A weak Gordon type condition for absence of eigenvalues of one-dimensional Schrödinger

operators

Christian Seifert and Hendrik Vogt

Abstract

We study one-dimensional Schrödinger operators with complex measures as potentials and present an improved criterion for absence of eigenvalues which involves a weak local periodicity condition. The criterion leads to sharp quantitative bounds on the eigenvalues. We apply our result to quasiperiodic measures as potentials.

MSC2010: 34L15, 34L40, 81Q10, 81Q12

Key words: Schrödinger operators, eigenvalue problem, quasiperiodic potential

1 Introduction

In this paper we study Schrödinger operators of the form

H_{μ} = −Δ + μ

in L_{2}(ℝ), where the potential μ is locally a complex Radon measure. We are going to study absence of eigenvalues of H_{μ} under the condition that μ can be approximated by periodic measures in a suitable sense. Roughly speaking, we require that there are arbitrarily large periods p > 0 such that the three “pieces” 𝟙_{[−p,0]}μ, 𝟙_{[0,p]}μ and 𝟙_{[p,2p]}μ look very similar.

The above so-called Gordon condition first appeared in [5]. Gordon treated the case of bounded potentials and measured the distance of the “pieces” in the L_{∞}-norm. In [2] this was generalised to uniformly locally integrable potentials and the L_{1,loc,unif}-norm. Recently, a similar result was shown in [7] for real measures as potentials, and the distance was measured in the total variation metric. We will extend these results in two directions.

The most important new aspect is that we work with a weaker Wasserstein type metric in the Gordon condition, thus allowing a wider range of applications. For linear combinations of Dirac measures this means that the positions of the Dirac deltas are allowed to vary, not only their coefficients as in [7].

The second extension concerns sharp quantitative results. We show that the operator H_{μ} has no eigenvalues z with |z| < E_{μ}, where E_{μ} is a constant given by an explicit expression in terms of μ. Moreover, we give an example that −E_{μ} can occur as an eigenvalue. We note that we can also deal with complex measures, thus obtaining non-selfadjoint operators.

Our result can be applied to quasiperiodic measures where the ratio of the periods satisfies a strong Liouville condition. Such potentials were also considered in [2, 7]; however, absence of eigenvalues was only shown under some additional hypotheses. Working with our weaker metric we can prove the result without any further assumptions.

The question of the absence of eigenvalues appears in the study of onedimensional quasicrystals. Here, it is crucial to obtain a good criterion when working with a weak metric to measure the distances of potentials (i.e., measures). The results in [8, Chapters 4 and 6] suggest that the topology of vague convergence of measures is appropriate in the study of continuum quasicrystals. Our Wasserstein type metric is precisely of this quality, see Lemma 2.4.

We now describe in more detail the Schrödinger operator H_{μ} we are going to study. We say that

μ: {B ⊆ ℝ; B is a bounded Borel set} → ℂ

is a local measure if 𝟙_{K}μ := μ(· ∩ K) is a complex Radon measure for any compact set K ⊆ ℝ. Then there exist a (unique) nonnegative Radon measure ν on ℝ and a measurable function σ: ℝ → ℂ such that |σ| = 1 ν-a.e. and 𝟙_{K}μ = 𝟙_{K}σν for all compact sets K ⊆ ℝ. The total variation of μ is defined by |μ| := ν. Let M_{loc}(ℝ) be the space of all local measures on ℝ.

A local measure μ ∈ M_{loc}(ℝ) is called uniformly locally bounded if

∥μ∥_{unif} := sup_{a∈ℝ} |μ|((a, a + 1]) < ∞.

Let M_{loc,unif}(ℝ) denote the space of all uniformly locally bounded local measures. The space M_{loc,unif}(ℝ) naturally extends L_{1,loc,unif}(ℝ) to measures.

Given μ ∈ M_{loc,unif}(ℝ), we now define the operator H_{μ} in L_{2}(ℝ) as the maximal operator associated with −Δ+μ (in the distributional sense). More precisely,

D(H_{μ}) := {u ∈ L_{2}(ℝ) ∩ C(ℝ); −u′′ + uμ ∈ L_{2}(ℝ)},
H_{μ}u := −u′′ + uμ,

i.e., for u, f ∈ L_{2}(ℝ) one has u ∈ D(H_{μ}), H_{μ}u = f if and only if

∫_{ℝ} fφ = −∫_{ℝ} uφ′′ + ∫_{ℝ} uφ dμ (φ ∈ C_{c}^{∞}(ℝ)).

The operator H_{μ} can also be defined via the form method [6] or along the lines of Sturm-Liouville differential operators [1]; see Remark 3.5 for details. In Theorem 3.6 we will show that the three approaches yield the same operator (cf. [8, Corollary 1.3.7] for the self-adjoint case). This result may be of independent interest.

Notation. For μ ∈ M_{loc,unif}(ℝ) and s, t ∈ ℝ we write

∫_{s}^{t} … dμ = \begin{cases}∫_{(s,t]} … dμ & if t ⩾ s, \\ − ∫_{(t,s]} … dμ & if t < s. \end{cases}

The paper is organized as follows. In Section 2 we introduce the seminorms on the space of measures we will work with. Estimates on solutions of the eigenvalue equation are provided in Section 3. These will then be used to prove the equivalence of various possibilities to define the Schrödinger operator. In Section 4 we estimate the difference of solutions to different potentials. Our main theorem is stated and proved in Section 5. We also give an application to quasiperiodic potentials there. Finally, in Section 6 we construct an example to show that our results are sharp.

2 Seminorms on measures

In this section we define suitable seminorms on M_{loc,unif}(ℝ). The Lipschitz continuous functions on ℝ will be denoted by W_{∞}^{1}(ℝ).

Definition. For μ ∈ M_{loc,unif}(ℝ) and a set I ⊆ ℝ (which will usually be an interval) we define

∥μ∥_{I} := sup{|∫ u dμ|; u ∈ W_{∞}^{1}(ℝ), spt u ⊆ I, diam spt u ⩽ 2, ∥u′∥_{∞} ⩽ 1}.

Choosing supports of length at most 2 yields nice values for “typical” measures.

Example 2.1. (a) For the Lebesgue measure λ on ℝ we have ∥λ∥_{ℝ} = 1.

(b) For the Dirac measure δ_{t} at t ∈ ℝ we have ∥δ_{t}∥_{ℝ} = 1. Moreover, ∥∑_{n∈ℤ} δ_{n}∥_{ℝ} = 1.

(c) Let ε ∈ (0, 1) and μ := δ_{0} − δ_{ε} + δ_{2} − δ_{2+ε}. Then ∥μ∥_{ℝ} = ε, but there exists u ∈ W_{∞}^{1}(ℝ) with diam spt u = 2 + ε and ∥u′∥_{∞} ⩽ 1 such that ∫ u dμ = 2ε. Thus, increasing the diameter of spt u in the definition of ∥μ∥_{I} only slightly can change the norm by a factor of 2.

Next we explain what happens if one further increases diam spt u.

Remark 2.2. Let μ ∈ M_{loc,unif}(ℝ), n ∈ ℕ, and let u ∈ W_{∞}^{1}(ℝ) such that spt u ⊆ [−n, n], ∥u′∥_{∞} ⩽ 1. Then

|∫ u dμ| ⩽ n^{2}∥μ∥_{[−n,n]}.

Indeed, there exist u_{k} ∈ W_{∞}^{1}(ℝ) with diam spt u_{k} ⩽ 2 (k = 1 − n, …, n − 1) such that u = ∑ u_{k} and ∑ ∥u′_{k}∥_{∞} ⩽ n^{2}: let

u_{k}(t) = \begin{cases}u(t) − u(k − 1) · (k − t) & if k − 1 ⩽ t ⩽ k, \\ u(k) · (k + 1 − t) & if k < t ⩽ k + 1 \end{cases}

for k < 0,

u_{0}(t) = \begin{cases}u(t) − u(−1) · (−t) & if −1 ⩽ t ⩽ 0, \\ u(t) − u(1) · t & if 0 < t ⩽ 1, \end{cases}

and u_{k} for k > 0 defined similarly as for k < 0. Then ∥u′_{k}∥_{∞} ⩽ n − |k| for all k since |u(t)| ⩽ n − |t| for all |t| ⩽ n, and ∑_{k=1−n}^{n−1}(n − |k|) = n^{2}.

Remark 2.3. (a) It is easy to see that ∥·∥_{ℝ} defines a norm on M_{loc,unif}(ℝ) satisfying ∥μ∥_{ℝ} ⩽ ∥μ∥_{unif}: for u ∈ W_{∞}^{1}(ℝ) with spt u ⊆ [a − 1, a + 1] and ∥u′∥_{∞} ⩽ 1 (with some a ∈ ℝ) one has

|u(s)| ⩽ max{1 − |s − a|, 0} = ∫_{a−1}^{a} 𝟙_{(t,t+1]}(s) dt (s ∈ ℝ)

and hence

|∫ u dμ| ⩽ ∫_{a−1}^{a} ∫ 𝟙_{(t,t+1]} d|μ| dt ⩽ ∥μ∥_{unif}.

