On dispersive estimates for one-dimensional Klein

Abstract

We improve previous results on dispersive decay for 1D Klein–Gordon equation. We develop a novel approach, which allows us to establish the decay in more strong norms and weaken the assumption on the potential.

Keywords

Klein–Gordon equation dispersive estimates scattering

1. Introduction

We are concerned with one-dimensional Klein–Gordon equation $\begin{matrix} (1.1) & \ddot{ψ} (x, t) = K ψ : = (\partial_{x}^{2} - m^{2} + V (x)) ψ (x, t), (x, t) \in R^{2}, m > 0, \end{matrix}$ with real potential V. In vector form, equation (1.1) reads $\begin{matrix} (1.2) & i \dot{Ψ} (t) = K Ψ (t), \end{matrix}$ where $\begin{matrix} (1.3) & Ψ (t) = (\begin{matrix} ψ (t) \\ \dot{ψ} (t) \end{matrix}), K = i (\begin{matrix} 0 & 1 \\ Δ - m^{2} + V & 0 \end{matrix}) . \end{matrix}$ We assume that $V (x)$ is a continuous function, and $\begin{matrix} (1.4) & | V (x) | ⩽ C {⟨ x ⟩}^{- β}, ⟨ x ⟩ = {(1 + | x |^{2})}^{1 / 2}, x \in R \end{matrix}$ with some $β > 0$ .

Our goal is to prove dispersive decay estimates for these equations. This is a well-studied area and our main contribution is to improve previous results. To formulate them, we introduce the weighted Sobolev spaces $W_{σ}^{ℓ, p}$ , $σ \in R$ , associated with the norm $\begin{matrix} ‖ ψ ‖_{W_{σ}^{ℓ, p}} = \sum_{k = 0}^{ℓ} {‖ {⟨ x ⟩}^{σ} ψ^{(k)} ‖}_{L^{p}}, 1 ⩽ p ⩽ \infty, \end{matrix}$ where $L^{p} = L^{p} (R)$ . Denote $L_{σ}^{p} = W_{σ}^{0, p}$ , $H_{σ}^{ℓ} : = W_{σ}^{ℓ, 2}$ , so $L_{σ}^{2} : = H_{σ}^{0}$ . Of course, the case $σ = 0$ corresponds to the usual $W^{ℓ, p}$ spaces without weight. Denote $\begin{matrix} F_{σ} = H_{σ}^{1} \oplus L_{σ}^{2}, L_{σ}^{1} = W_{σ}^{1, 1} \oplus L_{σ}^{1}, L_{σ}^{\infty} = W_{σ}^{1, \infty} \oplus L_{σ}^{\infty} . \end{matrix}$ We recall (e.g., [6]) that for V, satisfying (1.4) with $β > 1$ , the spectrum of $K$ consists of a purely absolutely continuous part, covering $Γ : = (- \infty, - m) \cup (m, \infty)$ , plus a finite number of eigenvalues located in $R ∖ Γ$ . In addition, there could be resonances at the edges $\pm m$ of the continuous spectrum of $K$ .

Recall that the edges points $λ = \pm m$ are resonances if the equation $K ψ = - m^{2} ψ$ has nonzero solution $ψ \in L_{- 1 / 2 -}^{\infty} (R) ∖ L^{2} (R)$ (see [4,10]). In this case the operator ${(K + m^{2})}^{- 1}$ is unbounded. In particular, the points $λ = \pm m$ are resonances for free Klein–Gordon equation with $V (x) \equiv 0$ .

Our first result reads as follows:

Theorem 1.1.
Let ( 1.4 ) hold with $β > 2$ . Then the following decay holds $\begin{matrix} (1.5) & {‖ e^{- i t K} P_{c} ‖}_{F_{σ} \to F_{- σ}} = O (t^{- 1 / 2}), t \to \infty, σ > 1, \end{matrix}$ where $P_{c} = P_{c} (K)$ is the orthogonal projection in $L^{2} (R)$ onto the continuous spectrum of $K$ .

Note that for the free Klein–Gordon equation with $V = 0$ the estimate (1.5) follows from explicit formula for the time evolution (see e.g. [4]). Let us emphasize that we do not require additional decay of V in the case when edges of the continuous spectrum are resonances.

In the second result we restrict ourselves to the non-resonant case.
Theorem 1.2.
Let ( 1.4 ) hold with $β > 3$ . Then, in the non-resonant case, the following decay holds $\begin{matrix} (1.6) & {‖ e^{- i t K} P_{c} ‖}_{F_{σ} \to F_{- σ}} = O (t^{- 3 / 2}), t \to \infty, σ > 3 / 2 . \end{matrix}$

The decay (1.6) was obtained in [4,7] under the more restrictive conditions $σ > 5 / 2$ and $β > 5$ and under an additional assumption on the decay of $\nabla V (x)$ . Now we extend this result to $σ > 3 / 2$ assuming (1.4) with $β > 3$ only.

