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Besov spaces for Schrodinger operators with barrier potentials

Let H be a Schrodinger operator with barrier potential on the real line. We define the Besov spaces for H by developing the associated Littlewood-Paley theory. This theory depends on the decay estimates of the spectral operator in the high and low energies. We also prove a Mikhlin-Hormander type multiplier theorem on these spaces, including the Lp boundedness result. Our approach has potential applications to other Schrodinger operators with short-range potentials, as well as in higher dimensions.


Introduction
Let H = −△+V be a Schrödinger operator on R, where the potential V is real-valued and belongs to L 1 ∩ L 2 . H is the Hamiltonian in the corresponding time-dependent Schrödinger equation where the solution is uniquely determined by the initial state: ψ(t, x) = e −itH f (x), t ≥ 0.
In [21] Jensen and Nakamura introduced Besov spaces associated with H on R d and showed that e −itH maps B s+2β,q p (H) into B s,q p (H) if s ≥ 0, 1 ≤ p, q ≤ ∞ and β > d| 1 2 − 1 p |, under the condition that V = V + − V − so that V + ∈ K loc d and V − ∈ K d , K d being the Kato class. In this paper we generalize the definition of Besov spaces to α ∈ R, 0 < p, q < ∞ and show, in the case of barrier potential, that such a definition is independent of the choice of the dyadic system {Φ, ϕ j }.
For α ∈ R, 0 < p < ∞, 0 < q ≤ ∞, define the quasi-norm for f ∈ L 2 as The Besov spaces associated with H, denoted by B α,q p (H) is defined to be the completion of the subspace L 2 0 := {f ∈ L 2 : f B α,q p < ∞} of L 2 .
As in the Fourier case and the Hermite case [39,40,10,11] we address the Besov space theory associated with H by considering the Schrödinger operator H = −△ + V , where V = ε 2 χ [−1,1] , ε > 0 (called the barrier potential) is one of the simplest discontinuous potential models in quantum mechanics.
Peetre's maximal operator plays an important role in the theory of function spaces. In order to establish a Peetre type maximal inequality for H, we need the decay estimates of the kernel of ϕ j (H) as well as of its derivative. Based on an integral expression of this kernel we obtain the decay estimates by exploiting the analytic behavior of the eigenfunctions e(x, ξ) as ξ approaches ∞ (high energy) and 0 (low energy) in various cases. When the support of Φ contains the origin, we are in the so-called "local energy" case, which usually is harder to deal with for general potentials. We use certain "matching" method to put together integrals of the "same type", so that each of the resulting integrals is the Fourier transform of a Schwartz function. This method seems interesting and may have applications to other potentials.
Our first main result (Theorem 3.7) is an equivalence theorem for B α,q p (H), which tells that f φ B α,q p and f ψ B α,q p are equivalent quasinorms on B α,q p (H), where {φ j }, {ψ j } are two given smooth dyadic systems.
Using functional calculus, Jensen and Nakamura [21,22] obtained smooth multiplier results for general potentials. For barrier potential we prove a sharp spectral multiplier theorem on B α,q p (H) (Theorem 6.5 and Theorem 6.6).
The remaining part of the paper is organized as follows. In §2 we give explicit solutions to the eigenfunction equation. In §3 we give norm characterization of B α,q p (H) using Peetre type maximal functions. The proof is based on the decay estimates for the kernel of ϕ j (H). A detailed proof of these decay estimates are included in §4 and §5. In §6 we prove a Hörmander type multiplier theorem for H. In §7, we give identifications of these new spaces B α,q p (H) with the ordinary Besov spaces for certain range of parameters α, p, q.
Acknowledgment The authors would like to thank A. Jensen for his useful comments on the identification of Besov spaces.
