WKB AND SPECTRAL ANALYSIS OF ONE-DIMENSIONAL SCHRÖDINGER OPERATORS WITH SLOWLY VARYING POTENTIALS

MICHAEL CHRIST AND ALEXANDER KISELEV

Abstract. Consider a Schrödinger operator on L^{2} of the line, or of a half line with appropriate boundary conditions. If the potential tends to zero and is a finite sum of terms, each of which has a derivative of some order in L^{1} + L^{p} for some exponent p < 2, then an essential support of the the absolutely continuous spectrum equals ℝ^{+}. Almost every generalized eigenfunction is bounded, and satisfies certain WKB-type asymptotics at infinity. If moreover these derivatives belong to L^{p} with respect to a weight |x|^{γ} with γ > 0, then the Hausdorff dimension of the singular component of the spectral measure is strictly less than one.

1. Introduction

The semiclassical WKB method, first proposed in [31, 15, 4], is one of the most widely used methods for approximating the wave function of a particle, and is a part of most textbooks on quantum mechanics. The principal requirement for its applicability is that the electric field potential V (x) vary slowly, allowing certain terms involving its derivatives to be neglected, provided that the difference E −V (x), where E is the energy, is bounded away from zero. The literature on mathematically rigorous applications of the WKB idea is enormous. As an example we refer to [5, 12] for the construction of modified wave operators for long-range decaying potentials; see [21] for many further references. A typical requirement, for example, is that the potential satisfy both |V (x)| ≤ C(1 + |x|)^{−1/2−ϵ} and |D^{α}V (x)| ≤ C(1 + |x|)^{−3/2−ϵ} for any |α| = 1; more slowly decaying potentials require stronger decay assumptions for derivatives of higher order.

A series of works were devoted to studying oscillating potentials of Wigner-Von Neumann type, like ax^{−δ} sin(bx^{β}), and their generalizations. See [2, 3, 32] for some of the results and further references.

In this paper, we extend the scope of the WKB method to analyze the asymptotic behavior at infinity of generalized eigenfunctions in one dimension, in the case where the potential may decay only very slowly or not at all, but where at least one derivative decays at a moderate rate, including (1 + |x|)^{−α} for 1/2 < α < 1. We establish WKB-type asymptotics for almost every energy (with respect to Lebesgue measure), even though there can occur an everywhere dense set of energies for which such asymptotics fail to hold; the decay rates hypothesized are very nearly optimal in this respect. This paper is one of a series of works [7, 8] treating such asymptotics with unstable parameter dependence.

Date: August 7, 2000. Revised October 16, 2000.

The first author was supported in part by NSF grant DMS-9970660 and performed part of this research while on appointment as a Miller Research Professor in the Miller Institute for Basic Research in Science.

The second author was supported in part by NSF grant DMS-9801530.

As a corollary, we deduce the existence of absolutely continuous spectrum, and an upper bound on the dimension of the singular continuous spectrum. Thus certain parts of the spectrum are stable under perturbations of the free Hamiltonian (V = 0) by suitably slowly varying potentials.

Denote by ℓ^{p}(L^{1})(ℝ) the Banach space of all (equivalence classes of) measurable functions from ℝ to ℝ for which the norm

∥f∥_{ℓ^{p}(L^{1})} = ( ∑_{k=−∞}^{∞} ( ∫_{k}^{k+1} |f(x)| dx)^{p})^{1/p}

is finite. This Banach space contains L^{1} + L^{p}. If p ≤ q, then ℓ^{p}(L^{1}) ⊂ ℓ^{q}(L^{1}).

Throughout the paper, we assume the following conditions (the hypotheses of Theorems 1.3 and 1.4 are more restrictive). Let n ≥ 0 be a nonnegative integer, and let p ∈ [1, 2) be an exponent. Let V be a measurable, real-valued function defined on the real line ℝ. Suppose that V admits a decomposition V = V_{0} +V_{n} where^{1} V_{0} ∈ ℓ^{p}(L^{1}), V_{n} is continuous and bounded, and d^{n}V_{n}/dx^{n} ∈ ℓ^{p}(L^{1}), in the sense of distributions.

Define esslimsup_{x→+∞} V (x) to be lim sup_{x→+∞} V_{n}(x). Define essliminf_{x→+∞} V (x), and the corresponding quantities with +∞ replaced by −∞, likewise. Let H = −d^{2}/dx^{2} + V (x). In the following statement, “almost every” means with respect to Lebesgue measure. Theorem 1.1. Under the above hypotheses, for almost every E > esslimsup_{x→+∞} V (x), each solution of the generalized eigenfunction equation Hu = Eu is a bounded function of x ∈ [0, +∞).

In addition, for almost every E in this same interval, WKB-type asymptotics hold in the sense that there exists a generalized eigenfunction u(x, E) satisfying u(x, E) · e^{−iΨ(x,E)} → 1 as x → +∞, where Ψ is a certain complex-valued function constructed from the potential by an explicit, though somewhat complicated, recipe dependent on the index n in the hypotheses. The exponent Ψ has relatively tame behavior, in contrast to u(x, E); ∂Ψ/∂x and all its partial derivatives with respect to x, E are uniformly bounded for x ∈ [0, ∞) and E in any compact subinterval of (esslimsup_{x→+∞} V (x), ∞). The complex conjugate of u is a second, linearly independent solution, with corresponding asymptotics.

Theorem 1.2. Under the above hypotheses, for the Schrödinger operator H = −\frac{d^{2}}{dx^{2}} + V (x) acting on L^{2}([0, ∞)), with some self-adjoint boundary condition at the origin, an essential support for the absolutely continuous spectrum of H is [esslimsup_{x→+∞} V (x), ∞). Moreover, the essential spectrum coincides with [essliminf_{x→+∞} V (x), +∞), and is purely singular in the interval from essliminf_{x→+∞} V (x) to esslimsup_{x→+∞} V (x).

1This includes any potential decomposable as ∑_{k=0}^{n} V_{k} where d^{k}V_{k}/dx^{k} ∈ ℓ^{p}(L^{1})(ℝ) for each k ≥ 0, and where ∑_{k=1}^{n} V_{k} is bounded.

Let H = −\frac{d^{2}}{dx^{2}} + V (x) acting on L^{2}(ℝ), and define A_{±} = esslimsup_{x→±∞} V (x), A = max(A_{+}, A_{−}), and a = min(A_{+}, A_{−}). Then an essential support for the absolutely continuous spectrum of H is [a, +∞]. The absolutely continuous spectrum has multiplicity two in [A, ∞), and has multiplicity one in [a, A]. The essential spectrum of H coincides with [essliminf_{|x|→∞} V (x), ∞), and is purely singular in the interval from essliminf_{|x|→∞} V (x) to a.