Low energy decay in $L^{1} \to L^{\infty}$ norms was obtained in [3] (Theorem 4.1 below). The approach of [3] depends on the fact that the scattering matrix is in the Wiener algebra (i.e. its Fourier transform is integrable). This result implies the decays (1.5) and (1.6) for low energy part of solutions. However, the decay in $L^{1} \to L^{\infty}$ norms for the operator $e^{- i t K} P_{c}$ does not hold even in the case $V = 0$ , in contrast to the Schrödinger operator. Namely, for the free Schrödinger equation the decay holds since its propagator is bounded for $| t | ⩾ 1$ and decays uniformly: $\begin{matrix} (1.7) & {‖ e^{- i t Δ} ‖}_{L^{1} \to L^{\infty}} = sup_{x, y \in R^{3}} | \frac{e^{i | x - y |^{2} / 4 t}}{\sqrt{4 π i t}} | \sim | t |^{- 1 / 2}, | t | ⩾ 1 . \end{matrix}$ On the other hand, the propagator $U (t, x, y)$ of the free Klein–Gordon equation does not decay for $| x - y | \sim | t |$ . Namely, for $t > 0$ , $\begin{matrix} (1.8) & U (t, x, y) = \frac{1}{2} θ (t - | x - y |) J_{0} (m \sqrt{t^{2} - | x - y |^{2}}) \sim \frac{1}{2}, | x - y | \sim t . \end{matrix}$

Here $J_{0}$ is the Bessel function of order 0, and θ is the Heavyside function. This difference reflects the distinct character of the wave propagation for the relativistic and nonrelativistic equations. Namely, the singularities of solutions to the Schrödinger equations are concentrated at $t = 0$ and disappear at infinity for $t \neq 0$ due to infinite speed of propagation. On the other hand, in the case of Klein–Gordon equation, the singularities move with bounded velocities, thus they are present forever in the space.

As shown in [3], the decay of solution $ψ (t)$ to (1.1) in $L^{\infty}$ -norm is possible only if the initial data $(ψ (0), \dot{ψ} (0)$ belong to generalized Sobolev space $W^{\frac{3}{2}, 1} \oplus W^{\frac{1}{2}, 1}$ .

Our approach relies on the Born expansion for high energy part of the dynamical group $U (t) = U (ζ, t) = e^{- i t K} ζ (K^{2})$ , where $ζ \in C^{\infty} (R)$ , $ζ (ω) = 0$ for $ω ⩽ m^{2} + 1$ , and $ζ (ω) = 1$ for $ω ⩾ m^{2} + 2$ , $\begin{array}{l} U (t) = & U_{0} (t) - i \int_{0}^{t} U_{0} (t - s) V U_{0} (s) d s \\ - \int_{0}^{t} U_{0} (t - s) V (\int_{0}^{s} U_{0} (s - u) V U_{0} (u) d u) d s + W (t), \end{array}$ where $U_{0} (t) = e^{- i t K_{0}} ζ (K_{0}^{2})$ , and $V$ is the matrix (4.5). First, we show that $\begin{matrix} {‖ U_{0} (t) ‖}_{F_{σ} \to F_{- σ}} ⩽ C {(1 + | t |)}^{- σ} \end{matrix}$ for any $σ ⩾ 0$ . Then the decay of the next two terms (under appropriate values of β in (1.4)) follows by standard estimates for the convolutions. Finally, we prove the decay in $L^{1} \to L^{\infty}$ norms for remainder $\begin{matrix} W (t) = \frac{i}{2 π} \int_{Γ} e^{- i ω t} ({(R_{0} (ω + i 0) V)}^{3} R (ω + i 0) - {(R_{0} (ω - i 0) V)}^{3} R (ω - i 0)) ζ (ω^{2}) d ω, \end{matrix}$ where $R$ and $R_{0}$ stand for the resolvents of the operators $K$ and $K_{0}$ , respectively. For the proof we use the representation of $R$ via Jost solutions (formula (3.3) below) and properties of the scattering matrix obtained in [3].

Note that dispersion estimates of type (1.5)–(1.6) play an important role in proving asymptotic stability of solitons and of scattering asymptotics in the associated one-dimensional nonlinear Klein–Gordon equations [8].
2. Free Klein–Gordon equation

Here we consider the case $V = 0$ . Denote $\begin{matrix} K_{0} = i (\begin{matrix} 0 & 1 \\ Δ - m^{2} & 0 \end{matrix}) \end{matrix}$ The resolvent $R_{0} (ω) = {(K_{0} - ω)}^{- 1}$ of the operator $K_{0}$ can be expressed in terms of the resolvent $R_{0} (ω) = {(- \partial_{x}^{2} - ω)}^{- 1}$ of the Schrödinger operator as $\begin{matrix} R_{0} (ω) = (\begin{matrix} 0 & 0 \\ - i & 0 \end{matrix}) + (\begin{matrix} ω & i \\ - i ω^{2} & ω \end{matrix}) R_{0} (ω^{2} - m^{2}) . \end{matrix}$ For the dynamical group $e^{- i t K_{0}}$ of the free Klein–Gordon equation the spectral representation holds: $\begin{array}{l} e^{- i t K_{0}} & = \frac{1}{2 π i} \int_{Γ} e^{- i t ω} (R_{0} (ω + i 0) - R_{0} (ω - i 0)) d ω \\ (2.1) & = \frac{1}{2 π i} \int_{Γ} e^{- i ω t} (\begin{matrix} ω & i \\ - i ω^{2} & ω \end{matrix}) (R_{0} ({(ω + i 0)}^{2} - m^{2}) - R_{0} ({(ω - i 0)}^{2} - m^{2})) d ω . \end{array}$ Recall, that $R_{0} (ω)$ is an integral operator with the integral kernel $\begin{matrix} [R_{0} (ω)] (x, y) = - \frac{e^{i \sqrt{ω} | x - y |}}{2 i \sqrt{ω}} . \end{matrix}$ Hence, the integral kernel of $e^{- i t K_{0}}$ reads $\begin{matrix} (2.2) & [e^{- i t K_{0}}] (x, y) = \frac{1}{2 π} \int_{- \infty}^{\infty} M_{t} (k) e^{i | y - x | k} d k, \end{matrix}$ where $\begin{matrix} (2.3) & M_{t} (k) = (\begin{matrix} cos (t \sqrt{k^{2} + m^{2}}) & \frac{sin (t \sqrt{k^{2} + m^{2}})}{\sqrt{k^{2} + m^{2}}} \\ - \sqrt{k^{2} + m^{2}} sin (t \sqrt{k^{2} + m^{2}}) & cos (t \sqrt{k^{2} + m^{2}}) \end{matrix}) . \end{matrix}$