(b) For μ ∈ M_{loc,unif}(ℝ), the translates μ(· + ε) converge to μ in the ∥·∥_{ℝ}-norm as ε → 0, so ∥·∥_{ℝ} is weaker than ∥·∥_{unif}: for ε > 0 one has

∥μ − μ(· + ε)∥_{ℝ} ⩽ 3ε∥μ∥_{unif}.

Indeed, for ε ⩾ 1 this is clear by part (a), and for 0 < ε < 1 it follows from the estimate

|∫_{ℝ} u d(μ − μ(· + ε))| ⩽ ∫_{ℝ} |u − u(· − ε)| d|μ| ⩽ ∫_{spt u+(0,ε)} ε∥u′∥_{∞} d|μ|.

The next lemma shows in particular that the norm ∥·∥_{ℝ} induces the vague topology on {μ ∈ M_{loc,unif}(ℝ); ∥μ∥_{unif} ⩽ R, spt μ ⊆ [−R, R]}, for any R > 0.

Lemma 2.4. Let (μ_{n}) be a sequence in M_{loc,unif}(ℝ) and μ ∈ M_{loc,unif}(ℝ). Then the following are equivalent:

(i) μ_{n} → μ vaguely, (ii) (𝟙_{I}μ_{n}) is ∥·∥_{unif}-bounded in M_{loc,unif}(ℝ) and ∥μ_{n} − μ∥_{I} → 0, for all

bounded intervals I ⊆ ℝ.

Proof. Without loss of generality assume that μ = 0. For a bounded open interval I ⊆ ℝ we can regard {𝟙_{I}μ; μ ∈ M_{loc,unif}(ℝ)} as the dual space of C_{0}(I). Note that then μ_{n} → 0 vaguely if and only if (i’) 𝟙_{I}μ_{n} → 0 in the weak^{∗} topology, for all bounded open intervals I ⊆ ℝ. We now fix a bounded open interval I ⊆ ℝ.

(i’) ⇒ (ii): As a null sequence in the weak^{∗} topology, (𝟙_{I}μ_{n}) is bounded and hence ∥·∥_{unif}-bounded; moreover, (𝟙_{I}μ_{n}) tends to 0 uniformly on compact subsets of C_{0}(I). By the Arzelà-Ascoli theorem,

{u ∈ W_{∞}^{1}(ℝ); spt u ⊆ I, diam spt u ⩽ 2, ∥u′∥_{∞} ⩽ 1}

is compact in C_{0}(I), so we conclude that ∥μ_{n}∥_{I} → 0.

(ii) ⇒ (i’): It follows from Remark 2.2 that ∫ u dμ_{n} → 0 for all Lipschitz functions u in C_{0}(I). Since the Lipschitz functions are dense in C_{0}(I) and (𝟙_{I}μ_{n}) is bounded in C_{0}(I)′, this implies ∫ u dμ_{n} → 0 for all u ∈ C_{0}(I).

An important property of the norm ∥·∥_{ℝ} is that unlike ∥·∥_{unif} it allows approximation of measures in M_{loc,unif}(ℝ) by absolutely continuous measures with C^{∞}-densities.

Proposition 2.5. Let μ ∈ M_{loc,unif}(ℝ). Then there exists a sequence (μ_{n}) in M_{loc,unif}(ℝ) such that μ_{n} has a C^{∞}-density and ∥μ_{n}∥_{unif} ⩽ ∥μ∥_{unif} for all n ∈ ℕ, ∥μ − μ_{n}∥_{ℝ} → 0, and lim sup_{n→∞} |μ_{n}|(I) ⩽ |μ|(I) for all compact intervals I ⊆ ℝ.

Proof. For n ∈ ℕ let ψ_{n} ∈ C_{c}^{∞}(ℝ) satisfy spt ψ_{n} ⊆ (−\frac{1}{n}, \frac{1}{n}), ∫ ψ_{n} = 1, ψ_{n}(−·) = ψ_{n} and ψ_{n} ⩾ 0. In particular, (ψ_{n}) is an approximation of the identity. For n ∈ ℕ we define

f_{n}(x) := ∫ ψ_{n}(x − y) dμ(y) = ψ_{n} ∗ μ(x) (x ∈ ℝ);

then f_{n} ∈ C^{∞}(ℝ). Set μ_{n} := f_{n}λ.

Let u ∈ W_{∞}^{1}(ℝ), diam spt u ⩽ 2, ∥u′∥_{∞} ⩽ 1. Then

|∫ u d(μ − μ_{n})| = |∫ (u(y) − ψ_{n} ∗ u(y)) dμ(y)|.

Moreover,

diam spt(ψ_{n} ∗ u) ⩽ 2 + \frac{2}{n}, ∥u − ψ_{n} ∗ u∥_{∞} ⩽ \frac{1}{n}

since ∥u′∥_{∞} ⩽ 1 and spt ψ_{n} ⊆ (−\frac{1}{n}, \frac{1}{n}). Thus, ∥μ − μ_{n}∥_{ℝ} → 0.

For a, b ∈ ℝ, a < b we compute, applying Fubini’s theorem twice and substituting x = t + y,

|μ_{n}|([a, b]) = ∫_{a}^{b} |f_{n}(x)| dx ⩽ ∫_{a}^{b}∫ ψ_{n}(x − y) d|μ|(y) dx
= ∫ ∫_{a−t}^{b−t} d|μ|(y) ψ_{n}(t) dt = ∫ |μ|((a − t, b − t])ψ_{n}(t) dt.

On the one hand, choosing b = a + 1 yields

∥μ_{n}∥_{unif} = sup_{a∈ℝ} |μ_{n}|([a, a + 1]) ⩽ ∥μ∥_{unif}.

On the other hand we obtain

lim sup_{n→∞} |μ_{n}|([a, b]) ⩽ lim sup_{n→∞} ∫ |μ|((a − \frac{1}{n}, b + \frac{1}{n}])ψ_{n}(t) dt = |μ|([a, b]).

Lemma 2.6. Let μ ∈ M_{loc,unif}(ℝ) and ψ ∈ W_{∞}^{1}(ℝ), and let I ⊆ ℝ be an interval. Then

∥ψμ∥_{I} ⩽ (∥ψ∥_{∞} + ∥ψ′∥_{∞})∥μ∥_{I ∩spt ψ}.

Proof. Let u ∈ W_{∞}^{1}(ℝ) satisfy spt u ⊆ I, diam spt u ⩽ 2 and ∥u′∥_{∞} ⩽ 1. Then ψu ∈ W_{∞}^{1}(ℝ), spt(ψu) ⊆ I ∩ spt ψ, diam spt(ψu) ⩽ 2 and

∥(ψu)′∥_{∞} ⩽ ∥ψ∥_{∞} + ∥ψ′∥_{∞}.

Hence, the assertion follows.

For μ ∈ M_{loc,unif}(ℝ) we define φ_{μ}: ℝ → ℂ by

φ_{μ}(t) := ∫_{0}^{t} dμ = \begin{cases}μ((0, t]) & if t ⩾ 0, \\ −μ((t, 0]) & if t < 0. \end{cases}

The following result provides an alternative way for computing the ∥·∥_{I}- seminorm of a given measure.

Proposition 2.7. Let μ ∈ M_{loc,unif}(ℝ) and a ∈ ℝ. If μ is real, then

∥μ∥_{[a−1,a+1]} = min_{c∈ℝ} ∫_{a−1}^{a+1} |φ_{μ}(t) − c| dt.

If μ is complex, then

∥μ∥_{[a−1,a+1]} ⩽ min_{c∈ℂ} ∫_{a−1}^{a+1} |φ_{μ}(t) − c| dt ⩽ 2∥μ∥_{[a−1,a+1]}.

(Note that the minimum exists since the integral depends continuously on c and tends to ∞ as |c| → ∞.)

Proof. Let u ∈ W_{∞}^{1}(ℝ) with spt u ⊆ [a − 1, a + 1] and ∥u′∥_{∞} ⩽ 1. Then

∫ u dμ = ∫_{a−1}^{a+1} (− ∫_{t}^{a+1} u′(s) ds) dμ(t)
= − ∫_{a−1}^{a+1} ∫_{a−1}^{s} dμ(t) u′(s) ds = − ∫_{a−1}^{a+1} u′(s)μ((a − 1, s]) ds.

Since ∫_{a−1}^{a+1} u′(s) ds = 0, it follows that

∫ u dμ = − ∫_{a−1}^{a+1} u′(s)(φ_{μ}(s) − c) ds

for all c ∈ ℂ. Thus,

|∫ u dμ| ⩽ ∫_{a−1}^{a+1} |φ_{μ}(s) − c| ds

and hence

∥μ∥_{[a−1,a+1]} ⩽ min_{c∈ℂ} ∫_{a−1}^{a+1} |φ_{μ}(s) − c| ds.