As is well known [28], the almost everywhere boundedness of the generalized eigenfunctions asserted in Theorem 1.1 implies the assertions of Theorem 1.2 concerning the absolutely continuous spectrum; see [10] for an alternative approach to this implication. The assertions concerning the absence of absolutely continuous spectrum are proved as was done for the case n = 1 in [8].

The first related result in one dimension was proved by Weidmann [30], who showed that if the potential is a sum of an L^{1} function and a function whose derivative is in L^{1}, then the spectrum on the positive semi-axis is purely absolutely continuous. Later, Behncke [1] and Stolz [27] established a result similar to our Theorems 1.1,1.2 for p = 1; the spectrum above esslimsup_{|x|→∞} V (x) is also purely absolutely continuous in that case. Even earlier, Ben-Artzi [2] proved a power-decaying version of the Behncke-Stolz result. The price to be paid for a more general class of potentials is that the WKB asymptotics can actually fail to hold for a dense set of energies having positive Hausdorff dimension [24], and moreover, the point spectrum can be dense in [0, ∞) [18, 25].

The conclusions of Theorems 1.1 and 1.2 are false for p > 2; Pearson [20] has exhibited potentials satisfying V (x) → 0 as |x| → ∞, for which every derivative of V belongs to L^{p} for every p > 2, yet the spectrum on the positive semi-axis is purely singular; see also [14]. Theorem 1.1 remains open for p = 2.

For 1 < p ≤ 2, the conclusion concerning the absolutely continuous spectrum had been independently conjectured by Molchanov, Novitskii and Vainberg [17], who proved by a different method that a support for the absolutely continuous spectrum coincides with (0, ∞), provided that d^{n}V/dx^{n} ∈ L^{2}, under the supplementary hypothesis that V ∈ L^{n+2} [17]. Certain partial results had also been obtained by Killip [13]. For n = 0, the main conclusion of Theorem 1.2 was obtained by Deift and Killip [11] under the hypothesis V ∈ L^{2} + L^{1}.

For potentials with more rapidly decaying derivatives, our conclusions can be strengthened. Define p′ = p/(p − 1).

Theorem 1.3. Suppose that n ≥ 0, 1 ≤ p ≤ 2, 0 < γ, and γp′ ≤ 1. Let V be a measurable, real-valued function defined on ℝ. Suppose that V = V_{0} + V_{n} where V_{n} is bounded and continuous, and both (1 + |x|)^{γ}V_{0} and (1 + |x|)^{γ}d^{n}V_{n}/dx^{n} belong to ℓ^{p}(L^{1}). Then every solution of Hu = Eu is a bounded function of x ∈ ℝ^{+}, for all E > esslimsup_{x→+∞} V (x), except for a set of values of E having Hausdorff dimension ≤ 1 − γp′.

Moreover, except for a set of energies having Hausdorff dimension ≤ 1 − γp′, there exists a generalized eigenfunction satisfying u(x, E) exp(−iΨ(x, E)) → 1 as x → +∞.

Theorem 1.4. Suppose that n ≥ 0, 1 ≤ p ≤ 2, 0 < γ, and γp′ ≤ 1. Let V be a measurable real-valued function defined on ℝ, satisfying the hypotheses of Theorem 1.3. Consider the Schrödinger operator −\frac{d^{2}}{dx^{2}} + V (x) acting on L^{2}([0, ∞)), with some self-adjoint boundary condition at the origin. Then the Hausdorff dimension of the singular component of its spectral measure in the interval (lim sup_{x→+∞} V (x), ∞) does not exceed 1 − γp′.

For a Schrödinger operator acting instead on L^{2}(ℝ), the singular component of its spectral measure has dimension ≤ 1−γp′ in the interval from min[lim sup_{x→+∞} V (x), lim sup_{x→−∞} V (x)] to +∞.

For n = 0, this was proved by Remling [23] under a power decay hypothesis V (x) = O(|x|^{−α}). Remling [24] also constructed examples for which WKB asymptotics fail to hold, for a set of energies of dimension equal to 2(1 − α), precisely consistent with the above bound 1 − γp′, but it remains unproven that singular continuous spectrum of positive dimension can actually occur for power decaying potentials. Theorem 1.4 is a consequence of Theorem 1.3; see for instance [10] for a general criterion which yields this implication.

Theorem 1.1 can be viewed as a nonlinear analogue of a basic property of the Fourier transform. Menshov, Paley, and Zygmund showed (in different versions) that if 1 ≤ p < 2 and V ∈ L^{p}(ℝ), then sup_{x} | ∫_{0}^{x} e^{−iλy}V (y) dy| is finite for almost every λ ∈ ℝ; the nonlinear analogue is the almost everywhere boundedness of the generalized eigenfunctions of the Schrödinger operator with potential V. Now if instead, V is a bounded function for which d^{n}V/dx^{n} ∈ L^{p}, then again the conclusion of Menshov, Paley, and Zygmund is valid, and is a direct consequence of the case n = 0 by the relation { \atop d^{n}V/dx^{n}}(λ) = i^{n}λ^{n}V̂(λ) (or integration by parts). Sums of finitely many functions V are equally easily handled. For the nonlinear analogue, we know of no such simple way to deduce the case n > 0 from n = 0, nor to conclude that V_{1} + V_{2} can be handled if V_{1}, V_{2} can be separately treated.

Likewise, Theorem 1.3 is related to the fact that if (1 + |x|)^{γ} · f ∈ ℓ^{p}(L^{1})(ℝ) for some p ∈ [1, 2], then lim_{y→∞} ∫_{0}^{y} e^{−ixλ}f(x) dx exists for all λ except for an exceptional set of Hausdorff dimension ≤ min(1−γp′, 0). This fact follows from a simpler version of our analysis.

In an earlier paper [8], we proved Theorems 1.1 and 1.2 for n = 1, developing a general method for treating certain kinds of multilinear expansions on which the present paper is also based. The principal new ingredient needed to make our multilinear machinery applicable for n > 1 is a sufficiently good WKB-type approximation exp[iΨ(x, E)] to the generalized eigenfunctions. A second novelty is that whereas for n = 1 we had been led [8] to multilinear operators mapping functions of x to functions of E, we now obtain operators mapping functions of (x, E) to functions of E; however, this second point had also arisen in [8], in the proof of a somewhat artificial supplementary theorem, in which V itself was allowed to depend on E. In the present paper we discuss the new points in detail, and merely outline the less novel remainder of the argument.

2. Simplifying a first-order system

Our first step is to write an ordinary differential equation −g′′ + Ug = 0 as a first-order system, and to record certain transformations that bring the system into a nearly diagonalized form. Let ϕ(x) be a complex-valued function to be determined later. Introduce the quantity

E(x) = iϕ′′ − (ϕ′)^{2} − U .

Write the equation for g as y′ = My where

(2.1)  y(x) = (\begin{matrix}g \\ g′ \end{matrix})  and  M = (\begin{matrix}0 & 1 \\ U & 0 \end{matrix}) .