Proposition 2.1.
$\begin{matrix} (2.4) & {‖ e^{- i K_{0} t} ‖}_{F_{σ} \to F_{- σ}} ⩽ C {(1 + t)}^{- 1 / 2}, t ⩾ 0, σ > 1 / 2 . \end{matrix}$

The proposition follows from two lemmas below.
Lemma 2.2.
For any $χ \in C_{0}^{\infty} (R)$ the decay holds $\begin{matrix} (2.5) & {‖ e^{- i K_{0} t} χ (K_{0}^{2}) ‖}_{L^{1} \to L^{\infty}} ⩽ C {(1 + t)}^{- 1 / 2}, t ⩾ 0 . \end{matrix}$
Lemma 2.3.
Let ζ be a smooth function such that $ζ (ω) = 0$ for $ω ⩽ m^{2} + 1$ and $ζ (ω) = 1$ for $ω ⩾ m^{2} + 2$ . Then the decay holds $\begin{matrix} (2.6) & {‖ e^{- i K_{0} t} ζ (K_{0}^{2}) ‖}_{F_{σ} \to F_{- σ}} ⩽ C {(1 + t)}^{- σ}, σ ⩾ 0, t ⩾ 0 . \end{matrix}$
Proof of Lemma 2.2.
Similarly to (2.2), $\begin{matrix} [e^{- i t K_{0}} χ (K_{0}^{2})] (x, y) = \frac{1}{2 π} \int_{- \infty}^{\infty} M_{t} (k) e^{i | y - x | k} χ (k^{2} + m^{2}) d k . \end{matrix}$ Note, that for an integral operator T, with the kernel $[T] (x, y)$ , $\begin{matrix} ‖ T ‖_{L^{1} \to L^{\infty}} = sup_{‖ f ‖_{L^{\infty}} = 1, ‖ g ‖_{L^{1}} = 1} ⟨ f, T g ⟩ = sup_{x, y} | [T] (x, y) | . \end{matrix}$ Hence, it suffices to prove that for $j = - 1, 0, 1$ and $p = 0, 1$ , $\begin{matrix} (2.7) & sup_{x, y} | \int_{- \infty}^{\infty} e^{i ω (k) t} e^{i (y - x) k} ω^{j} (k) k^{p} χ (k^{2} + m^{2}) d k | ⩽ C t^{- 1 / 2}, t ⩾ 1 . \end{matrix}$ Here we denote $ω (k) : = \sqrt{k^{2} + m^{2}}$ . The integral in (2.7) is an oscillatory integral with the phase function $ϕ (k) = ω (k) + \frac{y - x}{t} k$ , satisfying the condition $\begin{matrix} | ϕ^{″} (k) | = \frac{m^{2}}{ω^{3} (k)} ⩾ C (m, χ), (k^{2} + m^{2}) \in supp χ . \end{matrix}$ Hence, (2.5) follows by the van der Corput lemma [11, page 332]. □
Proof of Lemma 2.3.
One has $\begin{matrix} (2.8) & [e^{- i t K_{0}} ζ (K_{0}^{2})] (x, y) = \frac{1}{2 π} \int_{- \infty}^{\infty} M_{t} (k) e^{i | y - x | k} ζ (k^{2} + m^{2}) d k . \end{matrix}$ Denote by $B_{j} (t)$ , $j = 0, 1, - 1$ , integral operators with the integral kernels $\begin{matrix} [B_{j} (t)] (x, y) = \int_{- \infty}^{\infty} e^{i ω (k) t} e^{i k (x - y)} ζ (k^{2} + m^{2}) ω^{j} (k) d k . \end{matrix}$ In this notation, (2.6) is equivalent to $\begin{matrix} (2.9) & {‖ B_{0} (t) ‖}_{L_{σ}^{2} \to L_{- σ}^{2}} + {‖ B_{1} (t) ‖}_{H_{σ}^{1} \to L_{- σ}^{2}} + {‖ B_{- 1} (t) ‖}_{L_{σ}^{2} \to H_{- σ}^{1}} ⩽ C {(1 + t)}^{- σ}, σ ⩾ 0 \end{matrix}$ by (2.3) and (2.8). We also took into account that $\begin{matrix} {‖ B_{0} (t) ‖}_{H_{σ}^{1} \to H_{- σ}^{1}} = {‖ B_{0} (t) ‖}_{L_{σ}^{2} \to L_{- σ}^{2}}, \end{matrix}$ since ${(B_{0} (t) f)}^{'} (x) = (B_{0} (t) f^{'}) (x)$ .