In case μ is real, we choose c ∈ ℝ to be a “median” of φ_{μ}, i.e., both φ_{μ} ⩾ c and φ_{μ} ⩽ c hold on subsets of [a − 1, a + 1] with Lebesgue measure at least 1. Then we can construct a real-valued function u ∈ W_{∞}^{1}(ℝ) with spt u ⊆ [a − 1, a + 1] and ∥u′∥_{∞} ⩽ 1 such that

∫_{a−1}^{a+1} |φ_{μ}(s) − c| ds = − ∫_{a−1}^{a+1} u′(s)(φ_{μ}(s) − c) ds = ∫ u dμ.

Therefore,

min_{c∈ℝ} ∫_{a−1}^{a+1} |φ_{μ}(s) − c| ds ⩽ ∥μ∥_{[a−1,a+1]}.

In case μ is complex, we have μ = μ_{r} + iμ_{i} with real measures μ_{r}, μ_{i} ∈ M_{loc,unif}(ℝ). Then by the above we find c_{r} ∈ ℝ and a real-valued function u ∈ W_{∞}^{1}(ℝ) such that

∫_{a−1}^{a+1} |φ_{μ_{r}}(s) − c_{r}| ds = ∫ u dμ_{r} = Re ∫ u dμ ⩽ ∥μ∥_{[a−1,a+1]}.

In the same way we obtain ∫_{a−1}^{a+1} |φ_{μ_{i}}(s)−c_{i}| ds ⩽ ∥μ∥_{[a−1,a+1]} for some c_{i} ∈ ℝ. Since φ_{μ} = φ_{μ_{r}} + iφ_{μ_{i}}, it follows that

∫_{a−1}^{a+1} |φ_{μ}(s) − (c_{r} + ic_{i})| ds ⩽ 2∥μ∥_{[a−1,a+1]}.

Remark 2.8. (a) According to Proposition 2.7, for any μ ∈ M_{loc,unif}(ℝ) and a ∈ ℝ there exists c_{a} = c_{μ,a} ∈ ℂ such that

∫_{a−1}^{a+1} |φ_{μ}(t) − c_{a}| dt ⩽ 2∥μ∥_{[a−1,a+1]}.  (1)

Moreover, c_{μ,0} can always be chosen such that

|c_{μ,0}| ⩽ ∥μ∥_{unif}

since |φ_{μ}(t)| ⩽ ∥μ∥_{unif} for t ∈ [−1, 1].

(b) We point out that |c_{μ,0}| can be large even if ∥μ∥_{ℝ} is small. As an example consider μ = δ_{ε} − δ_{−ε} for small ε > 0, where one has ∥μ∥_{ℝ} = 2ε and φ_{μ} = 𝟙_{ℝ\[−ε,ε)}.

Lemma 2.9. Let μ ∈ M_{loc,unif}(ℝ) and α, β ∈ ℤ, α ⩽ −1, β ⩾ 1. Then

∫_{k}^{k+1} |φ_{μ}(t) − c_{0}| dt ⩽ 2 max{k + 1, −k}∥μ∥_{[α,β]}

for all k ∈ [α, β − 1], with c_{0} as in (1).

Proof. For k = −1 and k = 0 the assertion follows from (1). We now show the assertion for k ⩾ 1; the proof of the case k ⩽ −2 is analogous.

Using a telescope sum expansion, we estimate

|c_{k} − c_{0}| ⩽ ∑_{j=1}^{k} ∫_{j−1}^{j}(|c_{j} − φ_{μ}(t)| + |φ_{μ}(t) − c_{j−1}|) dt

= ∫_{0}^{1} |φ_{μ}(t) − c_{0}| dt + ∑_{j=1}^{k−1} ∫_{j−1}^{j+1} |φ_{μ}(t) − c_{j}| dt + ∫_{k−1}^{k} |φ_{μ}(t) − c_{k}| dt.

By (1) we conclude that

∫_{k}^{k+1} |φ_{μ}(t) − c_{0}| dt ⩽ ∫_{k}^{k+1}(|φ_{μ}(t) − c_{k}| + |c_{k} − c_{0}|) dt

⩽ ∑_{j=0}^{k} ∫_{j−1}^{j+1} |φ_{μ}(t) − c_{j}| dt ⩽ (k + 1) · 2∥μ∥_{[α,β]}.

3 Estimates on solutions

In this section we provide estimates on functions that satisfy the eigenvalue equation for H_{μ} (but that are not necessarily in L_{2}). These estimates enable us to show that the different versions for defining −Δ + μ actually lead to the same operator.

Definition. For μ ∈ M_{loc,unif}(ℝ) and z ∈ ℂ we say that u ∈ L_{1,loc}(ℝ) is a (generalized) solution of the equation H_{μ}u = zu if u ∈ C(ℝ) and

−u′′ + μu = zu

in the sense of distributions.

Remark 3.1. Let μ ∈ M_{loc,unif}(ℝ) and z ∈ ℂ.

(a) Note that u is a solution of H_{μ}u = zu if and only if u is a solution of H_{μ−zλ}u = 0. Therefore, it is no loss of generality to assume z = 0 in this and the following section.

(b) Let u be a solution of the equation H_{μ}u = 0. Then u′′ is a local measure, so u′ has locally bounded variation (and therefore its one-sided limits exist everywhere), and

u′(t^{+}) − u′(s^{+}) = ∫_{s}^{t} u(r) dμ(r)

for all s, t ∈ ℝ. In particular, any solution of H_{μ}u = 0 is locally Lipschitz continuous.

Lemma 3.2. Let μ ∈ M_{loc,unif}(ℝ), and let u be a solution of the equation H_{μ}u = 0. Then

|u(t)| + |u′(t^{+})| ⩽ (|u(0)| + |u′(0^{+})|)e^{ω(|t|+1)} (t ∈ ℝ),

with ω = ∥μ∥_{unif} + 1.

Proof. Below we will use Gronwall’s inequality in the following form: let φ: [0, ∞) → [0, ∞) be Borel measurable and locally bounded, ε > 0 and 0 ⩽ ν ∈ M_{loc}([0, ∞)) (a nonnegative Radon measure on [0, ∞)), with

φ(t) ⩽ ε + ∫_{[0,t)} φ(s) dν(s) (t ⩾ 0).

Then φ(t) ⩽ εe^{ν([0,t))} for all t ⩾ 0; see [4, Thm. 5.1, p. 498].

By Remark 3.1(b) we have

(\begin{matrix}u(t) \\ u′(t^{+}) \end{matrix}) = (\begin{matrix}u(0) \\ u′(0^{+}) \end{matrix}) + (\begin{matrix}∫_{0}^{t} u′(s^{+}) ds \\ ∫_{0}^{t} u(s) dμ(s) \end{matrix})

for all t ∈ ℝ. Thus, for φ(t) := |u(t)| + |u′(t^{+})| and ν := λ + |μ| we obtain

φ(t) ⩽ φ(0) + ∫_{(t,0]} φ(s) dν(s) (t < 0).

By Gronwall’s inequality we infer that

φ(t) ⩽ φ(0)e^{ν((t,0])} (t ⩽ 0).

Thus, for t ⩽ 0 the assertion follows since ∥ν∥_{unif} ⩽ 1+∥μ∥_{unif} = ω and hence ν((t, 0]) ⩽ ω(|t| + 1).

Now let t > 0. As above we can estimate

φ_{−}(s) := |u(s)| + |u′(s^{−})| ⩽ φ(0) + ∫_{(0,s)} φ_{−}(r) dν(r),

and Gronwall’s inequality yields

|u(s)| + |u′(s^{−})| ⩽ (|u(0)| + |u′(0^{+})|)e^{ν((0,s))} (s > 0).

For s ↓ t the assertion follows since ν((0, t]) ⩽ ω(|t| + 1).

Remark 3.3. One could refine the above proof to obtain an estimate with ω = 2∥μ∥_{unif}^{1/2}, at the expense of a multiplicative constant in the right hand side. By a bootstrap-like argument we will even improve the bound to ω = ∥μ∥_{unif}^{1/2} in Proposition 4.5.

In the following elementary lemma we provide estimates on the derivative of a solution in terms of norms of the solution itself.

Lemma 3.4. Let μ ∈ M_{loc,unif}(ℝ), and let u ∈ C(ℝ) be a solution of H_{μ}u = 0. Then for any interval I = [a, b] ⊆ ℝ of length 1 one has

∥u′|_{I}∥_{2} ⩽ ∥u′|_{I}∥_{∞} ⩽ M_{μ}∥u|_{I}∥_{∞} ⩽ \sqrt{3} M_{μ}^{3/2}∥u|_{I}∥_{2},

where M_{μ} := ∥μ∥_{unif} + 2. In particular, u ∈ L_{2}(ℝ) implies u′ ∈ L_{2}(ℝ), and u ∈ C_{0}(ℝ) implies that u′(t^{+}) → 0 as t → ±∞.

Proof. The first inequality is clear. We have

|∫_{a}^{b} u′(t+) dt| = |u(b) − u(a)| ⩽ 2∥u|_{I}∥_{∞},

so there exists t_{0} ∈ I with |u′(t_{0}^{+})| ⩽ 2∥u|_{I}∥_{∞}. For t ∈ I it follows with Remark 3.1(b) that

|u′(t^{+})| ⩽ |u′(t_{0}^{+})| + |∫_{t_{0}}^{t} u(s) dμ(s)| ⩽ (2 + ∥μ∥_{unif})∥u|_{I}∥_{∞},

which proves the second inequality.