Substitute y = Az where

A = (\begin{matrix}e^{iϕ} & e^{−iϕ̄} \\ iϕ′e^{iϕ} & −iϕ̄′e^{−iϕ̄} \end{matrix}) .

Then z′ = Bz where B = A^{−1}MA − A^{−1}A′. We have

A^{−1} = (ϕ′ + ϕ̄′)^{−1} (\begin{matrix}ϕ̄′e^{−iϕ} & −ie^{−iϕ} \\ ϕ′e^{iϕ̄} & ie^{iϕ̄} \end{matrix})

MA = (\begin{matrix}iϕ′e^{iϕ} & −iϕ̄′e^{−iϕ̄} \\ Ue^{iϕ} & Ue^{−iϕ̄} \end{matrix})

MA − A′ = (\begin{matrix}0 & 0 \\ −Ee^{iϕ} & −Ēe^{−iϕ̄} \end{matrix})

and therefore

B = (ϕ′ + ϕ̄′)^{−1} (\begin{matrix}iE & i\overline{E}e^{−i(ϕ+ϕ̄)} \\ −iEe^{i(ϕ+ϕ̄)} & −i\overline{E} \end{matrix}) .

Let ρ be another auxiliary function to be specified later, and substitute z = Λu where

Λ = (\begin{matrix}e^{ρ} & 0 \\ 0 & eρ̄ \end{matrix})

to obtain

(2.2)  u′ = Du

with D = Λ^{−1}BΛ − Λ^{−1}Λ′. Since

BΛ = (2 Re ϕ′)^{−1} · (\begin{matrix}iEe^{ρ} & iĒe^{−i(2 Re ϕ)+ρ̄} \\ −iEe^{i(2 Re ϕ)+ρ} & −iĒeρ̄ \end{matrix}) ,

we have

D = (2 Re ϕ′)^{−1} · (\begin{matrix}iE − (2 Re ϕ′)ρ′ & iĒexp(−2i Re ϕ − 2i Im ρ) \\ −iE exp(2i Re ϕ + 2i Im ρ) & −iĒ − (2 Re ϕ′)ρ̄′ \end{matrix}) .

Choosing

(2.3)  ρ(x) = i ∫_{0}^{x} \frac{E}{2 Re (ϕ′)}

eliminates the diagonal entries, yielding

D = (2 Re ϕ′)^{−1} (\begin{matrix}0 & iĒexp(−2i Re ϕ − 2i Im ρ) \\ −iE exp(2i Re ϕ + 2i Im ρ) & 0 \end{matrix}) .

More succinctly,

(2.4)  D = (\begin{matrix}0 & \frac{i\overline{E}}{2 Re ϕ′} e^{−ih} \\ \frac{−iE}{2 Re ϕ′} e^{ih} & 0 \end{matrix})

where

(2.5)  h(x) = 2 Re ϕ + ∫_{0}^{x} \frac{Re E}{Re ϕ′}

is purely real-valued.

Define

(2.6)  Ψ = ϕ − iρ = ϕ(x) + \frac{1}{2} ∫_{0}^{x} \frac{E}{Re ϕ′} .

Then

(2.7)  iΨ′ = \frac{−ϕ′′ − iU + i|ϕ′|^{2}}{2 Re ϕ′} .

Indeed,

(iϕ′ + ρ′) · 2 Re ϕ′ = iϕ′ · 2 Re ϕ′ + i[−(ϕ′)^{2} + iϕ′′ − U]
= iϕ′ · 2 Re ϕ′ − i(ϕ′)^{2} − ϕ′′ − iU
= −ϕ′′ − iU + i(Re ϕ′ + i Im ϕ′)(2 Re ϕ′)
− i[(Re ϕ′)^{2} − (Im ϕ′)^{2} + 2i Re ϕ′ · Im ϕ′]
= −ϕ′′ − iU + i[2(Re ϕ′)^{2} − (Re ϕ′)^{2} + (Im ϕ′)^{2}] .

A solution u of u′ = Du gives rise to a solution y of y′ = My by the substitution y = AΛu, that is,

(2.8)  y = (\begin{matrix}e^{iΨ} & e^{−iΨ̄} \\ iϕ′e^{iΨ} & −iϕ̄′e^{−iΨ̄} \end{matrix}) u .

Let M, E, Ψ, D be related to U, ϕ as above. Lemma 2.1. Let a potential U be given. Suppose that there exists a continuous complex-valued function ϕ such that log | Re ϕ′| is a bounded function on [0, ∞). Then the function Ψ defined by (2.6) has bounded imaginary part.

If in addition ϕ′ and each solution of u′ = Du are bounded on [0, ∞), then each solution of y′ = My is likewise bounded.

Proof. (2.7) implies that \frac{d}{dx} Re (iΨ) = −\frac{1}{2} \frac{d}{dx} log | Re (ϕ′)| , whence the first conclusion. The second then follows from (2.8). □

3. Splitting the potential

Let V be as in Theorem 1.1. Fix an auxiliary function η ∈ C_{0}^{∞}(ℝ) having compact support, identically equal to one in some neighborhood of the origin, and whose inverse Fourier transform is real-valued. Decompose

(3.1)  V = W + Ṽ where Ŵ(ξ) = V̂(ξ) · η(ξ).

Then W, Ṽ are real-valued. This decomposition will reduce matters to the analysis of sums of only two types of potentials.

The proof of the next elementary observation is left to the reader. Lemma 3.1. W ∈ C^{∞} ∩ L^{∞}, and for every k ≥ 1, d^{k}W (x)/dx^{k} → 0 as |x| → ∞. Moreover d^{k}W/dx^{k} ∈ L^{p} for every k ≥ n, while Ṽ ∈ ℓ^{p}(L^{1}).

The following routine refinement will be needed; we include a proof for the reader’s convenience. Lemma 3.2. For each 1 ≤ k < n, d^{k}W/dx^{k} ∈ L^{q_{k}} ∩ L^{∞} where q_{k} = pn/k.

Proof. Let 1 ≤ k < n. We claim that if f is bounded and f^{(n)} ∈ L^{p}(ℝ), then f^{(k)} ∈ L^{q_{k}}. To begin, observe that there exists C < ∞ such that for any smooth function f and any interval I,

(3.2)  ∥f^{(k)}∥_{L^{∞}(I)} ≤ C|I|^{−k}∥f∥_{L^{∞}(I)} + C|I|^{n−k−1}∥f^{(n)}∥_{L^{1}(I)}.