First we prove (2.9) for integer values of σ. Denote $\hat{f} (k) = \int e^{- i k y} f (y) d y$ the Fourier transform of $f (y)$ . Then $\begin{matrix} (2.10) & (B_{j} (t) f) (x) = \int_{- \infty}^{\infty} e^{i ω (k) t} e^{i k x} ζ (k^{2} + m^{2}) \hat{f} (k) ω^{j} (k) d k, j = 0, 1, - 1 . \end{matrix}$ Parseval’s identity implies $\begin{matrix} {‖ B_{0} (t) f ‖}_{L^{2}} = \frac{1}{\sqrt{2 π}} {‖ e^{i ω (k) t} ζ (k^{2} + m^{2}) \hat{f} (k) ‖}_{L^{2}} ⩽ C ‖ \hat{f} ‖_{L^{2}} = C ‖ f ‖_{L^{2}}, \end{matrix}$ and (2.9) with $σ = 0$ for the first term follows. Further, integration by parts gives $\begin{array}{l} (B_{0} (t) f) (x) = & \frac{i}{t} \int e^{i ω (k) t} \partial_{k} (e^{i k x} \frac{ω (k)}{k} ζ (k^{2} + m^{2}) \hat{f} (k)) d k \\ = & - \frac{x}{t} \int e^{i ω (k) t + i k x} \frac{ω (k)}{k} ζ (k^{2} + m^{2}) \hat{f} (k) d k \\ + \frac{i}{t} \int e^{i ω (k) t + i k x} \frac{ω (k)}{k} ζ (k^{2} + m^{2}) \partial_{k} \hat{f} (k) d k \\ + \frac{i}{t} \int e^{i ω (k) t + i k x} \hat{f} (k) \partial_{k} (\frac{ω (k)}{k} ζ (k^{2} + m^{2})) d k \\ (2.11) & = & g_{1} (x, t) + g_{2} (x, t) + g_{3} (x, t), t ⩾ 1 . \end{array}$ Evidently, $\begin{matrix} {‖ g_{3} (t) ‖}_{L^{2}} = \frac{1}{t \sqrt{2 π}} {‖ e^{i ω (k) t} \hat{f} (k) \partial_{k} (\frac{ω (k)}{k} ζ (k^{2} + m^{2})) ‖}_{L^{2}} ⩽ \frac{C}{t} ‖ \hat{f} ‖_{L^{2}} = \frac{C}{t} ‖ f ‖_{L^{2}} . \end{matrix}$ Similarly, $\begin{matrix} {‖ {⟨ x ⟩}^{- 1} g_{1} (x, t) ‖}_{L^{2}} ⩽ \frac{C}{t} ‖ f ‖_{L^{2}}, {‖ g_{2} (x, t) ‖}_{L^{2}} ⩽ \frac{C}{t} {‖ \partial_{k} \hat{f} (k) ‖}_{L^{2}} ⩽ \frac{C}{t} {‖ ⟨ x ⟩ f (x) ‖}_{L^{2}} . \end{matrix}$ Hence, (2.9) with $σ = 1$ for the first term follows. Similarly, (2.9) with $σ = 2, 3, \dots$ follows for the first term by σ- times integration by parts. Note, that $\begin{array}{l} {(B_{- 1} (t) f)}^{'} (x) = \int_{- \infty}^{\infty} e^{i ω (k) t} e^{i k x} ζ (k^{2} + m^{2}) \hat{f} (k) \frac{i k}{ω (k)} d k . \\ (B_{1} (t) f) (x) = \int_{- \infty}^{\infty} e^{i ω (k) t} e^{i k x} ζ (k^{2} + m^{2}) \hat{f^{'}} (k) \frac{ω (k)}{i k} d k . \end{array}$ □

Hence, the remaining terms in the left hand side of (2.9) can be estimated similarly for integer σ. Finally, (2.9) for orbitrary $σ ⩾ 0$ follows by a suitable version of Stein–Weiss interpolation theorem:
Lemma 2.4 (cf. [1, Theorem 5.4.1]).

Let $σ_{0} \neq σ_{1}$ , and let $T : L_{σ_{j}}^{2} \to L_{- σ_{j}}^{2}$ with the norm $M_{j} = sup {⟨ T f, g ⟩ : ‖ f ‖_{L_{σ_{j}}^{2}} = ‖ g ‖_{L_{σ_{j}}^{2}} = 1}$ , $j = 0, 1$ . Then for $σ = σ (θ) = (1 - θ) σ_{0} + θ σ_{1}$ , $θ \in (0, 1)$ , one has $\begin{matrix} T : L_{σ}^{2} \to L_{- σ}^{2} with the norm M ⩽ M_{0}^{1 - θ} M_{1}^{θ} . \end{matrix}$

For the convenience of readers we give the proof in the Appendix.