Let now s_{0} ∈ I such that |u(s_{0})| = ∥u|_{I}∥_{∞}. Then by the above we have |u(s_{0} + t)| ⩾ (1 − M_{μ}|t|)∥u|_{I}∥_{∞} for all t ∈ ℝ with s_{0} + t ∈ I. Therefore,

∥u|_{I}∥_{2}^{2} ⩾ ∫_{0}^{1/M_{μ}}(M_{μ}t)^{2}∥u|_{I}∥_{∞}^{2} dt = \frac{1}{3M_{μ}} ∥u|_{I}∥_{∞}^{2},

and the last inequality follows.

Remark 3.5. As mentioned in the introduction, different methods for defining −Δ + μ appear in the literature.

(a) In [1] the operator is defined along the lines of Sturm-Liouville differential operators. For u ∈ W_{1,loc}^{1}(ℝ) one defines A_{μ}u ∈ L_{1,loc}(ℝ) by

(A_{μ}u)(t) := u′(t) − ∫_{0}^{t} u(s) dμ(s)

for a.a. t ∈ ℝ. (Here and in the following, we choose the continuous representative of u in the integral.)

Then we define H_{μ,SL} as a realization of −Δ + μ by

D(H_{μ,SL}) := {u ∈ L_{2}(ℝ) ∩ W_{1,loc}^{1}(ℝ); A_{μ}u ∈ W_{1,loc}^{1}(ℝ), (A_{μ}u)′ ∈ L_{2}(ℝ)},
H_{μ,SL}u := −(A_{μ}u)′.

(b) In [6] the form method is used to define the operator. It is wellknown that μ defines an infinitesimally form small perturbation of the classical Dirichlet form τ_{0} given by

D(τ_{0}) := W_{2}^{1}(ℝ), τ_{0}(u, v) := ∫ u′\overline{v}′,

i.e., for all δ > 0 there exists C_{δ} > 0 such that

|∫ |u|^{2} dμ| ⩽ δτ_{0}(u, u) + C_{δ}∥u∥_{2}^{2} (u ∈ D(τ_{0})).

Thus,

D(τ_{μ}) := W_{2}^{1}(ℝ), τ_{μ}(u, v) := ∫ u′\overline{v}′ + ∫ u\overline{v} dμ

defines a closed sectorial form τ_{μ} in L_{2}(ℝ). (For the second integral note that u\overline{v}μ defines a complex Radon measure for any u, v ∈ W_{2}^{1}(ℝ).)

We denote by H_{μ,F} the m-sectorial operator associated with τ_{μ}, i.e.,

D(H_{μ,F}) = {u ∈ D(τ_{μ}); ∃f_{u} ∈ L_{2}(ℝ) ∀v ∈ D(τ_{μ}): τ_{μ}(u, v) = (f_{u} | v)},
H_{μ,F}u = f_{u}.

Theorem 3.6. With the above notation one has H_{μ} = H_{μ,SL} = H_{μ,F}.

Proof. It is easy to see that H_{μ,F} ⊆ H_{μ}; we show the converse operator inclusion. Since H_{μ,F} is sectorial, there exists z ∈ ℂ such that z + H_{μ,F} is injective and surjective. Let u be in the kernel of z + H_{μ}; then u is a solution of H_{μ+zλ}u = 0. Since u ∈ L_{2}(ℝ), Lemma 3.4 yields u′ ∈ L_{2}(ℝ). For φ ∈ C_{c}^{∞}(ℝ) we obtain

(H_{μ}u | φ) = − (u | φ′′) + ∫ u\overline{φ} dμ = (u′ | φ′) + ∫ u\overline{φ} dμ = τ_{μ}(u, φ).

Since C_{c}^{∞}(ℝ) is dense in W_{2}^{1}(ℝ), it follows that u ∈ D(H_{μ,F}) and H_{μ,F}u = H_{μ}u. Hence also (z + H_{μ,F})u = 0. Thus u = 0, so we have shown that z + H_{μ} is injective. Since surjective mappings cannot have proper injective extensions, we obtain H_{μ,F} = H_{μ}.

Applying Fubini’s theorem as in [7, Lemma 1.3], we obtain H_{μ,F} ⊆ H_{μ,SL}. For u ∈ D(H_{μ,SL}) and φ ∈ C_{c}^{∞}(ℝ) we compute, with the help of Fubini’s theorem,

(H_{μ,SL}u | φ) = ∫ u′\overline{φ}′ − ∫ ∫_{0}^{t} u(s) dμ(s) \overline{φ}′(t) dt = − (u | φ′′) + ∫ u\overline{φ} dμ.

Thus, u ∈ D(H_{μ}) and H_{μ}u = H_{μ,SL}u. Hence, we obtain H_{μ,SL} ⊆ H_{μ} = H_{μ,F} and therefore the equality of all three operators.

4 Estimates on the difference of two solutions

Throughout this section let μ_{1}, μ_{2} ∈ M_{loc,unif}(ℝ), and let u_{1}, u_{2} be solutions to

H_{μ_{1}}u_{1} = 0 and H_{μ_{2}}u_{2} = 0,

respectively. We set

v := u_{1} − u_{2} and ν := μ_{1} − μ_{2},

and we aim for an estimate on v on an interval I in terms of ∥ν∥_{I}.

We first recall the definition and properties of the transfer matrices associated with a measure μ.

Remark 4.1. Let μ ∈ M_{loc,unif}(ℝ), and let u be a solution of the equation H_{μ}u = 0. Fix s ∈ ℝ. By Lemma 3.2, u is uniquely determined by (u(s), u′(s^{+})) (see also [1, Theorem 2.3]). Thus, for t ∈ ℝ we can define the transfer matrix mapping the solution of H_{μ}u = 0 at s to the solution at t, i.e.,

T_{μ}(t, s): (\begin{matrix}u(s) \\ u′(s^{+}) \end{matrix}) ↦ (\begin{matrix}u(t) \\ u′(t^{+}) \end{matrix}) .

Note that T_{μ}(t, t) = (\begin{matrix}1 & 0 \\ 0 & 1 \end{matrix}) and T_{μ}(t, r) = T_{μ}(t, s)T_{μ}(s, r) for all r, s, t ∈ ℝ.

Let u_{N,μ}(·, s), u_{D,μ}(·, s) be the solutions of H_{μ}u = 0 satisfying Neumann and Dirichlet boundary conditions at s, respectively, i.e.,

u_{N,μ}(s, s) = 1,  u_{D}(s, s) = 0,
∂_{1}u_{N,μ}(s^{+}, s) = 0, ∂_{1}u_{D,μ}(s^{+}, s) = 1.

Then

T_{μ}(t, s) = (\begin{matrix}u_{N,μ}(t, s) & u_{D,μ}(t, s) \\ ∂_{1}u_{N,μ}(t^{+}, s) & ∂_{1}u_{D,μ}(t^{+}, s) \end{matrix})

and det T_{μ}(t, s) = 1 for all t ∈ ℝ, cf. [1, Proposition 2.5] and also [7, Remark 2.7]. It follows that

T_{μ}(s, t) = T_{μ}(t, s)^{−1} = (\begin{matrix}∂_{1}u_{D,μ}(t^{+}, s) & −u_{D,μ}(t, s) \\ −∂_{1}u_{N,μ}(t^{+}, s) & u_{N,μ}(t, s) \end{matrix}) ,  (2)

in particular, u_{D,μ}(s, t) = −u_{D,μ}(t, s) for all t ∈ ℝ. As a shorthand notation we will also write

u_{D,μ}(t) := u_{D,μ}(t, 0), u_{N,μ}(t) := u_{N,μ}(t, 0).

Next we provide a variation of constants type formula.

Lemma 4.2. Let s, t ∈ ℝ. Then

(\begin{matrix}v(t) \\ v′(t^{+}) \end{matrix}) = T_{μ_{1}}(t, s) (\begin{matrix}v(s) \\ v′(s^{+}) \end{matrix}) + ∫_{s}^{t} T_{μ_{1}}(t, r) (\begin{matrix}0 \\ u_{2}(r) \end{matrix}) dν(r).

Proof. (i) We will use the following integration by parts type formula: let μ ∈ M_{loc,unif}(ℝ), u a solution of H_{μ}u = 0 and ũ ∈ W_{1,loc}^{1}(ℝ). Then

∫_{s}^{t} ũ(r)u(r) dμ(r) = ũ(t)u′(t^{+}) − ũ(s)u′(s^{+}) − ∫_{s}^{t} ũ′(r)u′(r) dr,

see [8, Lemma 1.3.3] (note that there ∫_{s}^{t} = ∫_{[s,t]} for s < t, so the boundary values slightly change).

(ii) Without loss of generality let s = 0. We are going to show that

(\begin{matrix}u_{2}(0) \\ u′_{2}(0^{+}) \end{matrix}) − T_{μ_{1}}(t, 0)^{−1} (\begin{matrix}u_{2}(t) \\ u′_{2}(t^{+}) \end{matrix}) = ∫_{0}^{t} T_{μ_{1}}(r, 0)^{−1} (\begin{matrix}0 \\ u_{2}(r) \end{matrix}) dν(r).