By scaling, it suffices to establish this for intervals of length one. Suppose that the right-hand side is ≤ 1, and the left-hand side is large. f^{(n−1)} may be decomposed as a polynomial of degree zero, plus a function whose supremum over I is a priori bounded. Iterating this, f^{(k)} may be decomposed as a polynomial of degree ≤ n − k, plus a function whose supremum over I is a priori bounded. Since the L^{∞} norm of f^{(k)} is large, this last polynomial must be large in L^{∞}(I) norm. Consequently f^{(k−1)} may be decomposed as a polynomial of degree ≤ n − k + 1 with large norm in L^{∞}(I), plus a function whose L^{∞}(I) norm is a priori bounded. Iterating this, we eventually conclude that f itself equals a large polynomial of bounded degree, plus a function whose supremum norm is small; hence ∥f∥_{L^{∞}(I)} is large, a contradiction.

Fix 1 < p < ∞, let q = np/k, let f be any C^{n} function with f^{(n)} ∈ L^{p}, and normalize so that ∥f∥_{L^{∞}} = 1. Any point x ∈ ℝ belongs to some interval I satisfying

(3.3)  |I|^{1−n} = ∥f^{(n)}∥_{L^{1}(I)};

for if I is taken to be centered at x then the right-hand side is a nondecreasing function of |I|, while the left-hand side decreases to zero as |I| → ∞. Choose and fix a covering {I_{j}} of ℝ consisting of such intervals, such that no point of ℝ belongs to more than two intervals I_{j}. By (3.2) and the normalization, ∥f^{(k)}∥_{L^{∞}(I_{j})} ≤ C|I|^{n−k−1}∥f^{(n)}∥_{L^{1}(I_{j})} for each I_{j}. By (3.2), (3.3), and Hölder’s inequality, for each J = I_{j},

∥f^{(k)}∥_{L^{q}(J)} ≤ C|J|^{1/q}∥f^{(k)}∥_{L^{∞}(J)} ≤ C|J|^{1/q}|J|^{n−k−1}∥f^{(n)}∥_{L^{1}(J)}

≤ C|J|^{1/q}|J|^{n−k−1}∥f^{(n)}∥_{L^{1}(J)}^{k/n} · |J|^{(1−n)(n−k)/n}

≤ C|J|^{1/q}|J|^{n−k−1}|J|^{(1−n)(n−k)/n}|J|^{\frac{p−1}{p} \frac{k}{n}} ∥f^{(n)}∥_{L^{p}(J)}^{k/n} = C∥f^{(n)}∥_{L^{p}(J)}^{k/n} .

Raising this to the power q and summing over j completes the proof. □

The result can alternatively be proved via complex interpolation.

4. Higher-order WKB approximations

We next show how to construct useful approximations exp(iΨ), exp(−iΨ̄) to the generalized eigenfunctions associated to V, by constructing the auxiliary function ϕ of §2. Consider a Schrödinger equation −g′′ + (W − λ^{2})g = 0 with some potential W (x). If g = e^{iΦ}, this equation becomes (Φ′)^{2} − iΦ′′ + W − λ^{2} = 0. Writing F = Φ′, this becomes

(4.1)  F^{2} − iF′ + W − λ^{2} = 0.

Throughout this discussion we assume that the real-valued function λ^{2} − W (x) is uniformly bounded below by some fixed strictly positive number, and moreover that the quantities F′_{k}, to be introduced shortly, are uniformly small. It will be a consequence of our construction that this is the case, under the hypotheses of Theorem 1.1, for all potentials W and all k to which this discussion is applied. The notation \sqrt{λ^{2} − W (x) + iF′_{k}(x)} thus refers, without ambiguity, to that branch of the square root function close to the positive square root of the positive quantity λ^{2} − W (x).

We approximately solve the preceding equation by recursion: set F_{0}(x, λ) = λ and for k ≥ 0,

(4.2)  F_{k+1} = \sqrt{λ^{2} − W + iF′_{k}} .

Define the error

(4.3)  E_{k} = F_{k}^{2} − iF′_{k} + W − λ^{2}.

Substituting the recursion formula for F_{k}^{2} gives the alternative expression

(4.4)  E_{k} = iF′_{k−1} − iF′_{k} .

A different expression will be more useful for our purpose. (4.3) can be rewritten as iF′_{k} = F_{k}^{2} +(W −λ^{2})−E_{k}; substituting this into (4.2) gives F_{k+1} = \sqrt{F_{k}^{2} − E_{k}}. Thus from (4.4) we deduce

(4.5)  E_{k+1} = i\frac{d}{dx}(F_{k} − \sqrt{F_{k}^{2} − E_{k}}) = i\frac{d}{dx} \frac{E_{k}}{F_{k} + \sqrt{F_{k}^{2} − E_{k}}} .

The first few functions F_{k}, E_{k} are:

F_{0} = λ
E_{0} = W
F_{1} = \sqrt{λ^{2} − W}
E_{1} = \frac{i}{2}(λ^{2} − W)^{−1/2}W′
F_{2} = [(λ^{2} − W) − \frac{i}{2}(λ^{2} − W)^{−1/2}W′]^{1/2}

E_{2} = −\frac{1}{2} \frac{d}{dx} [W′(λ^{2} − W)^{−1} ( 1 + \sqrt{1 − \frac{i}{2}(λ^{2} − W)^{−3/2}W′} )^{−1}]

In our application, the function ϕ′ of §2 will be chosen to equal F_{n} for some n. Our asymptotic expression for generalized eigenfunctions will thus be

u(x, λ) ∼ exp(iΨ(x, λ)) (4.6)

where

Ψ′_{n} = F_{n} − \frac{Ṽ + E_{n}}{2 Re F_{n}} . (4.7)

The first two functions Ψ′_{n} are:

Ψ′_{0} = λ − (2λ)^{−1}V
Ψ′_{1} = (λ^{2} − W)^{1/2} − \frac{1}{2}(λ^{2} − W)^{−1/2}Ṽ − \frac{i}{4}(λ^{2} − W)^{−1}W′ .

Ψ′_{2} is already rather complicated.

In the next lemma, W is a bounded, continuous real-valued function of x ∈ ℝ, and K is an arbitrary compact subinterval of (0, ∞), which is to remain fixed, for the remainder of the proof of Theorem 1.1; all assertions are uniform in λ ∈ K. We will always assume that sup_{x} max [W (x), 0] is as small as may be desired, and that d^{k}W/dx^{k} is likewise small in L^{∞} ∩ L^{q_{k}} for 1 ≤ k < n and in L^{p} for k ≥ n. In our eventual application, this can be achieved by replacing the original potential V by a suitable potential that equals V on [N, ∞) for sufficiently large N. Lemma 4.1. Suppose that W ∈ L^{∞}, that d^{k}W/dx^{k} ∈ L^{p}(ℝ, dx) for every k ≥ n, and that the supremum of max [W (x), 0] is sufficiently small. Then F_{n} ∈ L^{∞}, and Re (F_{n} −λ) ≥ −δ where δ > 0 may be taken to be as small as desired. The remainder term E_{n} belongs to L^{p}(ℝ, dx), and moreover ∂^{m}E_{n}/∂λ^{m} ∈ L^{p}(ℝ, dx) for every m, uniformly in λ ∈ K.