These generalized eigenfunctions have a physical interpretation in quantum mechanics, where ξ 2 is viewed as a energy parameter; they represent the transmission and reflection waves when a particle passes through the potential. The coefficients T, R are called the transmission coefficient and the reflection coefficient (cf. [17], p.4179, also [14]). Under the condition that V is in L 1 ∩ L 2 , we show in [41,42] Then F is a unitary operator from H ac onto L 2 and its adjoint is given by for g ∈ L 2 Therefore ϕ(H)| Hac = F * ϕ(ξ 2 )F . If H has no point spectrum and all generalized eigenfunctions of H are uniformly bounded in x and ξ, then for f ∈ L 2 , A variant of the formula (6) can be found in [17] for short-range potentials defined as a measure. In 3D similar formula is used by Tao [36] in a scattering problem.
Since the barrier potential V = ε 2 χ [−1,1] is in L 1 ∩ L 2 and the eigenfunctions of H are uniformly bounded (see subsection 2.3), the formula (6) is valid for V . Note that the corresponding point spectrum is empty.
2.2. Dyadic system and Besov spaces The existence of such functions can be justified by consulting e.g., [16]. The almost orthogonal relation (iii) for the system ϕ j (ξ) := ϕ(2 −j ξ), ψ j (ξ) := ψ(2 −j ξ) allows us to give an expansion of f in L 2 as As in the Fourier case, let 0 < p, q ≤ ∞, α ∈ R we define the B α,q p quasi-norm as in (2) for f ∈ L 2 . Note that when 0 < p < 1 or 0 < q < 1, we can always define a metric d on B α,q p , so that the metric space (B α,q p , d) is topologically isomorphic to the quasi-normed space. In fact, Lemma 3.10.1 in [5] tells that every quasi-normed linear space is metrizable. 1] , also see, e.g., [14].
We also need decay estimates for the derivative of the kernel.
. a) If j > J, then for each n there is a constant C n : where N means the the same as in Lemma 3.1 (a). b) If −∞ < j ≤ J, then for each n there is a constant C n : Lemma 3.4. Let Φ be as in Lemma 3.2. Then for each n Proofs of Lemma 3.1−3.4 are given in §4 and §5, which are elementary calculus but quite lengthy. These lemmas are essential for us to establish a Peetre type maximal inequality.
Given s > 0 define the Peetre maximal functions for H by: if j > J, where the minimum is taken over 0 ≤ ℓ ≤ 2N and N = max{1, [s]+2 4 }. Similarly, define for j ≤ J, We have used the abbreviation ϕ * j f := ϕ * j,s f . Notice that (9) ϕ * j f (x) ≥ |ϕ j (H)f (x)|. In the following we slightly abuse the notation ϕ * 0 f = Φ * f , etc, in case of no confusion.
where the maximum is taken over 0 ≤ k ≤ 2N and both ±.
It follows that provided n − s > 1. Thus one may take n = [s] + 2.
For j ≤ J similarly we obtain the following inequalities, using Lemma 3.3(b) and Lemma 3.4 in place of Lemma 3.3(a), This proves Lemma 3.5. ✷ We are ready to show Peetre maximal inequality for H. Let M be the Hardy-Littlewood maximal operator: where the supreme is taken over all intervals I containing x. Lemma 3.6. Let 0 < r < ∞. There exists a constant C > 0 independent of 0 < ε ≤ 1 such that where ǫ > 0 can be chosen arbitrarily small.
Remark 1. It is well known that M is bounded on L p , 1 < p < ∞. Lemma 3.6 implies that if s = 1/r, then (10) ϕ * j f p ≤ c ϕ j (H)f p , 0 < p ≤ ∞ by taking ǫ small enough and 0 < r < p (s = 1/r > 1/p). Remark 2. For j ≤ J, the inequality in Lemma 3.6 takes a simpler form Compare the analogue in Fourier case [39] and Hermite case [10].
Proof. Let g(x) ∈ C 1 (R). As in [39], the mean value theorem gives by taking δ = 2 −j/2 ǫ and using Lemma 3.5. This proves the lemma. ✷ A direct consequence of Lemma 3.6 is the Peetre maximal function characterization of the spaces B α,q p (H).
Proof. In view of (9), it is sufficient to show that but this follows from (10) immediately. Next we show that B α,q p is independent of the generating functions, i.e., given two systems p . This concludes the proof of Theorem 3.7 (That B α,q p are quasi-Banach spaces follows directly from the definition). ✷ As expected from Lemma 3.6 we can define the homogeneous Besov spaces and obtain a maximal function characterization as well.
Let The proof is completely implicit in that of Theorem 3.7 and hence omitted.