Multiplying this identity by T_{μ_{1}}(t, 0) yields the assertion since

T_{μ_{1}}(t, 0) (\begin{matrix}v(0) \\ v′(0^{+}) \end{matrix}) = (\begin{matrix}u_{1}(t) \\ u′_{1}(t^{+}) \end{matrix}) − T_{μ_{1}}(t, 0) (\begin{matrix}u_{2}(0) \\ u′_{2}(0^{+}) \end{matrix})

and T_{μ_{1}}(t, 0)T_{μ_{1}}(r, 0)^{−1} = T_{μ_{1}}(t, r).

(iii) Using (2) we compute

∫_{0}^{t} T_{μ_{1}}(r, 0)^{−1} (\begin{matrix}0 \\ u_{2}(r) \end{matrix}) dν(r) = (\begin{matrix}−∫_{0}^{t} u_{D,μ_{1}}(r)u_{2}(r) dν(r) \\ ∫_{0}^{t} u_{N,μ_{1}}(r)u_{2}(r) dν(r) \end{matrix}) .

Integrating by parts as in (i) we obtain

(\begin{matrix}−∫_{0}^{t} u_{D,μ_{1}}u_{2} dν \\ ∫_{0}^{t} u_{N,μ_{1}}u_{2} dν \end{matrix}) = (\begin{matrix}−∫_{0}^{t} u_{2}u_{D,μ_{1}} dμ_{1} + ∫_{0}^{t} u_{D,μ_{1}}u_{2} dμ_{2} \\ ∫_{0}^{t} u_{2}u_{N,μ_{1}} dμ_{1} − ∫_{0}^{t} u_{N,μ_{1}}u_{2} dμ_{2} \end{matrix})

= (\begin{matrix}−(u_{2}(t)u′_{D,μ_{1}}(t^{+}) − u_{2}(0)) + (u_{D,μ_{1}}(t)u′_{2}(t^{+}) − 0) \\ (u_{2}(t)u′_{N,μ_{1}}(t^{+}) − 0) − (u_{N,μ_{1}}(t)u′_{2}(t^{+}) − u′_{2}(0^{+})) \end{matrix})

= (\begin{matrix}u_{2}(0) \\ u′_{2}(0^{+}) \end{matrix}) − (\begin{matrix}u′_{D,μ_{1}}(t^{+}) & −u_{D,μ_{1}}(t) \\ −u′_{N,μ_{1}}(t^{+}) & u_{N,μ_{1}}(t) \end{matrix}) (\begin{matrix}u_{2}(t) \\ u′_{2}(t^{+}) \end{matrix}) ,

which by (2) completes the proof.

Lemma 4.3. Let c ∈ ℝ. Suppose that u_{2}(0) = u_{1}(0), u′_{2}(0^{+}) = u′_{1}(0^{+}) + cu_{1}(0). Then

v(t) = ∫_{0}^{t} \frac{d}{ds}(u_{D,μ_{1}}(t, s)u_{2}(s)) · (c − φ_{ν}(s)) ds (t ∈ ℝ).

Proof. Let t ∈ ℝ. Note that v(0) = 0, v′(0^{+}) = −cu_{2}(0). By Lemma 4.2 we thus obtain

v(t) = −u_{D,μ_{1}}(t)cu_{2}(0) + ∫_{0}^{t} u_{D,μ_{1}}(t, r)u_{2}(r) dν(r).

Since u_{D,μ_{1}}(t, t) = 0, we have

u_{D,μ_{1}}(t, r)u_{2}(r) = − ∫_{r}^{t} \frac{d}{ds}(u_{D,μ_{1}}(t, s)u_{2}(s)) ds.

By Fubini’s theorem it follows from the above two identities that

v(t) = −u_{D,μ_{1}}(t)cu_{2}(0) − ∫_{0}^{t}∫_{0}^{s} dν(r) \frac{d}{ds}(u_{D,μ_{1}}(t, s)u_{2}(s)) ds
= ∫_{0}^{t}(c − φ_{ν}(s)) \frac{d}{ds}(u_{D,μ_{1}}(t, s)u_{2}(s)) ds.

Now we can prove the desired estimate on v = u_{1} − u_{2} in terms of weak seminorms of ν = μ_{1} − μ_{2}.

Proposition 4.4. Suppose that u_{2}(0) = u_{1}(0), u′_{2}(0^{+}) = u′_{1}(0^{+}) + c_{ν,0}u_{1}(0), where c_{ν,0} is as in (1). Let α, β ∈ ℤ, α ⩽ −1, β ⩾ 1. Let c, ω > 0 such that

|∂_{1}u_{D,μ_{1}}(t^{+}, s)| ⩽ ce^{ω|t−s|}  (s, t ∈ ℝ).  (3)

Then

|v(t)| ⩽ Cce^{ω|t|}∥u_{2}|_{[α,β]}∥_{∞}∥ν∥_{[α,β]} (t ∈ [α, β]),

with a constant C > 0 depending only on ω and ∥μ_{2}∥_{unif}.

Observe that (3) is always satisfied with ω = ∥μ_{1}∥_{unif} + 1 and c = e^{ω}; this follows from Lemma 3.2 since u_{D,μ_{1}}(s, s) = 0 and ∂_{1}u_{D,μ_{1}}(s^{+}, s) = 1 for all s ∈ ℝ.

Proof. From u_{D,μ_{1}}(s, s) = 0 and the assumed estimate on ∂_{1}u_{D,μ_{1}} we derive |u_{D,μ_{1}}(t, s)| ⩽ \frac{c}{ω}e^{ω|t−s|} for all s, t ∈ ℝ. Moreover, by Lemma 3.4 we have ∥u′_{2}|_{[α,β]}∥_{∞} ⩽ (∥μ_{2}∥_{unif} + 2)∥u_{2}|_{[α,β]}∥_{∞}. Recalling from (2) that u_{D,μ_{1}}(t, s) = −u_{D,μ_{1}}(s, t), we thus obtain

\begin{aligned}|\frac{d^{+}}{ds} (u_{D,μ_{1}}(t, s)u_{2}(s))| & = |−∂_{1}u_{D,μ_{1}}(s^{+}, t)u_{2}(s) + u_{D,μ_{1}}(t, s)u′_{2}(s^{+})| \\ & ⩽ C_{0}ce^{ω|t−s|}∥u_{2}|_{[α,β]}∥_{∞} \end{aligned} (4)

for all s, t ∈ [α, β], with C_{0} = 1 + \frac{1}{ω}(∥μ_{2}∥_{unif} + 2).

Let t ∈ [0, β]. By Lemma 4.3 we have

|v(t)| ⩽ ∫_{0}^{t} |\frac{d^{+}}{ds} (u_{D,μ_{1}}(t, s)u_{2}(s))| · |φ_{ν}(s) − c_{ν,0}| ds.

By (4) and Lemma 2.9 we conclude that

|v(t)| ⩽ C_{0}c∥u_{2}|_{[α,β]}∥_{∞} ∑_{k=1}^{β} ∫_{k−1}^{k} e^{ω(t−s)}|φ_{ν}(s) − c_{ν,0}| ds

⩽ C_{0}c∥u_{2}|_{[α,β]}∥_{∞} ∑_{k=1}^{β} e^{ω(t+1−k)} · 2k∥ν∥_{[α,β]} ⩽ Cc∥u_{2}|_{[α,β]}∥_{∞}e^{ωt}∥ν∥_{[α,β]},

with C := C_{0} ∑_{k=1}^{∞} 2ke^{−ω(k−1)}. The proof in the case t ∈ [α, 0) is analogous.

Using the previous result we can sharpen the estimate in Lemma 3.2.

Proposition 4.5. Let μ ∈ M_{loc,unif}(ℝ) and let u be a solution of the equation H_{μ}u = 0. Then for t ∈ ℝ one has

(∥μ∥_{unif}|u(t)|^{2} + |u′(t^{+})|^{2})^{1/2} ⩽ (∥μ∥_{unif}|u(0)|^{2} + |u′(0^{+})|^{2})^{1/2}e^{∥μ∥_{unif}^{1/2}(|t|+1/2)}.

Proof. The case μ = 0 is trivial, since then u is affine. So, let μ ≠ 0.

(i) First, assume that μ = ρλ with a density ρ ∈ C(ℝ). Then u ∈ C^{2}(ℝ), u′′ = ρu. Let ω > 0. Then for φ(t) := ω^{2}|u(t)|^{2} + |u′(t)|^{2} we obtain

|φ′(t)| = |2 Re((ω^{2} + ρ)u(t)\overline{u′(t)})|
⩽ (ω^{2} + |ρ|)(ω|u(t)|^{2} + \frac{1}{ω}|u′(t)|^{2}) = (ω + \frac{1}{ω}|ρ|)φ(t).