Proof. Write W^{(s)} = ∂^{s}W/∂x^{s}. It is a direct consequence of the recursion (4.2) that each F_{k}(x, λ) may be expressed as a smooth function of λ, W (x), W′(x), …, W^{(k−1)}(x). Moreover, for k ≥ 1, F_{k} − \sqrt{λ^{2} − W} will be as small as may be desired in supremum norm, provided that W′, …, W^{(k−1)} and max_{x}(W (x), 0) are all sufficiently small in the senses detailed above.

We say that a monomial ∏_{m}(W^{(m)})^{d_{m}} has weight d = ∑_{m} d_{m} · m/n. Such a monomial belongs to L^{r}(ℝ, dx), where r^{−1} = d/p, by Lemma 3.2.

Like F_{k+1}, E_{k} may be expressed as a smooth function of λ, W (x), W′(x), …, W^{(k)}(x). More precisely, E_{k} can be expressed as a finite sum of terms H(W, W′, …, W^{(k−1)}) · P (W, W′, …, W^{(k)}) where H is a smooth function in a neighborhood of (−∞, ϵ) × {0, 0, …, 0} ⊂ ℝ × ℝ^{k−1} for a fixed constant ϵ = ϵ(K) > 0, and P is a monomial of weight exactly k/n. In particular, there can be at most one factor of the highestorder derivative W^{(k)}. This description of E_{k} follows by induction on k from the above description of F_{k}, together with the recursion (4.5). Consequently E_{n} ∈ L^{p}, by Lemma 3.2 and Hölder’s inequality. □

One consequence is that the real parts of both F_{k} and F_{k}^{2} will be bounded below by a fixed strictly positive constant.

5. Combining Ingredients

Fix n ≥ 1 and assume that V satisfies the hypotheses of Theorem 1.1 with this index n. For E > A_{+} = esslimsup_{x→+∞} V (x), rewrite V (x) − E = [V (x) − A_{+}] − λ^{2} where λ > 0. We will henceforth replace V by V − A_{+}, and may thus assume that esslimsup_{x→+∞} V (x) = 0. By modifying this new V only on an interval (−∞, N], we may then assume that esslimsup_{x→+∞} V (x) is smaller than any preassigned positive quantity. Such a modification has no effect on the asymptotic behavior of the generalized eigenfunctions, as x → +∞.

Now we combine the splitting V = W + Ṽ of the potential, the generalized WKB approximation of the preceding section, and the computations in §2. Decompose V = W + Ṽ as in (3.1). Then max(0, sup_{x} W (x)) may likewise be taken to be arbitrarily small.

Let F_{n}(x, λ) be constructed from W, by iterating the recursion (4.2) to order n, and let E_{n} = F_{n}^{2} − iF′_{n} + (W − λ^{2}). As in (4.7), define

(5.1)  Ψ(x, λ) = ∫_{0}^{x} (F_{n} − \frac{Ṽ + E_{n}}{2 Re F_{n}} ) (s, λ) ds .

Also define ϕ(x, λ) = ∫_{0}^{x} F_{n} .

Ψ has bounded imaginary part. Indeed, by Lemma 4.1, log | Re ϕ′(x, λ)| = log | Re F_{n}| is a bounded function of x, for each λ ∈ K, provided that V is sufficiently small, in the sense described in the first paragraph of this section. By Lemma 2.1, exp(iΨ(x, λ)) is therefore a bounded function of x ∈ ℝ^{+}, uniformly for every λ ∈ K.

The two functions exp(±iΨ) are linearly independent, and the same holds for any two perturbations of them that are sufficiently small in the supremum norm near +∞. Indeed, the main constituent F_{n} of ∂Ψ/∂x has a real part that is bounded below by a positive constant. The other part, (Ṽ + E_{n})/2 Re F_{n}, tends to zero in the sense that its L^{1} norm on an arbitrary interval [N, N + 1] approaches zero as N → ∞. Linear independence thus follows from an elementary argument.

The remainder of the paper is devoted to the proof of the following result, which was mentioned in the Introduction but not formulated precisely there.

Theorem 5.1. Under the hypotheses of Theorem 1.1, for almost every λ > 0 = A_{+}, there exists a generalized eigenfunction u(x, λ) such that

(5.2)  u(x, λ)e^{−iΨ(x,λ)} → 1

as x → +∞. u and its complex conjugate are linearly independent.

Under the stronger hypotheses of Theorem 1.3, the set of all λ for which (5.2) fails to hold has Hausdorff dimension less than or equal to 1 − γp′.

Because Ψ has bounded imaginary part, (5.2) implies boundedness of all generalized eigenfunctions associated to the spectral parameter λ^{2}, and hence Theorem 5.1 implies the main conclusions of the theorems formulated in the Introduction.

To make use of the results in §2, set U = V − λ^{2}. Then

E(x, λ) = iϕ′′ − (ϕ′)^{2} − (V − λ^{2}) = −E_{n}(x, λ) − Ṽ (x) .

We have ∂^{m}E/∂λ^{m} ∈ ℓ^{p}(L^{1})(ℝ, dx) for every m ≥ 0, uniformly for each λ in any compact subset of ℝ, and the same goes for E/ Re ϕ′ = E/ Re F_{n}. Define

(5.3)  F(x, λ) = −iE/2 Re F_{n} .

We know that F(·, λ) ∈ ℓ^{p}(L^{1}) uniformly for all λ in any compact subset K of (0, ∞), and moreover the same goes for ∂_{λ}^{r}F(·, λ) for all r. According to (2.5),

h(x, λ) = 2 Re F_{n} + ∫_{0}^{x} \frac{Re E}{Re F_{n}} .

In order to demonstrate the first conclusion of Theorem 1.1, it suffices to show that for almost every λ, there exists a ℂ^{2}–valued solution u of u′ = Du such that

u(x) − (\begin{matrix}1 \\ 0 \end{matrix}) → 0 as x → +∞. We will deduce this from analytic machinery built

up in earlier papers [8, 9]. In the final section of the paper, we will indicate the modifications needed to treat all λ outside a set of appropriately bounded Hausdorff dimension, under the stronger hypotheses of Theorem 1.3.

6. A nonlinear Hausdorff-Young inequality

Fix n, let 1 ≤ p ≤ 2, and a compact subinterval K ⊂ (0, ∞). Let h(x, λ) be defined as in §5. Consider the operator

Sf(λ) = ∫_{ℝ} e^{ih(x,λ)}f(x) dx.