High and low energy estimates
We give proofs of Lemma 3.1 and Lemma 3.3 for the decay estimates of the kernel , which means that the spectrum of ϕ j (H) is bounded away from 0.
When j > J = 4 + [2 log 2 ε], we treat K j (x, y), the kernel of the operator ϕ j (H), as an oscillatory integral as ξ → ∞. When j ≤ J, we use the asymptotic property (as ξ → 0) of eigenfunctions e(x, ξ) to get estimates for the kernel.
−1/λ . We break the estimate of I + into two parts: where we used denotes a function whose derivatives of arbitrary order ≥ 0 has estimates O(ξ −m ), as ξ → ∞.
Note that If we write sin 2K = (2i) −1 (e i2K − e −i2K ), the integral in each term of the sum I + N is bounded by a linear combination of the form, with 0 ≤ ℓ ≤ 2p, 0 ≤ p ≤ N − 1, The following estimates will be used often.

Integration by parts yields
It follows that (14) I Also, follows via integration by parts and the estimates 4N ). Combining (14) and (15) we obtain For I − , we denote − = −1/2λ −1/λ , then Hence we obtain that if x > 1, y > 1, Case 2b. |x| ≤ 1, |y| ≤ 1. Let notation be the same as in Case 1a. by symmetry it is enough to deal with I + .
From the expression of B + , B ′ + we have . We estimate these terms separately. For instance, Using the identity we can write The integral in each term of the sum I + 2,N is bounded by a linear combination of the form + ψ(λξ)e i(x+y−2)K e ±i2Kℓ (2Kξ) −2p (1 − ξ 2 /K 2 )dξ.

Integration by parts gives us
The other terms I + 1 , I + 3 , I + 4 also verify the above inequality (possibly with x + y replaced by x − y). And so does I + and I − .
We have This completes the proof of Lemma 3.1(a). ✷

Proof of Lemma 3.3 (a).
Note that ∂ ∂x e(x, ξ) exist for all ξ = ±ε, 0 and are uniformly bounded in x ∈ R and ξ in any bounded set.
The term I − (x, y) satisfies the same inequality by the relation I − (x, y) = I + (−x, −y). ✷

Proof of Lemma 3.3 (b).
The same argument in proving Lemma 3.1(b) is valid for the proof of Lemma 3.3(b). The interested reader can fill in the details.

Local energy estimates
Let Φ ∈ C ∞ have support contained in {ξ : |ξ| ≤ 1}. Then the spectrum of Φ(H) includes the low energy, a neighborhood of 0. We use the term "local energy" to distinguish from the low energy case, where the support of ϕ j (j ≤ J) keeps away from 0. Since 0 ∈ supp Φ and e(x, ξ) is not continuous at the origin ξ = 0 (!), we need to treat the corresponding kernel more carefully. The proof is more delicate and requires a "matching" method.