3. Scattering properties of Schrödinger operator

Next we recall a few facts from scattering theory [2,3] of the Schrödinger operator $H = - (\partial_{x}^{2} + V)$ . Under the assumption $V \in L_{1}^{1}$ there exist Jost solutions $f_{\pm} (x, k) = e^{\pm i k x} h_{\pm} (x, k)$ of $\begin{matrix} H f = k^{2} f, k \in \overline{C_{+}}, \end{matrix}$ normalized according to $\begin{matrix} h_{\pm} (x, k) \sim 1, x \to \pm \infty . \end{matrix}$ Denote by $A$ the Banach algebra of Fourier transforms of integrable functions $\begin{matrix} A = {f (k) : f (k) = \int_{R} e^{i k p} \hat{f} (p) d p, \hat{f} (\cdot) \in L^{1} (R)} \end{matrix}$ with the norm $‖ f ‖_{A} = ‖ \hat{f} ‖_{L^{1}}$ , and by $A_{1}$ the corresponding unital Banach algebra $\begin{matrix} A_{1} = {f (k) : f (k) = c + \int_{R} e^{i k p} \hat{g} (p) d p, \hat{g} (\cdot) \in L^{1} (R), c \in C} \end{matrix}$ with the norm $‖ f ‖_{A_{1}} = | c | + ‖ \hat{g} ‖_{L^{1}}$ . Evidently, $A$ is a subalgebra of $A_{1}$ . Then $\begin{matrix} (3.1) & h_{\pm} (x, \cdot) - 1, h_{\pm}^{'} (x, \cdot) \in A, \forall x \in R . \end{matrix}$ Let $\begin{matrix} W (φ (x, k), ψ (x, k)) = φ (x, k) ψ^{'} (x, k) - φ^{'} (x, k) ψ (x, k) \end{matrix}$ be the usual Wronskian, and set $\begin{array}{l} W (k) = W (f_{-} (x, k), f_{+} (x, k)), W_{\pm} (k) = W (f_{\mp} (x, k), f_{\pm} (x, - k)) . \\ T (k) = \frac{2 i k}{W (k)}, R_{\pm} (k) = \mp \frac{W_{\pm} (k)}{W (k)}, \end{array}$ Recall that T and $R_{\pm}$ are the entries of the scattering matrix, that is, the transmission and reflection coefficients. For $V \in L_{1}^{1}$ , $\begin{matrix} T (k) - 1 \in A, R_{\pm} (k) \in A \end{matrix}$ by [3, Theorem 2.1]. Introduce the function $\begin{matrix} (3.2) & ψ (x, y, k) = h_{+} (y, k) h_{-} (x, k) T (k), y ⩾ x, \end{matrix}$ and $ψ (x, y, k) = ψ (y, x, k)$ for $y < x$ . Then the kernel of the resolvent $R (ω) = {(H - ω)}^{- 1}$ for $ω = k^{2} \pm i 0$ , $k > 0$ , can be expressed as $\begin{matrix} (3.3) & [R (k^{2} \pm i 0)] (x, y) = \mp \frac{e^{\pm i k (y - x)} ψ (x, y, \pm k)}{2 i k} . \end{matrix}$

Lemma 3.1 (cf. [3, Lemma 2.3]).

Let $V \in L_{1}^{1}$ . Then $ψ (x, y, \cdot) \in A_{1}$ , and the following estimate holds $\begin{matrix} (3.4) & {‖ ψ (x, y, \cdot) ‖}_{A_{1}} ⩽ C, \end{matrix}$ with some constant C, which does not depend on x and y.

Finally, in [3] it was proved that for $V \in L_{2}^{1}$ , $\begin{matrix} (3.5) & {‖ ξ (k) \partial_{k} ψ (x, y, k) ‖}_{A} ⩽ C (1 + | x |) (1 + | y |), \end{matrix}$ where $ξ (k)$ is a smooth function such that $ξ (k) = 0$ for $| k | ⩽ 1$ , and $ξ (k) = 1$ for $| k ⩾ 2$ .

4. Perturbed Klein–Gordon equation

The resolvent $R (ω)$ of the operator (1.3) associated with the Klein–Gordon equation (1.1) can be expressed in terms of the resolvent $R (ω)$ of the Schrödinger operator H as $\begin{matrix} (4.1) & R (ω) = (\begin{matrix} 0 & 0 \\ - i & 0 \end{matrix}) + (\begin{matrix} ω & i \\ - i ω^{2} & ω \end{matrix}) R (ω^{2} - m^{2}) . \end{matrix}$ For the one-parameter group $e^{- i t K}$ of (1.2) and for any $χ \in C_{0}^{\infty}$ the spectral representation of type (2.1)–(2.2) holds (see [3, Formula (5.4)]): $\begin{array}{l} e^{- i t K} P_{c} χ (K^{2}) & = \frac{1}{2 π i} \int_{Γ} e^{- i t ω} (R (ω + i 0) - R (ω - i 0)) χ (ω^{2}) d ω \\ (4.2) & = \frac{1}{2 π} \int_{R} M_{t} (k) e^{i | y - x | k} ψ (x, y, k) χ (k^{2} + m^{2}) d k, \end{array}$ where the functions $M_{t} (k)$ and $ψ (x, y, k)$ are defined in (2.3) and (3.2). Representation (4.2), Lemma 3.4, and the version of the van der Corput lemma [3, Lemma 5.4] imply