Lemma 6.1. Let V satisfy the hypotheses of Theorem 1.1, for a certain n ≥ 0 and p < ∞, with sufficiently small norms. Let s ≤ 2, and let q = s/(s−1) be the exponent conjugate to s. Then there exists C < ∞ such that for any f ∈ L^{1}(ℝ),

(6.1)  ∥Sf∥_{L^{q}(K,dλ)} ≤ C∥f∥_{ℓ^{s}(L^{1})(ℝ)} .

Proof. First consider the case s = 2. Let ζ be a real-valued, nonnegative, smooth auxiliary function that is strictly positive on K and is supported in a small neighborhood of K. Consider

∫_{K} | ∫_{ℝ} e^{ih(x,λ)}f(x) dx|^{2} ζ(λ) dλ

= ∫_{K} [∫_{ℝ} e^{ih(x,λ)}f(x) dx] [∫_{ℝ} e^{−ih(y,λ)}f̄(y) dy] ζ(λ) dλ

= ∫∫_{ℝ×ℝ} ∫_{K} e^{i(h(x,λ)−h(y,λ))}ζ(λ) dλ f(x)f̄(y) dx dy.

The hypotheses and earlier lemmas imply that |h(x, λ) − h(y, λ)| is bounded above and below by positive constants times |x − y|, provided that |x − y| is sufficiently large, uniformly in λ.

We claim that the inner integral is bounded by C(1 + |x − y|)^{−2}; the conclusion (6.1) then follows directly from this, for q = 2. When |x−y| is bounded, the estimate is trivial. When |x − y| is large, multiply and divide by ∂_{λ}((h(x, λ) − h(y, λ)), and integrate by parts with respect to λ in the inner integral, integrating exp(i(h(x, λ) − h(y, λ)) · ∂_{λ}(h(x, λ) − h(y, λ)). Multiply and divide, then integrate by parts once more, to conclude the proof for s = 2. The case s = 1 is trivial, and the general case then follows by interpolation. □

In our application, S will act on F(x, λ), which itself depends on λ. This will necessitate some modification; see the proof of Proposition 7.1.

7. Summation of the solution series

In order to prove Theorem 5.1, we now combine the WKB-type Ansatz developed in §§2-5 with Lemma 6.1 and with the machinery developed in [8, 9]. We seek to solve the equation u′ = Du, and to find a solution such that u(x) → (1 0)^{t} as x → +∞. Writing the equation as

u(x) = (\begin{matrix}1 \\ 0 \end{matrix}) − ∫_{x}^{∞} D(y)u(y) dy ,

we obtain the formal series solution

(7.1) u(x) = (\begin{matrix}1 \\ 0 \end{matrix})

+ ∑_{k=1}^{∞}(−1)^{k} ∫ ⋯ ∫_{x≤t_{1}≤t_{2}···≤t_{k}<∞} D(t_{1})D(t_{2}) ⋯ D(t_{k}) (\begin{matrix}1 \\ 0 \end{matrix}) dt_{k} ⋯ dt_{2} dt_{1} .

Introduce multilinear operators

(7.2) T_{m}(f_{1}, …, f_{m})(x, λ) =  ∫_{x≤t_{1}≤t_{2}≤···≤t_{m}} ∏_{k=1}^{m} exp(i(−1)^{m−k}h(t_{k}, λ)) f_{k}(t_{k}) dt_{k}

for each m ≥ 1. Then the series solution is formally

u(x) = (\begin{matrix}1 \\ 0 \end{matrix}) + (\begin{matrix}∑_{m=1}^{∞} T_{2m}(F_{2m,1}, …, F_{2m,2m}) \\ ∑_{m=0}^{∞} T_{2m+1}(F_{2m+1,1}, …, F_{2m+1,2m+1}) \end{matrix})

where each F_{m,j} equals either F(x, λ) or its complex conjugate; the precise rule is of no consequence for our estimates. Even the operator T_{1} is highly nonlinear, since both h and F are nonlinear functions of the potential V.

It suffices to show that for almost every λ ∈ K, these two series converge, and define bounded functions of x ∈ ℝ. The proof below, together with arguments in [8], then demonstrates that the sum of the vector-valued series does define a solution of u′ = Du such that u(x) → (1 0)^{t} as x → +∞, hence gives rise to a bounded solution, not identically vanishing, of the original generalized eigenfunction equation Hf = λ^{2}f.

This type of result was treated in our analysis [8] of the case n = 1. The principal new twist here is that the functions F on which our multilinear operators act, now depend on λ. This situation is much like that considered in Theorem 1.3 of [8], where V itself was allowed to depend on λ.

Let

Ω = ∑_{r=0}^{1} ∥∂_{ρ}^{r}F(·, ρ)∥_{ℓ^{p}(L^{1})}^{p}.

As proved in [8], there exist sets E_{j}^{m} ⊂ ℝ, indexed by 1 ≤ m < ∞, 1 ≤ j ≤ 2^{m}, satisfying:

• ℝ = ∪_{j}E_{j}^{m} for every m.

• E_{j}^{m} ∩ E_{j′}^{m} = ∅ for every j < j′.
• If j < j′, x ∈ E_{j}^{m}, and x′ ∈ E_{j′}^{m}, then x < x′.
• For every m, j, E_{j}^{m} = E_{(2j−1)}^{m+1} ∪ E_{2j}^{m+1}.
• For every m, j, ∥∂_{ρ}^{r}F(·, ρ)χ_{j}^{m}∥_{ℓ^{p}(L^{1})}^{p} ≤ 2^{−m}Ω for r = 0, 1.

We denote by χ_{E} the characteristic function of the set E, and introduce the special notation χ_{j}^{m} for the characteristic function of the interval E_{j}^{m}. Fix such sets E_{j}^{m}, for the remainder of §7.

Define a multilinear operator M_{n}, acting on n functions g_{k} of (x, λ), by

M_{n}(g_{1}, …, g_{n})(x, x′, λ) = ∫_{x≤t_{1}≤···≤t_{n}≤x′} ∏_{k=1}^{n} [g_{k}(t_{k}, λ) dt_{k}] .

In the special case when there is a single function g such that each g_{k} is an element of the set {g, ḡ}, we write simply M_{n}(g)(x, x′, λ). In our application, g(x, λ) will essentially be equal to e^{ih} · F. Define

M_{n}^{∗}(g_{1}, … g_{n})(λ) = sup_{x≤x′∈ℝ}|M_{n}(g_{1}, … g_{n})(x, x′, λ)|
M_{n}^{∗}(g)(λ) = sup_{x≤x′∈ℝ}|M_{n}(g)(x, x′, λ)| ,

(7.3)  G̃(g, λ) = ∑_{r=0}^{1} ∑_{m=1}^{∞} m (∑_{j=1}^{2^{m}} | ∫_{E_{j}^{m}} g(x, λ) dx|^{2})^{1/2} .