It follows that φ(t) ⩽ φ(s) exp(ω(t − s) + \frac{1}{ω} ∫_{s}^{t} |ρ(r)| dr) and hence

ω^{2}|u(t)|^{2} + |u′(t)|^{2} ⩽ (ω^{2}|u(s)|^{2} + |u′(s)|^{2}) exp(ω(t − s) + \frac{1}{ω}|μ|([s, t]))

for all s, t ∈ ℝ, s < t.

(ii) By Proposition 2.5 there exists a sequence (μ_{n})_{n∈ℕ} in M_{loc,unif}(ℝ) such that μ_{n} has a smooth density and ∥μ_{n}∥_{unif} ⩽ ∥μ∥_{unif} for all n ∈ ℕ, ∥μ_{n} − μ∥_{ℝ} → 0 and lim sup_{n→∞} |μ_{n}|(I) ⩽ |μ|(I) for all compact intervals I ⊆ ℝ. Lemma 2.4 implies μ_{n} → μ vaguely and, hence, 𝟙_{[a,b]}μ_{n} → 𝟙_{[a,b]}μ weakly for all a, b ∈ ℝ such that μ({a}) = μ({b}) = 0. (Note that μ has at most countable many atoms.)

(iii) For n ∈ ℕ let u_{n} be the solution of H_{μ_{n}}u_{n} = 0 such that u_{n}(0) = u(0) and u′_{n}(0^{+}) = u′(0^{+}) + c_{μ−μ_{n},0}u(0). Then (u_{n}) is uniformly bounded on any compact interval by Lemma 3.2 and Remark 2.8(a), and Proposition 4.4 implies that u_{n} → u locally uniformly. Let s, t ∈ ℝ such that μ({s}) = μ({t}) = 0. Then

u′_{n}(t) − u′_{n}(s) = ∫_{s}^{t} u_{n}(r) dμ_{n}(r) → ∫_{s}^{t} u(r) dμ(r) = u′(t^{+}) − u′(s^{+})

as n → ∞ since u_{n} → u uniformly on [s, t] and 𝟙_{[s,t]}μ_{n} → 𝟙_{[s,t]}μ weakly. It follows from Lemma 3.4 that (u′_{n}) is uniformly bounded on [0, 1]. Thus, integrating s from 0 to 1 we conclude by Lebesgue’s dominated convergence theorem that

u′_{n}(t) − (u_{n}(1) − u_{n}(0)) → u′(t^{+}) − (u(1) − u(0))

and hence u′_{n}(t) → u′(t^{+}).

(iv) Let t > s > 0 such that μ({s}) = μ({t}) = 0. By (i) we obtain

ω^{2}|u_{n}(t)|^{2} + |u′_{n}(t)|^{2} ⩽ (ω^{2}|u_{n}(s)|^{2} + |u′_{n}(s)|^{2}) exp(ω(t − s) + \frac{1}{ω}|μ_{n}|([s, t]))

for all n ∈ ℕ. For n → ∞ we infer, using part (iii) and the estimate lim sup |μ_{n}|([s, t]) ⩽ |μ|([s, t]), that

ω^{2}|u(t)|^{2} + |u′(t^{+})|^{2} ⩽ (ω^{2}|u(s)|^{2} + |u′(s^{+})|^{2}) exp(ω(t − s) + \frac{1}{ω}|μ|([s, t])).

Finally, let t > 0. Then there exist sequences (s_{n}) in [0, t) and (t_{n}) in [t, ∞) such that s_{n} → 0, t_{n} → t and μ({s_{n}}) = μ({t_{n}}) = 0 for all n ∈ ℕ, and from the above we deduce that

ω^{2}|u(t)|^{2} + |u′(t^{+})|^{2} ⩽ (ω^{2}|u(0)|^{2} + |u′(0^{+})|^{2}) exp(ωt + \frac{1}{ω}|μ|((0, t])).

Choosing ω = ∥μ∥_{unif}^{1/2} yields the assertion for t ⩾ 0 since then |μ|((0, t]) ⩽ ω^{2}(t + 1). The case t < 0 is proved analogously.

5 Gordon’s Theorem

We are now in the position to formulate the weak Gordon condition and to prove our main theorem. We will then apply the result to quasiperiodic measures.

Definition. Let μ ∈ M_{loc,unif}(ℝ) and C > 0. We say that μ is a weak Gordon measure with weight C if there exists a sequence (p_{m}) in (0, ∞) with p_{m} → ∞ such that

lim_{m→∞} e^{Cp_{m}}∥μ − μ(· + p_{m})∥_{[−p_{m},p_{m}]} = 0.

The following lemma relates our notion of weak Gordon measures with previous versions for Gordon potentials/measures (see [5, 2, 7]).

Lemma 5.1. Let μ ∈ M_{loc,unif}(ℝ), C > 0. Then the following are equivalent: (i) μ is a weak Gordon measure with weight C. (ii) There exist sequences (p_{m}) in (0, ∞) and (μ_{m}) in M_{loc,unif}(ℝ) such that

μ_{m} is periodic with period p_{m} (m ∈ ℕ), p_{m} → ∞ and

e^{Cp_{m}}∥μ − μ_{m}∥_{[−p_{m},2p_{m}]} → 0 (m → ∞).

Moreover, the measures μ_{m} in (ii) can be chosen such that

𝟙_{[a_{m},p_{m}−a_{m}]}μ_{m} = 𝟙_{[a_{m},p_{m}−a_{m}]}μ, ∥μ_{m}∥_{unif} ⩽ (1 + \frac{1}{2a_{m}} )∥μ∥_{unif}

for all m ∈ ℕ, with 0 < a_{m} ⩽ \frac{p_{m}}{2} and inf_{m∈ℕ} a_{m} > 0.

Proof. (i) ⇒ (ii): Choose a sequence (p_{m}) as in the above definition, and let (a_{m}) be as in the additional statement of the lemma. Let m ∈ ℕ, and let ψ_{m}: ℝ → ℝ be the piecewise affine function with spt ψ_{m} = [−a_{m}, p_{m} + a_{m}], ψ_{m} = 1 on [a_{m}, p_{m} − a_{m}] and ∥ψ′∥_{∞} = \frac{1}{2a_{m}}. Then ∑_{k∈ℤ} ψ_{m}(· + kp_{m}) = 𝟙_{ℝ}, the measure

μ_{m} := ∑_{k∈ℤ}(ψ_{m}μ)(· + kp_{m})

is p_{m}-periodic and 𝟙_{[a_{m},p_{m}−a_{m}]}μ_{m} = 𝟙_{[a_{m},p_{m}−a_{m}]}μ.

With I_{m} := [−p_{m}, 2p_{m}] we obtain

∥μ − μ_{m}∥_{I_{m}} ⩽ ∑_{k∈ℤ} ∥ψ_{m}(· + kp_{m})(μ − μ(· + kp_{m}))∥_{I_{m}}

⩽ (1 + \frac{1}{2a_{m}} ) ∑_{k∈ℤ} ∥μ − μ(· + kp_{m})∥_{I_{m}∩[−a_{m}−kp_{m},p_{m}+a_{m}−kp_{m}]}

by Lemma 2.6. For |k| > 2 the set I_{m} ∩[−a_{m} −kp_{m}, p_{m} +a_{m} −kp_{m}] is empty; for |k| ⩽ 2 one easily shows the estimate

∥μ − μ(· + kp_{m})∥_{I_{m}∩[−a_{m}−kp_{m},p_{m}+a_{m}−kp_{m}]} ⩽ |k| · ∥μ − μ(· + p_{m})∥_{[−p_{m},p_{m}]},

where for |k| = 2 one has to use the triangle inequality. It follows that ∥μ − μ_{m}∥_{I_{m}} ⩽ 6(1 + \frac{1}{2a_{m}} )∥μ − μ(· + p_{m})∥_{[−p_{m},p_{m}]}, which implies (ii).

Finally, assume without loss of generality that p_{m} ⩾ 2 for all m ∈ ℕ. Then for all a ∈ ℝ we have

|μ_{m}|((a, a + 1]) ⩽ ∑_{k∈ℤ} ∥ψ_{m}(· + kp_{m})|_{(a,a+1]}∥_{∞}∥μ∥_{unif} ⩽ (1 + \frac{1}{2a_{m}} )∥μ∥_{unif},

due to the particular construction of ψ_{m} and the fact that at most 3 summands in the second term are nonzero.

(ii) ⇒ (i): Since μ_{m} is p_{m}-periodic, we have

∥μ − μ(· + p_{m})∥_{[−p_{m},p_{m}]}

⩽ ∥μ − μ_{m}∥_{[−p_{m},p_{m}]} + ∥μ_{m}(· + p_{m}) − μ(· + p_{m})∥_{[−p_{m},p_{m}]}
⩽ 2∥μ − μ_{m}∥_{[−p_{m},2p_{m}]},

and the assertion follows.

Remark 5.2. The above lemma shows that a weak Gordon measure can (locally on three periods) be very well approximated by periodic measures (where the approximation depends on C). In [7] a local measure was called Gordon measure if the condition of Lemma 5.1(ii) is satisfied for all C > 0, where ∥·∥_{[−p_{m},2p_{m}]} is replaced by the total variation norm on [−p_{m}, 2p_{m}].