Theorem 4.1.
(Low energy decay)

i) Assume $V \in L_{1}^{1}$ . Then for any smooth function χ with bounded support the following decay holds $\begin{matrix} {‖ e^{- i t K} P_{c} χ (K^{2}) ‖}_{L^{1} \to L^{\infty}} = O (t^{- 1 / 2}), t \to \infty . \end{matrix}$ ii) Assume $V \in L_{2}^{1}$ . Then in non-resonant case the following decay holds $\begin{matrix} {‖ e^{- i t K} P_{c} χ (K^{2}) ‖}_{L_{1}^{1} \to L_{- 1}^{\infty}} = O (t^{- 3 / 2}), t \to \infty . \end{matrix}$

The detailed proof of the theorem can be found in [3]. The theorem immediately implies
Theorem 4.2.
i) Assume $V \in L_{1}^{1}$ . Then for any smooth function χ with bounded support and any $σ > 1 / 2$ the following decay holds $\begin{matrix} {‖ e^{- i t K} P_{c} χ (K^{2}) ‖}_{F_{σ} \to F_{- σ}} = O (t^{- 1 / 2}), t \to \infty . \end{matrix}$ ii) Assume $V \in L_{2}^{1}$ . Then in the non-resonant case the following decay holds $\begin{matrix} {‖ e^{- i t K} P_{c} χ (K^{2}) ‖}_{F_{σ} \to F_{- σ}} = O (t^{- 3 / 2}), t \to \infty, σ > 3 / 2 . \end{matrix}$

Now we prove high energy decay.
Theorem 4.3.
i) Let condition ( 1.4 ) holds with some $β > 2$ and let $ζ (x)$ be a smooth function such that $ζ (x) = 0$ for $x ⩽ m^{2} + 1$ and $ζ (x) = 1$ for $x ⩾ m^{2} + 2$ . Then for any $σ > 3 / 4$ , the decay holds $\begin{matrix} (4.3) & {‖ e^{- i t K} ζ (K^{2}) ‖}_{F_{σ} \to F_{- σ}} = O (t^{- 1 / 2}), t \to \infty . \end{matrix}$ ii) Let condition ( 1.4 ) holds with some $β > 3$ . Then for any $σ > 3 / 2$ , $\begin{matrix} (4.4) & {‖ e^{- i t K} ζ (K^{2}) ‖}_{F_{σ} \to F_{- σ}} = O (t^{- 3 / 2}), t \to \infty . \end{matrix}$
Proof.
Step i). Substituting the resolvent identity $\begin{matrix} R (λ) = R_{0} (λ) - R_{0} (λ) V R_{0} (λ) + {(R_{0} (λ) V)}^{2} R_{0} (λ) + {(R_{0} (λ) V)}^{3} R (λ) \end{matrix}$ into (4.1), we obtain $\begin{matrix} e^{- i t K} ζ (K^{2}) = U_{0} (t) + U_{1} (t) + U_{2} (t) + W (t), \end{matrix}$ where $U_{0} (t) = e^{- i t K_{0}} ζ (K_{0}^{2})$ , and $\begin{array}{l} U_{j} (t) = \frac{1}{2 π i} \int_{Γ} e^{- i ω t} ([{(R_{0} V)}^{j} R] (ω + i 0) - [{(R_{0} V)}^{j} R_{0}] (ω - i 0)) ζ (ω^{2}) d ω, j = 1, 2, \\ W (t) = \frac{1}{2 π i} \int_{Γ} e^{- i ω t} ([{(R_{0} V)}^{3} R] (ω + i 0) - [{(R_{0} V)}^{3} R] (ω - i 0)) ζ (ω^{2}) d ω . \end{array}$ Here we denote $\begin{matrix} (4.5) & V (x) = (\begin{matrix} 0 & 0 \\ i V (x) & 0 \end{matrix}) . \end{matrix}$ Due to Lemma 2.3, $\begin{matrix} (4.6) & {‖ U_{0} (t) ‖}_{F_{σ} \to F_{σ}} = O (t^{- σ}), t \to \infty, σ ⩾ 0 . \end{matrix}$ Let us prove that for $β > 2$ $\begin{matrix} (4.7) & {‖ U_{j} (t) ‖}_{F_{β / 2} \to F_{- β / 2}} = O (t^{- β / 2}), t \to \infty, j = 1, 2 . \end{matrix}$

For arbitrary $Ψ_{0} \in F_{β / 2}$ , denote $Ψ_{j + 1} (t) = U_{j} (t) Ψ_{0}$ , $j = 0, 1, 2$ . Lemma 4.4 (cf. [5, Lemma 3.8], [6, Lemma 36.6]).

The convolution representation holds $\begin{matrix} (4.8) & Ψ_{j + 1} (t) = i \int_{0}^{t} U_{0} (t - τ) V Ψ_{j} (τ) d τ, t \in R, j = 1, 2, \end{matrix}$ where the integral converges in $F_{- β / 2}$ .