It will be useful to regard G̃ as a linear operator. To do this, introduce the Banach space B consisting of all complex-valued sequences a = a(m, j) indexed by 1 ≤ m < ∞ and 1 ≤ j ≤ 2^{m}, for which ∑_{m} m (∑_{j} |a(m, j)|^{2})^{1/2} < ∞. Then G̃(g, λ) equals the norm in B of the sequence {∫_{E_{j}^{m}} g(x, λ) dx}.

In Proposition 4.2 of [9] and in the proof of Theorem 1.3 of that reference it is shown^{2} that

(7.4)  M_{n}^{∗}(g_{1}, …, g_{n})(λ) ≤ C^{n} ∏_{k=1}^{n} G̃(g_{k}, λ)

(7.5)  M_{n}^{∗}(g)(λ) ≤ C^{n} \frac{G̃(g, λ)^{n}}{\sqrt{n!}}

for some universal constant C < ∞. Moreover, there is likewise a factor of 1/\sqrt{n!} on the right-hand side of (7.4), if the number of distinct functions g_{k} is bounded by any fixed constant independent of n. One formal consequence of (7.5) is that the series solution u of u′ = Du satisfies

(7.6)  sup_{x} |u(x, λ)| ≤ C exp(CG̃(λ)^{2})

where G̃ is as defined in (7.3), with g(x, λ) = exp(ih(x, λ)) · F(x, λ). Proposition 7.1. Let n, V, p be as in the hypotheses of Theorem 1.1. Let F be as defined above, and suppose that for each s, k, F_{s,k} equals either F, or F̄. Then for each m ≥ 1, for almost every λ ∈ K,

lim_{x′→∞} ∫_{x≤t_{1}≤···≤t_{s}≤x′} [ ∏_{k=1}^{s} e^{±ih(t_{k},λ)}F_{s,k}(t_{k}, λ) dt_{k}]

exists for every x, and

T_{s}^{∗}(λ) = sup_{x} | ∫_{x≤t_{1}≤···≤t_{s}<∞} [ ∏_{k=1}^{s} e^{±ih(t_{k},λ)}F_{s,k}(t_{k}, λ) dt_{k}]|

is finite. Finally, ∑_{s=0}^{∞} T_{s}^{∗}(λ) is finite for almost every λ ∈ K. The plus and minus signs in the exponents are not specified; these assertions are valid for all choices of signs, with uniform bounds.

Proof. Fix any compact subinterval K of (0, ∞), and let q = p/(p − 1) > 2 be the exponent conjugate to p. By Lemma 6.1, the mapping f ↦ ∫_{ℝ} e^{±ih(x,λ)}f(x) dx maps ℓ^{p}(L^{1}) boundedly to L^{q}(K). Therefore by Proposition 3.3 of [8] and the remark following the proof of Theorem 1.1 of [9], the L^{q}(K, B) norm of {∫_{E_{j}^{m}} e^{±ih(x,λ)}f(x) dx} is majorized by a fixed constant times the ℓ^{p}(L^{1}) norm of f, provided that the collection of sets E_{j}^{m} is adapted to f, in the sense that ∥f · χ_{j}^{m}∥_{ℓ^{p}(L^{1})} ≤ 2^{−m}∥f∥_{ℓ^{p}(L^{1})}. Taking f(x) first equal to F(x, ρ) and then equal to ∂_{ρ}F(x, ρ), we conclude that for r = 0, 1, the L^{q}(K, B, dλ) norm of {∂_{ρ}^{r} ∫_{E_{j}^{m}} e^{±ih(x,λ)}F(x, ρ) dx} is majorized by C∥F∥_{ℓ^{p}(L^{1})} + C∥∂_{ρ}F∥_{ℓ^{p}(L^{1})}, hence by a finite constant, uniformly for all ρ ∈ K. Thus ∂_{ρ}^{r} ∫_{E_{j}^{m}} e^{±ih(x,λ)}F(x, ρ) dx ∈ L^{q}(K × K, B, dλ dρ) for r = 0, 1. Therefore by the

2A bound with a factor of 1/\sqrt{n!} also appears in the elementary theory of Volterra integral equations [29] p. 12; it appears not to be closely related to (7.5).

one-dimensional Sobolev embedding theorem, K ∋ ρ ↦ ∫_{E_{j}^{m}} e^{±ih(x,λ)}F(x, ρ) dx is a continuous B–valued function for almost every λ ∈ K, and the supremum over ρ of its B–norm belongs to L^{q}(K, dλ). Thus we conclude that

(7.7)  G̃(F(·, λ), λ) ∈ L^{q}(K, dλ).

From (7.5), it follows that the supremum over all pairs x, x′ of

| ∫_{x≤t_{1}≤···≤t_{s}≤x′} ∏ e^{±ih(t_{k},λ)}F_{s,k}(x, λ) dt_{k}|

is finite for almost every λ ∈ K. Existence of the limit, as x′ → ∞, then follows as in the proof of Proposition 4.1 of [8]. Summability with respect to s holds, because of the factor of 1/\sqrt{n!} in (7.5), as expressed in (7.6). □

8. A bound on the set of exceptional energies

Assume that V satisfies the hypotheses of Theorem 1.3 for some 1 < p ≤ 2, γ > 0. We seek an upper bound on the Hausdorff dimension of the set of all λ for which the WKB-type asymptotics fail to hold. Suppose that β > 1 − p′γ, where p′ = p/(p − 1). We may assume that 0 < β < 1, γ < 1 and p > 1; otherwise there is nothing to prove, or the result is already known [1]. Let H^{β} denote β–dimensional Hausdorff measure. Fix a compact subinterval K of (0, ∞). Throughout this section, V, F, h are functions of x ≥ 0.

By its construction, the exponent h(x, λ) defined in (2.5) satisfies |∂^{k}h(x, λ)/∂λ^{k}| ≤ C + Cx, uniformly in λ ∈ K, for every k ≥ 0. Indeed, ∂^{k+1}h/∂λ^{k}∂x is bounded. Likewise, (1 + x)^{γ} times any partial derivative of F(x, λ) with respect to λ belongs to ℓ^{p}(L^{1})(ℝ^{+}). This follows from natural analogues of Lemmas 3.1 and 3.2; in particular, (1 + x)^{γk/n}∂^{k}W (x, λ)/∂x^{k} ∈ L^{∞} ∩ L^{pk/n} for 0 ≤ k ≤ n, and the same holds for each of its partial derivatives with respect to λ.

By subtracting a constant from V, we can also assume that esslimsup_{x→+∞} V (x) equals 0. As in §7, consider the formal series solution (7.1) of u′ = Du. Define

T_{m}(F_{1}, …, F_{m})(x, x′, λ) = ∫_{x≤t_{1}≤t_{2}≤···≤t_{m}≤x′} ∏_{k=1}^{m} e^{±ih(t_{k},λ)}F_{k}(t_{k}) dt_{k}.

