(y)) = \frac{1}{2} − \frac{1}{2} cos(2θ_{n}(y)) and let Y_{n}^{(3)} indicate the −\frac{1}{2} cos(2θ) terms and Y_{n}^{(4)} the \frac{1}{2} terms. By an argument analogous to the one in the proof of Theorem 9.2 that used Lemma 8.5, ∑ Y_{n}^{(4)} ≗ 0 because k ∉ ℤπ.

As in (6.5), we get

log R(n) ≗ ∑_{j=0}^{n−1} \frac{(j + 1)^{−2α}E(X_{n}(ω)^{2})}{4k^{2}}(∫_{0}^{1} f(y) cos(2θ_{j}(y)) dy)^{2}.

As in the proof of Lemma 6.2, this last square is

\frac{1}{2}|∫_{0}^{1} f(y)e^{iky}|^{2}

plus a term that has cos(4θ_{j}(y)), which we can handle using Lemma 8.5. □

References

1. R. Carmona, One-dimensional Schrödinger operators with random or deterministic potentials, New spectral types, J. Funct. Anal. 51 (1983), 229–258.

2. V. Chulaevsky and T. Spencer, Positive Lyapunov exponents for a class of deterministic potentials, Commun. Math. Phys. 168 (1995), 455–466.

3. R. del Rio, S. Jitomirskaya, Y. Last, and B. Simon, Operators with singular continuous spectrum, IV. Hausdorff dimensions, rank one perturbations, and localization, J. d’Analyse Math. 69 (1996), 153–200.

4. R. del Rio, N. Makarov, and B. Simon, Operators with singular continuous spectrum, II. Rank one operators, Commun. Math. Phys. 165 (1994), 59–67.

5. R. del Rio, B. Simon, and G. Stolz, Stability of spectral types for Sturm-Liouville operators, Math. Research Lett. 1 (1994), 437–450.

6. F. Delyon, Apparition of purely singular continuous spectrum in a class of random Schrödinger operators, J. Statist. Phys. 40 (1985), 621–630.

7. F. Delyon, B. Simon, and B. Souillard, From power pure point to continuous spectrum in disordered systems, Ann. Inst. H. Poincaré 42 (1985), 283–309.

8. M.S.P. Eastham and H. Kalf, Schrödinger-type Operators With Continuous Spectra, Pitman Books, London, 1982.

9. T. Eggarter, Some exact results on electron energy levels in certain one-dimensional random potentials, Phys. Rev. B5 (1972), 3863–3865.

10. D.J. Gilbert, On subordinacy and analysis of the spectrum of Schrödinger operators with two singular endpoints, Proc. Roy. Soc. Edin. 112A (1989), 213–229.

11. D.J. Gilbert and D.B. Pearson, On subordinacy and analysis of the spectrum of onedimensional Schrödinger operators, J. Math. Anal. 128 (1987), 30–56.

12. A.Ya. Gordon, Pure point spectrum under 1-parameter perturbations and instability of Anderson localization, Commun. Math. Phys. 164 (1994), 489–505.

13. S.A. Gredeskul and L.A. Pastur, Behavior of the density of states in one-dimensional disordered systems near the edges of the spectrum, Theor. Math. Phys. 23 (1975), 132–139.

14. G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford University Press, Oxford, 1979.

15. W.A. Harris and D.A. Lutz, Asymptotic integration of adiabatic oscillator, J. Math. Anal. Appl. 51 (1975), 76–93.

16. S. Jitomirskaya and Y. Last, Dimensional Hausdorff properties of singular continuous spectra, Phys. Rev. Lett. 76 (1996), 1765–1769.

17. S. Jitomirskaya and Y. Last, Power law subordinacy and singular spectra, I. Half-line operators, in preparation.

18. J.P. Kahane, Some Random Series of Functions, Cambridge University Press, Cambridge, 1985.

19. A.A. Kiselev, Absolutely continuous spectrum of one-dimensional Schrödinger operators and Jacobi matrices with slowly decreasing potentials, Commun. Math. Phys. 179 (1996), 377–400.

20. A. Kiselev, C. Remling, and B. Simon, Effective perturbation methods for one dimensional Schrödinger operators, in preparation.

21. S. Kotani and N. Ushiroya, One-dimensional Schrödinger operators with random decaying potentials, Commun. Math. Phys. 115 (1988), 247–266.

22. Y. Last and B. Simon, Eigenfunctions, transfer matrices, and absolutely continuous spectrum of one-dimensional Schrödinger operators, preprint.

23. S. Molchanov, One-dimensional Schrödinger operators with sparse potentials, preprint

24. S. Naboko, Dense point spectra of Schrödinger and Dirac operators, Theor. Math. Phys. 68 (1986), 18–28.

25. S. Naboko and S.I. Yakovlev, On the point spectrum of discrete Schrödinger operators, Func. Anal. Appl. 26 (1992), 85–88.

26. L. Pastur and A. Figotin, Spectra of Random and Almost-Periodic Operators, Springer, Berlin, 1992.

27. D.B. Pearson, Singular continuous measures in scattering theory, Commun. Math. Phys. 60 (1978), 13–36.

28. D.B. Pearson, Pathological spectral properties, Mathematical Problems in Theoretical Physics (Proc. Internat. Conf. Math. Phys., Lausanne, 1979), pp. 49–51, Lecture Notes in Phys., 116, Springer, Berlin-New York, 1980.

29. C. Remling, A probabilistic approach to one-dimensional Schrödinger operators with sparse potentials, to appear in Commun. Math. Phys.

30. D. Ruelle, Ergodic theory of differentiable dynamical systems, Publ. Math. IHES 50 (1979), 275–306.

31. B. Simon, Some Jacobi matrices with decaying potentials and dense point spectrum, Commun. Math. Phys. 87 (1982), 253–258.

32. B. Simon, Spectral analysis and rank one perturbations and applications, CRM Lecture Notes Vol. 8 (J. Feldman, R. Froese, L. Rosen, eds.), pp. 109–149, Amer. Math. Soc., Providence, RI, 1995.

33. B. Simon, Bounded eigenfunctions and absolutely continuous spectra for onedimensional Schrödinger operators, Proc. Amer. Math. Soc. 124 (1996), 3361–3369.

34. B. Simon, Some Schrödinger operators with dense point spectrum, to appear in Proc. Amer. Math. Soc.

35. B. Simon and G. Stolz, Operators with singular continuous spectrum, V. Sparse potentials, Proc. Amer. Math. Soc. 124 (1996), 2073–2080.

36. G. Stolz, Bounded solutions and absolute continuity of Sturn-Liouville operators, J. Math. Anal. Appl. 169 (1992), 210–228.

37. J. Weidmann, Zur Spektral theorie von Sturm-Liouville Operatoren, Math. Z. 98 (1967), 268–302.