By Lemma 2.3, $‖ Ψ_{1} ‖_{F_{- β / 2}} ⩽ C {(1 + | t |)}^{- β / 2} ‖ Ψ_{0} ‖_{F_{β / 2}}$ . Applying Lemma 2.3 to the integrand in (4.8), we obtain for $σ = β / 2$ $\begin{matrix} {‖ U_{0} (t - τ) V Ψ_{1} (τ) ‖}_{F_{- σ}} ⩽ \frac{C ‖ V Ψ_{1} (τ) ‖_{F_{σ}}}{{(1 + | t - τ |)}^{σ}} ⩽ \frac{C_{1} ‖ Ψ_{1} (τ) ‖_{F_{- σ}}}{{(1 + | t - τ |)}^{σ}} ⩽ \frac{C_{2} ‖ Ψ_{0} ‖_{F_{σ}}}{{(1 + | t - τ |)}^{σ} {(1 + | τ |)}^{σ}} \end{matrix}$ Therefore, integrating here in τ, we obtain $\begin{matrix} {‖ Ψ_{2} (t) ‖}_{F_{- β / 2}} ⩽ C {(1 + | t |)}^{- β / 2} ‖ Ψ_{0} ‖_{F_{β / 2}} . \end{matrix}$ Similarly, $\begin{matrix} {‖ Ψ_{3} (t) ‖}_{F_{- β / 2}} ⩽ C {(1 + | t |)}^{- β / 2} ‖ Ψ_{0} ‖_{F_{β / 2}} . \end{matrix}$ Hence, (4.7) follows.

It remains to estimate the operator $W (t)$ with the kernel $\begin{matrix} (4.9) & [W (t)] (x, y) = \int_{R^{3}} \frac{V (u, w, z)}{16 π i} (\int_{R} ξ (k) M_{t} (k) \frac{e^{i k p (x, u, w, z, y)}}{k^{3}} ψ (z, y, k) d k) d u d w d z, \end{matrix}$ where we denote $p (x, u, w, z, y) = | x - u | + | u - w | + | w - z | + | z - y |$ , $ξ (k) = ζ (k^{2} + m^{2})$ , $V (u, w, z) = V (u) V (w) V (z)$ . We will apply the following version of [9, Lemma 2]:

Lemma 4.5 (cf. [3, Lemma 5.5]).

Let $η (k)$ , $k ⩾ 1$ , be a smooth function such that $| η^{(j)} (k) | ⩽ k^{- j}$ for $j = 0, 1$ . Then for any $g (k) \in A_{1}$ , $α > 3 / 2$ and $t ⩾ 1$ $\begin{matrix} (4.10) & sup_{p \in R} | \int_{1}^{\infty} η (k) \frac{e^{\pm i t ω (k) + i k p}}{k^{α}} g (k) d k | ⩽ C t^{- 1 / 2} ‖ g ‖_{A_{1}}, ω (k) = \sqrt{k^{2} + m^{2}} . \end{matrix}$

Applying this lemma with $g (k) = ψ (z, y, k)$ , $p = p (x, u, w, z, y)$ , and taking into account (3.4), we get $\begin{matrix} (4.11) & sup_{x, y} | [W^{i j} (t)] (x, y) | ⩽ C t^{- 1 / 2} ‖ V ‖_{L^{1}}^{3} ⩽ C_{1} t^{- 1 / 2}, t ⩾ 1, i, j = 1, 2 . \end{matrix}$ Here $W^{i j}$ are the entries of the matrix $W$ . Further, $\begin{array}{l} \partial_{x} [W (t)] (x, y) \\ = \int_{R^{3}} sgn (x - u) \frac{V (u, w, z)}{16 π} (\int_{R} ξ (k) M_{t} (k) \frac{e^{i k p (x, y, u, w, z)}}{k^{2}} ψ (z, y, k) d k) d u d w d z . \end{array}$ Hence, similarly to (4.11), $\begin{matrix} (4.12) & sup_{x, y} | \partial_{x} [W^{1 j} (t)] (x, y) | ⩽ C t^{- 1 / 2}, t ⩾ 1, j = 1, 2 . \end{matrix}$ Bounds (4.11) and (4.12) imply that $\begin{matrix} {‖ W (t) ‖}_{L^{1} \to L^{\infty}} ⩽ C t^{- 1 / 2}, t ⩾ 1 . \end{matrix}$ Together with (4.6) and (4.7) this gives (4.3).