Define intervals E_{j}^{m} ⊂ ℝ^{+} by the same construction used in §7, but applied to (1 + x)^{γ} · F(·, λ). To any function F (x, λ) associate the doubly indexed sequence of numbers

g(F)(λ) = { ∫_{E_{j}^{m}} e^{ih(t,λ)}F (t, λ) dt}_{m≥1, 1≤j≤2^{m}} .

Recall from §7 that the B–norm of a doubly indexed sequence a(m, j) is defined to be ∑_{m} m[∑_{j} |a(m, j)|^{2}]^{1/2}.

Define F^{(N)}(x, λ) = F(x, λ) for x ≥ N, and = 0 for x < N. A direct consequence of the definitions is that

(8.1)  T_{m}(F, …, F)(x, x′, λ) ≡ T_{m}(F^{(N)}, …, F^{(N)})(x, x′, λ) for all x, x′ ≥ N.

Throughout this discussion, the exponent h and the sets E_{j}^{m} appearing in the definitions of T_{m} and g are defined in terms of the original potential V; they are independent of N.

Define

Λ_{c} = {λ ∈ K : ∥g(F^{(N)})(λ)∥_{B} ≥ c for every N ≥ 0} .

We will prove that for any c > 0, H^{β}(Λ_{c}) = 0. Since by (7.5) and (8.1), whenever N ≥ M,

∑_{m=1}^{∞} sup_{x,x′≥N} |T_{m}(F^{(N)}, …, F^{(N)})(x, x′, λ)| ≤ C exp(C∥g(F^{(M)})(λ)∥_{B}^{2}),

this implies that for any c > 0,

(8.2)  H^{β}{λ ∈ K : lim sup_{x,x′→∞} ∑_{m=1}^{∞} |T_{m}(F(·, λ), …, F(·, λ))(x, x′, λ)| ≥ c} = 0.

As in the proofs of Proposition 7.1 of this paper, and Proposition 4.1 of [8], that suffices to establish convergence of the series defining u, and validity of the WKB asymptotics, for all λ outside a set whose H^{β} measure equals zero.

Let q = p′. We claim that g(F) belongs to the Sobolev space L_{γ}^{q} of all B–valued functions having γ derivatives in L^{q}, in a fixed neighborhood of K. To prove this, consider the analytic family of functions F_{z}(x, λ) = (1 + x)^{z}F(x, λ). For Re (z) = γ, g(F_{z}) ∈ L^{q}, by Lemma 6.1 and Proposition 7.1. For Re (z) = γ − 1, we have ∂_{λ}g(F_{z}) ∈ L^{q}(K, dλ). Indeed, when ∫_{E_{j}^{m}} e^{±ih(x,λ)}F_{z}(x, λ) dx is differentiated with respect to λ, the derivative falls either on F, or on the exponent h. In the former case, no harm is done, because each partial derivative of F with respect to λ satisfies the same bounds as does F itself; moreover, matters are improved by the factor of (1 + x)^{z} in the definition of F_{z}, since Re (γ − 1) ≤ 0. In the latter case, F is replaced by ±i∂_{λ}h·F. Since ∂^{k}h/∂λ^{k} = O(x) for every k ≥ 1, this results in an extra O(1 + x) factor; when combined with the factor of (1 + x)^{γ−1} in the definition of F_{z}, this means that we are applying g to a function all of whose λ–derivatives belong to ℓ^{p}(L^{1}) for each λ. Thus Lemma 6.1, in its λ–dependent version developed in the proof of Proposition 7.1, applies once more.

Moreover, the Sobolev norm of g(F^{(N)}) tends to zero as N → ∞. This follows from three facts. Firstly, in a fixed neighborhood of K, by the discussion in the preceding paragraph, for any fixed m, j, the scalar-valued function ∫_{E_{j}^{m}} e^{ih(t,λ)}F (t, λ) dt has

L_{γ}^{q} norm bounded by C ∑_{r=0}^{1} sup_{λ∈K} ∥(1 + |t|)^{γ}∂_{λ}^{r}F (t, λ)∥_{ℓ^{p}(L^{1})(ℝ^{+}, dt)}. Secondly, for F = F^{(N)}, the two ℓ^{p}(L^{1}) norms in this last expression tend to zero as N → ∞. Thirdly, the claim of the preceding paragraph remains valid if the norm on B is changed so that the norm of a sequence a(m, j) is ∑_{m} m^{2}[ ∑_{j} |a(m, j)|^{2}]^{1/2} (the weight m has been changed to m^{2}). Indeed, this remains true with any power of m, as follows from the proof of Proposition 7.1, the argument two paragraphs above, and [9].

Suppose now that for some c > 0, H^{β}(Λ_{c}) > 0. Then by Theorem II.1 of [6], there exists a finite positive measure μ with μ(Λ_{c}) > 0, satisfying μ(I) ≤ |I|^{β} for every interval I.

Let N be large. By a potential-theoretic characterization of Sobolev spaces [26], we conclude that ∥g(F^{(N)})(λ)∥_{B} ≤ J ∗ f_{N}(λ) = ∫_{ℝ} J(λ − ρ)f_{N}(ρ) dρ, where f_{N}, J are nonnegative, f_{N} ∈ L^{q}(ℝ), ∥f_{N}∥_{L^{q}} → 0 as N → ∞, J(ρ) ≤ C|ρ|^{γ−1} for all |ρ| ≤ 1, and J(ρ) ≤ C exp(−c|ρ|) for |ρ| ≥ 1.

A simple calculation using the hypothesis β > 1−p′γ in conjunction with the upper bound on μ(I) demonstrates that J ∗ μ ∈ L^{q′} = L^{p}. (Decompose J as a Schwartz function plus ∑_{r=0}^{∞} J_{r} where J_{r}(x) is supported where |x| ≤ 2^{−r}, and ∥J_{r}∥_{L^{∞}} ≤ C2^{r(1−γ)}. Estimate the L^{p} norm of μ ∗ J_{r} by interpolating between simple L^{1} and L^{∞} bounds, then sum over r.) Thus

∫ (J ∗ f_{N}) dμ = ∫ f_{N} · (J ∗ μ) dλ ≤ ∥f_{N}∥_{L^{q}} · ∥J ∗ μ∥_{L^{q′}},

which tends to zero as N → ∞. But by hypothesis and the definition of Λ_{c}, ∥g(F^{(N)})(λ)∥_{B} ≥ c for every λ ∈ Λ_{c} and every N. Therefore

∫ (J ∗ f_{N})dμ ≥ ∫ ∥g(F^{(N)})∥_{B} dμ ≥ cμ(Λ_{c}) > 0,

a contradiction.

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Michael Christ, Department of Mathematics, University of California, Berkeley, CA 94720-3840, USA

  E-mail address: mchrist@math.berkeley.edu

Alexander Kiselev, Department of Mathematics, University of Chicago, Chicago, Ill. 60637

  E-mail address: kiselev@math.uchicago.edu