The definition in [7] extends the previous definitions for bounded ([5]) and locally integrable potentials ([2]), so our notion generalizes and refines the Gordon condition from those papers.

We will need the following estimate for solutions to periodic measures.

Lemma 5.3 (see [7, Lemma 2.8]). Let μ be p-periodic and z ∈ ℂ. Let u be a solution of H_{μ}u = zu. Then

∥(\begin{matrix}u(0) \\ u′(0^{+}) \end{matrix})∥ ⩽ 2 max\{∥(\begin{matrix}u(t) \\ u′(t^{+}) \end{matrix})∥; t ∈ \{−p, p, 2p\}\}.

The following is our main result on absence of eigenvalues.

Theorem 5.4. Let μ ∈ M_{loc,unif}(ℝ) be a weak Gordon measure with weight C > 0. Then H_{μ} does not have any eigenvalues with modulus less than C^{2} − ∥μ∥_{unif}.

Proof. Let (p_{m}) and (μ_{m}) be as in Lemma 5.1(ii), without loss of generality p_{m} ⩾ 2 for all m ∈ ℕ. Let the additional statement of the lemma be satisfied with a sequence (a_{m}) satisfying p_{m} + a_{m} ∈ ℕ for all m ∈ ℕ, a_{m} → ∞ and \frac{a_{m}}{p_{m}} → 0.

Assume that z ∈ ℂ with |z| < C^{2} − ∥μ∥_{unif} is an eigenvalue of H_{μ}. Let u be a corresponding eigenfunction; then u ∈ W_{2}^{1}(ℝ) ⊆ C_{0}(ℝ).

For m ∈ ℕ let u_{m} be the solution of H_{μ_{m}}u_{m} = zu_{m} satisfying u_{m}(a_{m}) = u(a_{m}), u′_{m}(a_{m}^{+}) = u′(a_{m}^{+}). Then u_{m} = u on [a_{m}, p_{m} − a_{m}] since μ_{m} = μ on this interval. We now apply Proposition 4.4 with ν = (μ_{m} − zλ) − (μ − zλ). Note that 𝟙_{[a_{m},a_{m}+2]}ν = 0, so one has c_{ν,a_{m}+1} = 0 in (1). Setting ω_{m} := ∥μ_{m} − zλ∥_{unif}^{1/2} and taking into account Proposition 4.5, we obtain

|u(t) − u_{m}(t)| ⩽ C_{m}e^{ω_{m}|t−(a_{m}+1)|}∥μ − μ_{m}∥_{[−p_{m},a_{m}+1]} (m ∈ ℕ, t ∈ [−p_{m}, a_{m}])

with C_{m} > 0 depending only on ω_{m}, and similarly for t ∈ [p_{m} − a_{m}, 2p_{m}]. Therefore,

sup_{t∈[−p_{m},2p_{m}]} |(u − u_{m})(t)| ⩽ C_{m}e^{ω_{m}(p_{m}+a_{m}+1)}∥μ − μ_{m}∥_{[−p_{m},2p_{m}]}.  (5)

By Lemma 5.1 and the assumption on z we have

ω_{m}^{2} ⩽ ∥μ_{m}∥_{unif} + |z| ⩽ (1 + \frac{1}{2a_{m}} )∥μ∥_{unif} + |z| ⟶ ∥μ∥_{unif} + |z| < C^{2}.

Hence, for large m we obtain

ω_{m}(p_{m} + a_{m} + 1) = ω_{m}p_{m}(1 + \frac{a_{m}+1}{p_{m}}) ⩽ Cp_{m}.

We conclude that the right hand side in (5) tends to zero as m → ∞. Therefore, given ε > 0 there exists m_{0} ∈ ℕ such that |u(t) − u_{m}(t)| ⩽ ε for all m ⩾ m_{0} and t ∈ [−p_{m}, 2p_{m}]. Since u ∈ C_{0}(ℝ), we can choose m_{0} such that |u(t)| ⩽ ε for |t| ⩾ p_{m_{0}} − 1 =: t_{0}. Then |u_{m}| ⩽ 2ε on [−p_{m}, 2p_{m}] \ (−t_{0}, t_{0}). By Lemma 3.4 it follows that |u′_{m}| ⩽ 2ε(2 + ∥μ_{m}∥_{unif}) on that set.

Thus we have shown

(u_{m}(±p_{m}), u′_{m}(±p_{m}^{+})), (u_{m}(2p_{m}), u′_{m}(2p_{m}^{+})) → 0 (m → ∞).

By Lemma 5.3 we conclude that (u_{m}(0), u′_{m}(0^{+})) → 0 as m → ∞. For any compact interval I ⊆ ℝ, Lemma 3.2 yields u_{m} → 0 uniformly on I. Since u_{m} → u uniformly on I by (5), we obtain u = 0, a contradiction.

As an immediate consequence of Theorem 5.4 we obtain the following version of Gordon’s theorem.

Corollary 5.5. Let μ be a weak Gordon measure with weight C for all C > 0. Then H_{μ} does not have any eigenvalues.

By a scaling argument we can improve the eigenvalue bound C^{2} − ∥μ∥_{unif} from Theorem 5.4.

Corollary 5.6. Let μ ∈ M_{loc,unif}(ℝ) be a weak Gordon measure with weight C > 0, and let r > 0. Then H_{μ} does not have any eigenvalues with modulus less than C^{2} − ∥μ∥_{unif,r}, where

∥μ∥_{unif,r} := \frac{1}{r} sup_{a∈ℝ} |μ|((a, a + r]).

(Note that ∥μ∥_{unif,1} = ∥μ∥_{unif}.)

Proof. Let μ_{r} := rμ(r·), and define B_{r}: L_{2}(ℝ) → L_{2}(ℝ) by B_{r}u := u(\frac{·}{r}). A straightforward computation shows that B_{r}^{−1}H_{μ}B_{r} = r^{−2}H_{μ_{r}}.

We now show that μ_{r} is a weak Gordon measure with weight Cr. Let (p_{m}) be a sequence of periods for the weak Gordon measure μ. For u ∈ W_{∞}^{1}(ℝ) with spt u ⊆ [−\frac{p_{m}}{r} , \frac{p_{m}}{r}], diam spt u ⩽ 2 and ∥u′∥_{∞} ⩽ 1 we have

∫_{−p_{m}/r}^{p_{m}/r} u d(μ_{r} − μ_{r}(· + \frac{p_{m}}{r})) = ∫_{−p_{m}}^{p_{m}} ru(\frac{·}{r}) d(μ − μ(· + p_{m})).

Since u_{r} := ru(\frac{·}{r}) satisfies diam spt u_{r} ⩽ 2r and ∥u′_{r}∥_{∞} ⩽ 1, we infer by Remark 2.2 that

∥μ_{r} − μ_{r}(· + \frac{p_{m}}{r})∥_{[−p_{m}/r,p_{m}/r]} ⩽ (r + 1)^{2}∥μ − μ(· + p_{m})∥_{[−p_{m},p_{m}]}.

Multiplying by exp(Cr\frac{p_{m}}{r}) we deduce that μ_{r} is indeed a weak Gordon measure with weight Cr. Therefore, by Theorem 5.4, H_{μ_{r}} has no eigenvalues z with |z| < (Cr)^{2} − ∥μ_{r}∥_{unif}.

To conclude the assertion for H_{μ} = r^{−2}B_{r}H_{μ_{r}}B_{r}^{−1}, it remains to observe that ∥μ_{r}∥_{unif} = r^{2}∥μ∥_{unif,r}.

Remark 5.7. (a) The supremum of all C > 0 such that μ is a weak Gordon measure with weight C is given by

C_{μ} := − lim inf_{p→∞} \frac{1}{p} ln ∥μ − μ(· + p)∥_{[−p,p]}

if C_{μ} > 0.

(b) Assume that C_{μ} > 0. Then by part (a) and Corollary 5.6, H_{μ} does not have any eigenvalues z with

|z| < E_{μ} := C_{μ}^{2} − inf_{r>0} ∥μ∥_{unif,r}.

It is easy to see that ∥μ∥_{unif,r} ⩽ (1 + \frac{s}{r})∥μ∥_{unif,s} for all r > s > 0. Thus, lim_{r→∞} ∥μ∥_{unif,r} = inf_{r>0} ∥μ∥_{unif,r}.

In the next section we will show that −E_{μ} can occur as an eigenvalue, so the above bound is sharp.

(c) In the proof of Theorem 5.4 we have shown that H_{μ}u = zu does not have any solutions in C_{0}(ℝ), for z with small modulus. Hence, we also obtain absence of eigenvalues for H_{μ} considered as an operator in L_{p}(ℝ) with 1 ⩽ p < ∞.

We now apply our main result to quasiperiodic measures.

Example 5.8. Let μ_{1}, μ_{2} ∈ M_{loc,unif}(ℝ) be 1- and α-periodic, respectively, where α > 0 is irrational. Then μ := μ_{1} + μ_{2} is a quasiperiodic measure. Under a suitable assumption on α we are going to prove that H_{μ} has no eigenvalues. Analogous results have been shown in [2] under an additional assumption on the oscillation of V_{2}, where μ_{2} = V_{2}λ, and in [7] under a Hölder condition on the translates of μ_{2}. Using our weak Gordon condition, we can remove these additional assumptions on μ_{2} altogether.