Proof of Lemma 3.4.
With the convention The function ξ → ∂ ∂x e(x, ξ) is discontinuous at ξ = 0. As suggested by the treatment of K(x, y) we want to properly "match" different parts of the above integrals so ∂ ∂x K(x, y) can be written as a linear combination of the Fourier transform of C ∞ 0 functions. We only need to check five cases 1a, 1b, 1c, 2a, 2b. Estimates for the other cases follow readily from the relation −y). We outline the proofs for 1a, 1b and 2b only, since 1c and 2a can be dealt with similarly.
As in the case for K(x, y), we split each integral into two parts and let "Re" and "Im" denote the sum of integrals involving only these "reals" and "imaginaries" respectively. As a result , where we find, by noting that ∆ is odd, "Re" and "Im" have the same expressions as in (16) and (17) resp., except that Ψ should be replaced by ∆. Case 1b is so verified.
As a consequence of Theorem 6.5, we shall show that, m(H) initially defined for f ∈ L 2 , has a bounded linear extension to the Banach spaces B α,q p (H), 1 < p < ∞.
Proof of Theorem 6.5. Applying Calderón-Zygmund decomposition and using Lemma 6.4, we can get the weak (1, 1) result for m(H). Then the L p result, 1 < p < ∞, follows via Marcinkiewicz interpolation and duality. For completeness, we give the weak (1, 1) estimation. It is enough to assume f ∈ L 1 ∩ L 2 by density. Given f ∈ L 1 , s > 0, according to Calderón-Zygmund lemma there is a decomposition f = g + b with b = b k and a countable collection of disjoint open intervals I k such that i) |g(x)| ≤ Cs a.e. ii) Each b k is supported in I k , b k dx = 0 and Now let f ∈ L 1 ∩ L 2 , then b = b k converges both a.e. and in L 1 ∩ L 2 , by the definition of b k and properties (ii), (iii), where It follows from Lemma 6.4 and properties (ii) and (iv) that (27) and (28) we obtain the weak (1, 1) estimate. ✷ Proof of Theorem 6.6. For g ∈ L 2 ∩ B α,q p (H), where m j = m(ϕψ) j . Therefore it is sufficient to show that m j (H) are uniformly bounded on L p , 1 < p < ∞. But according to Theorem 6.5 this is true because m j = mψ j verify the obvious condition Proof of Lemma 6.2. Assuming zK j (·, y) 2 ≤ Cλ 1/2 , Schwartz inequality gives Next we need to show zK j (·, y) 2 ≤ Cλ 1/2 . Clearly, We can show that each of these three terms is ≤ C ε λ 1/2 . We shall prove the estimate for the first term only since the other two terms can be proved similarly. The discussion is divided into three cases: y > 1, |y| ≤ 1 and y < −1. Again here we indicate the proof for the case y > 1 only. Let y > 1, x > 1 and consider the high frequency case j > J : Integrating by parts we get By Plancherel formula, Estimation for the low frequency case j ≤ J can be obtained by following the same line (with a suitable modification when necessary) for the high frequency case above, except that we use certain asymptotic properties near the origin instead of ∞, (consult §4).
We are left with the first inequality (19) concerning the "size" of the kernel. The proof of (19) is similar to but easier than that of (20) and may be left as a dull exercise. This closes the proof of Lemma 6.2. ✷ Outline of the proof of Lemma 6.3. Lemma 6.3 can be proved in the same fashion as Lemma 6.2. Assuming (23), apply Schwartz inequality to get for all y |z|>t ∂ ∂y K j (x, y) dx ≤ {|x−y|>t}∩{|x+y|>t} (min |x ± y|) −2 dx 1/2 · z ∂ ∂y K j (·, y) 2 ≤ Ct −1/2 λ −1/2 .
Inequalities (22), (23) measure the L 2 -norm of ∂ ∂y K j (·, y) and z ∂ ∂y K j (·, y), which are derivative analogue of (19), (20) in Lemma 6.2. We only indicate here some points for (23) since (22) is easier to deal with. Consider the high frequency case j > J first. To prove (23) we break the function x → z ∂ ∂y K j (x, y) into three parts: its restriction to the sets {x > 1}, {|x| ≤ 1} and {x < −1}. As before we are able to show that the L 2 -norm of these restrictions (in x) is ≤ Cλ −1/2 .
For m = 2, we have good reason in doubting the verification of the domain condition for H m .
Conjecture. B α,q p (H) = B α,q p (H 0 ), α = 2. To see the reason we compare H 2 and H 2 0 . Write H 2 = H 2 0 + H 0 V + V H 0 + V 2 . If we take A = H 2 0 , B = H 0 V + V H 0 + V 2 , the only term that could cause problem is the term H 0 V , which involves formally Dirac delta distributions and their first derivatives. On the other hand, Theorem 3.2.2 in [2] tells that the domain of the operator H 0 +c 1 δ+c 2 δ ′ consists of functions u ∈ W 2 2 (R\{0}) with u satisfying certain boundary condition at the origin. So if (H + M) 2 is bounded from W 4,p to L p , p = 2, we would have that the domain of the commutator [V, H 0 ] is W 4,p , p = 2, which is not the case by the above theorem in [2].