Step ii). Now we rewrite (4.9) as follows $\begin{array}{l} [W (t)] (x, y) \\ (4.13) & = \sum_{\pm} \int_{R^{3}} \frac{V (u, w, z)}{16 π i} (\int_{R} ξ (k) A_{\pm} (k) \frac{e^{i t ω (k) + i k p (x, u, w, z, y)}}{k^{3}} ψ (z, y, k) d k) d u d w d z, \end{array}$ where $\begin{matrix} A_{\pm} (k) = (\begin{matrix} 1 & \mp \frac{i}{ω (k)} \\ \pm i ω (k) & 1 \end{matrix}), \end{matrix}$ Integrating by parts gives $\begin{array}{l} F_{\pm} (t, x, u, w, z, y) & : = \int ξ (k) A_{\pm} (k) \frac{e^{i t ω (k) + i k p (x, u, w, z, y)}}{k^{3}} ψ (z, y, k) d k \\ = \frac{i}{t} \int e^{i t ω (k)} \frac{\partial}{\partial k} [e^{i k p (x, u, w, z, y)} ξ (k) \frac{ω (k)}{k^{4}} A_{\pm} (k) ψ (z, y, k)] d k . \end{array}$ Recall that in the case $V \in L_{2}^{1}$ , $\begin{matrix} ‖ \frac{\partial}{\partial k} ψ (z, y, k) ‖_{A} ⩽ C (1 + | z |) (1 + | y |) \end{matrix}$ by (3.5). Moreover, $\begin{matrix} | x - u | + | u - w | + | w - z | + | z - y | ⩽ (1 + | x |) (1 + | y |) (1 + 2 | u |) (1 + 2 | w |) (1 + 2 | z |) . \end{matrix}$ Hence, $\begin{matrix} | F_{\pm} (t, x, u, w, z, y) | ⩽ C t^{- 3 / 2} (1 + | x |) (1 + | y |) (1 + | u |) (1 + | w |) (1 + | z |), t ⩾ 1 . \end{matrix}$ by Lemma 4.5. The last estimate and (4.13) imply $\begin{matrix} | [W^{i j} (t)] (x, y) | ⩽ C t^{- 3 / 2} (1 + | x |) (1 + | y |), t ⩾ 1, i, j = 1, 2 . \end{matrix}$ Similarly, $\begin{matrix} | \partial_{x} [W^{1 j} (t)] (x, y) | ⩽ C t^{- 3 / 2} (1 + | x |) (1 + | y |), t ⩾ 1, j = 1, 2 . \end{matrix}$ Therefore, $\begin{matrix} {‖ W (t) ‖}_{L_{1}^{1} \to L_{1}^{\infty}} ⩽ C t^{- 3 / 2}, t ⩾ 1 . \end{matrix}$ Together with (4.6) and (4.7) this gives (4.4). □

Footnotes

Acknowledgement

Research supported by the Austrian Science Fund (FWF) under Grant No. P 34177-N.

Proof of Lemma 2.4

Suppose that f, g are simple functions, and for $0 ⩽ Re z ⩽ 1$ define the functions $\begin{matrix} φ (z) : = φ (x, z) = {⟨ x ⟩}^{- σ + σ (z)} f (x), ψ (z) : = ψ (x, z) = {⟨ x ⟩}^{σ - σ (z)} g (x), \end{matrix}$ where $σ (z) : = (1 - z) σ_{0} + z σ_{1}$ . One has $\begin{array}{l} {‖ φ (i t) ‖}_{L_{- σ_{0}}^{2}}^{2} = \int {⟨ x ⟩}^{- σ_{0}} {| {⟨ x ⟩}^{- σ + 2 (1 - i t) σ_{0} + i t σ_{1})} f (x) |}^{2} d x = ‖ f ‖_{L_{- σ}^{2}}^{2} = 1, \\ {‖ φ (1 + i t) ‖}_{L_{- σ_{1}}^{2}}^{2} = \int {⟨ x ⟩}^{- σ_{1}} {| {⟨ x ⟩}^{- σ - 2 i t σ_{0} + (1 + i t) σ_{1})} f (x) |}^{2} d x = ‖ f ‖_{L_{- σ}^{2}}^{2} = 1 . \end{array}$ Similarly, $‖ ψ (i t) ‖_{L_{σ_{0}}^{2}} = ‖ ψ (1 + i t) ‖_{L_{σ_{1}}^{2}} = 1$ . The function $F (z) = ⟨ T φ (z), ψ (z) ⟩$ is analytic in $0 < Re z < 1$ , and bounded and continuous in $0 ⩽ Re z ⩽ 1$ . Moreover, $\begin{array}{l} | F (i t) | ⩽ {‖ T (φ (i t)) ‖}_{L_{- σ_{0}}^{2}} {‖ ψ (i t) ‖}_{L_{σ_{0}}^{2}} ⩽ M_{0}, \\ | F (1 + i t) | ⩽ {‖ T (φ (1 + i t)) ‖}_{L_{- σ_{1}}^{2}} {‖ ψ (1 + i t) ‖}_{L_{σ_{1}}^{2}} ⩽ M_{1} . \end{array}$ Besides, $φ (θ) = f$ , $ψ (θ) = g$ , and hence $F (θ) = ⟨ T f, g ⟩$ . Applying the Hadamard three-lines theorem (see [1, Lemma 1.1.2]), we obtain $\begin{matrix} M : = sup {⟨ T f, g ⟩ : ‖ f ‖_{L_{σ}^{2}} = ‖ g ‖_{L_{σ}^{2}} = 1} ⩽ M_{0}^{1 - θ} M_{1}^{θ} . \end{matrix}$

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On dispersive estimates for one-dimensional Klein–Gordon equations

Abstract

Keywords

1. Introduction

3. Scattering properties of Schrödinger operator

Lemma 3.1 (cf. [3, Lemma 2.3]).

4. Perturbed Klein–Gordon equation

Lemma 4.5 (cf. [3, Lemma 5.5]).

Footnotes

Acknowledgement

Proof of Lemma 2.4

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