As in [2, 7] we assume that there exist B > 0 and a sequence (\frac{p_{m}}{q_{m}}) in ℚ such that

|α − \frac{p_{m}}{q_{m}} | ⩽ Bm^{−q_{m}} (m ∈ ℕ);

in particular, α is a Liouville number. As for the Liouville numbers one sees that the set of all numbers α with the above property is a dense G_{δ} set.

Using the periodicity of μ_{1}, μ_{2} and Remark 2.3(b), we now show that μ is a weak Gordon measure with weight C, for all C > 0:

∥μ − μ(· + p_{m})∥_{[−p_{m},p_{m}]} = ∥μ_{2} − μ_{2}(· + p_{m} − αq_{m})∥_{[−p_{m},p_{m}]}

⩽ 3|p_{m} − αq_{m}|∥μ_{2}∥_{unif} = 3q_{m}|\frac{p_{m}}{q_{m}} − α|∥μ_{2}∥_{unif} and hence

e^{Cp_{m}}∥μ − μ(· + p_{m})∥_{[−p_{m},p_{m}]} ⩽ exp(Cq_{m} \frac{p_{m}}{q_{m}} ) · 3q_{m}Bm^{−q_{m}}∥μ_{2}∥_{unif} → 0

as m → ∞. By Corollary 5.5 it follows that H_{μ} does not have any eigenvalues.

6 Sharpness of the condition

In this section we show that our results are sharp in the following sense: we construct a measure μ ∈ M_{loc,unif}(ℝ) such that the operator H_{μ} has −1 as an eigenvalue, μ is a weak Gordon measure with weight C for all C < 1, and inf_{r>0} ∥μ∥_{unif,r} = 0. With the notation of Remark 5.7 we then have C_{μ} = 1 and E_{μ} = 1.

The construction will be such that the Gordon condition is satisfied with arbitrarily many periods, not just with three periods. In fact, we construct a symmetric eigenfunction to the eigenvalue −1 in the following way. We first fix u on an infinite discrete set T ⊆ ℝ and then choose the continuous continuation to ℝ satisfying u′′ = u on ℝ \ T. We will determine T and also u on T recursively.

Choose a sequence (l_{m})_{m∈ℕ} in (0, ∞) satisfying l_{m} → ∞. Set p_{0} := 0, T_{0} := {0} and u(0) := 1. Recursively we define, for m ∈ ℕ,

p_{m} := 2(m − 1)p_{m−1} + l_{m}
T_{m} := {jp_{m} + t; |j| ⩽ m, t ∈ T_{m−1}} ∩ [−mp_{m}, mp_{m}],
u(jp_{m} + t) := 2^{−m|j|}u(t) (jp_{m} + t ∈ T_{m} \ T_{m−1}).

Let T := ⋃_{m∈ℕ_{0}} T_{m}. Then u is fixed on the discrete set T. Consider the unique continuous continuation u: ℝ → (0, ∞) satisfying u′′ = u on ℝ \ T. It is easy to see that u ∈ L_{2}(ℝ). Moreover, note that the derivative of u has jumps precisely at T. Thus, u solves −u′′ + μu = −u for some pure-point measure μ with spt μ = T.

We now assume that

\frac{2(m − 1)p_{m−1} + m ln 2}{l_{m}} → 0 (m → ∞).  (6)

Since then 2^{m} ⩽ e^{l_{m}} for large m, a straightforward computation yields boundedness of the masses of single points (see also Remark 6.2 below). Furthermore, T = spt μ is uniformly discrete (i.e., there exists a minimal distance

\begin{matrix}−p_{1} \\ p_{1} \\ 1 \end{matrix}  \begin{matrix}−2p_{2} \\ −p_{2} \\ −p_{1} \\ p_{1} \\ p_{2} \\ 2p_{2} \\ 1 \end{matrix}

Figure 1: u defined on the intervals corresponding to T_{1} and to T_{2}.

between any two points in T). These two facts imply μ ∈ M_{loc,unif}(ℝ). Hence, −1 is an eigenvalue of H_{μ} (with eigenfunction u).

We will now investigate μ and show that μ is indeed a weak Gordon measure with weight C for all C < 1. Let s_{m} := (m − 1)p_{m−1} and t_{m} := s_{m} + l_{m} = p_{m} − s_{m}. By construction, the measures

𝟙_{(−mp_{m},(m−1)p_{m})}μ and 𝟙_{(−mp_{m},(m−1)p_{m})}(μ(· + p_{m}))

differ only at the points −s_{m} and −t_{m}, and there the corresponding point masses are interchanged. (It suffices to consider the open intervals here since the test functions for computing the ∥·∥_{I}-norm are zero at the boundary.) With the help of the next lemma we can compute the difference μ({−s_{m}}) − μ({−t_{m}}).

Lemma 6.1. Let l, L > 0, and let a, b, c, d > 0 satisfy \frac{a}{b} = \frac{d}{c}. Let u ∈ C[−l, L+l] such that u(−l) = a, u(0) = b, u(L) = c and u(L+l) = d. Assume that u is twice continuously differentiable on (−l, L + l) \ {0, L}, with u′′ = u. Then the measure μ defined by −u′′ + μu = −u satisfies spt μ ⊆ {0, L} and

μ(\{0\}) − μ(\{L\}) = \frac{2\frac{c}{b} − 2\frac{b}{c}}{e^{L} − e^{−L}} .

Proof. First observe that \frac{1}{b}u(−t) = \frac{1}{c}u(L + t) holds for t = 0 and for t = l, and hence for all t ∈ [−l, 0] by uniqueness of the boundary value problem for u′′ = u. It follows that \frac{u′(0^{−})}{u(0)} + \frac{u′(L^{+})}{u(L)} = 0. Therefore,

\begin{aligned}μ(\{0\}) − μ(\{L\}) & = \frac{u′(0^{+}) − u′(0^{−})}{u(0)} − \frac{u′(L^{+}) − u′(L^{−})}{u(L)} \\ & = \frac{1}{b} u′(0^{+}) + \frac{1}{c} u′(L^{−}). \end{aligned}  (7)

A straightforward computation yields

u(t) = \frac{c − e^{−L}b}{e^{L} − e^{−L}} e^{t} + \frac{e^{L}b − c}{e^{L} − e^{−L}} e^{−t}  (8)

for all t ∈ [0, L], so from (7) we deduce that

μ(\{0\}) − μ(\{L\}) = \frac{1}{b} · \frac{2c − (e^{L} + e^{−L})b}{e^{L} − e^{−L}} + \frac{1}{c} · \frac{(e^{L} + e^{−L})c − 2b}{e^{L} − e^{−L}}

= \frac{2\frac{c}{b} − 2\frac{b}{c}}{e^{L} − e^{−L}} .

Remark 6.2. If e^{−L} ⩽ \frac{c}{b} ⩽ e^{L}, then the two coefficients in the right hand side of (8) are non-negative, and this implies |\frac{u′(0^{+})}{u(0)} |, |\frac{u′(L^{−})}{u(L)} | ⩽ 1. (In the following we will have \frac{c}{b} = 2^{−m}.)

We now apply Lemma 6.1 with l = l_{1} and L = l_{m}, for m ⩾ 2. Note that u(t_{m}) = u(1p_{m} − s_{m}) = 2^{−m·1}u(−s_{m}) = 2^{−m}u(s_{m}). Therefore,

μ(\{s_{m}\}) − μ(\{t_{m}\}) = \frac{2 · 2^{−m} − 2 · 2^{m}}{e^{l_{m}} − e^{−l_{m}}} ,

and for m ⩾ 2 with l_{m} ⩾ 2 it follows that

∥μ − μ(· + p_{m})∥_{[−mp_{m},(m−1)p_{m}]} = \frac{2 · 2^{m} − 2 · 2^{−m}}{e^{l_{m}} − e^{−l_{m}}} ⩽ 4e^{m ln 2−l_{m}}.

For C < 1 we infer by (6) that

e^{Cp_{m}}∥μ − μ(· + p_{m})∥_{[−mp_{m},(m−1)p_{m}]} ⩽ 4e^{C2(m−1)p_{m−1}+m ln 2−(1−C)l_{m}} → 0.

In particular, μ is a weak Gordon measure with weight C, and hence C_{μ} ⩾ 1. Furthermore we have inf_{r>0} ∥μ∥_{unif,r} = 0 since μ ∈ M_{loc,unif}(ℝ) and

\frac{1}{r} sup_{a∈ℝ} #(spt μ ∩ (a, a + r]) → 0

as r → ∞, where # denotes the counting measure. Thus, E_{μ} = C_{μ}^{2} = 1.

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http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-102766

Christian Seifert and Hendrik Vogt Institut für Mathematik Technische Universität Hamburg-Harburg 21073 Hamburg, Germany

christian.sehugo@egon ifert@tu-darmstadt harburg.de
hendrik.vohugo@egon gt@tu-darmstadt harburg.de