Abstract

We show that for a one-dimensional Schrödinger operator with a potential, whose $(j+1)$-th moment is integrable, the $j$-th derivative of the scattering matrix is in the Wiener algebra of functions with integrable Fourier transforms. We use this result to improve the known dispersive estimates with integrable time decay for the one-dimensional Schrödinger equation in the resonant case.

1. Introduction

This paper is concerned with the one-dimensional Schrödinger equation

$$\begin{equation} \mathrm{i}\dot{\psi }(x,t)=H \psi (x,t), \quad H=-\frac{d^2}{dx^2} + V(x),\quad (x,t)\in {\mathbb{R}}^2, \tag{1.1}{} \end{equation}$$

with a real-valued potential $V$ contained in one of the spaces $L^1_{\sigma }=L^1_{\sigma }({\mathbb{R}})$, $\sigma \in {\mathbb{R}}$, associated with the norms

$$\begin{equation*} \Vert \psi \Vert _{L^p_{\sigma }}= \begin{cases} \left(\int _{{\mathbb{R}}}(1+|x|)^{p\sigma } |\psi (x)|^p dx\right)^{1/p}, & 1\le p <\infty ,\\ \sup _{x\in {\mathbb{R}}} (1+|x|)^{\sigma } |\psi (x)|, & p=\infty . \end{cases} \end{equation*}$$

We recall (e.g. 15 or 16, §9.7) that for $V\in L^1_1$ the operator $H$ has a purely absolutely continuous spectrum on $[0,\infty )$ and a finite number of eigenvalues in $(-\infty ,0)$. Associated with the absolutely continuous spectrum is the scattering matrix

$$\begin{equation*} \mathcal{S}(k) = \begin{pmatrix} T(k) & R_-(k)\\ R_+(k) & T(k) \end{pmatrix} \end{equation*}$$

which maps incoming to outgoing states at a given energy $\omega =k^2\ge 0$. Here $T$ is the transmission coefficient and $R_\pm$ are the reflection coefficients with respect to right and left incident. At the edge of the continuous spectrum $k=0$ the scattering matrix generically looks like

$$\begin{equation*} \mathcal{S}(0) = \begin{pmatrix} 0 & -1\\ -1 & 0 \end{pmatrix}. \end{equation*}$$

More precisely, this happens when the zero energy is non-resonant, that is, if the equation $H f_0 =0$ has no bounded (distributional) solution. In the resonant situation the behavior of the scattering matrix is more delicate. For $V\in L^1_1$ it is already non-trivial to establish continuity of the scattering matrix at $k=0$ (for $V\in L^1_2$ de l’Hospital’s rule suffices). This question arose around 1985 in an attempt to clarify whether the low-energy asymptotics of the scattering matrix, obtained for $V\in L_2^1$ in 6, are valid for $V\in L_1^1$. The problem was solved independently by Guseĭnov 12 and Klaus 13 (for a refined version see 1 and 9, Theorem 2.1). It also plays an important role in semiclassical analysis; see 5 and the references therein. Furthermore, if $V\in L^1_{j+1}$ with $j>0$, then, away from $0$, one can take derivatives of the scattering matrix up to order $j$, and in the non-resonant case it is again easy to see that they are continuous at $k=0$. This clearly raises the question about continuity at $k=0$ of the $j$-th derivative in the resonant case. We will establish this as one of our main results in Theorem 2.2. This result is new even for the first derivative. We remark that if $V$ decays exponentially, then $\mathcal{S}(k)$ is analytic in a neighborhood of $k=0$ and the full Taylor expansion can be obtained 23.

It is important to emphasize that this question is not only of interest in scattering theory but also plays a role in the solution of the Korteweg–de Vries equation via the inverse scattering transform (see e.g. 11 where continuity of higher derivatives is needed) and in deriving dispersive estimates for 1.1. The latter case has attracted considerable interest (e.g. 910 and the references therein) due to its importance for proving asymptotic stability of solitons for the associated non-linear evolution equations (see e.g. 414).

As an application of our results we will establish the following dispersive decay estimate with integrable time decay in the resonant case:

Theorem 1.1.

Suppose $V \in L_3^1({\mathbb{R}})$ and suppose there is a bounded solution $f_0$ of $H f_0=0$ satisfying the normalization $\lim _{x \to \infty } (|f_0(x)|^2 + |f_0(-x)|^2)=2$. Denote by $P_0:L^1_2 \to L^\infty _{-2}$ the operator given by the kernel $[P_0](x,y)=f_0(x)f_0(y)$. By $P_{ac}$ we denote the projector onto the absolutely continuous subspace of $H$. Then $\mathrm{e}^{-\mathrm{i}tH}P_{ac}$ extends to a bounded operator $L^1_2 \to L^\infty _{-2}$ satisfying the following decay estimate:

$$\begin{equation} \Vert \mathrm{e}^{-\mathrm{i}tH}P_{ac}-(4 \pi \mathrm{i}t)^{-\frac{1}{2}}P_0 \Vert _{L^1_2 \to L^\infty _{-2}}=\mathcal{O}(t^{-3/2}),\quad t\to \infty . \tag{1.2}{} \end{equation}$$

This theorem is an improvement of an earlier result by Goldberg 10 who established it for $V \in L_4^1({\mathbb{R}})$. If there is no resonance (i.e. no bounded solution) this result (with $P_0=0$) was shown for $V \in L_2^1({\mathbb{R}})$ in 9, Theorem 1.2. For extensions to discrete one-dimensional Schrödinger equations (Jacobi operators) see 78.

2. Low energy scattering

In this section we establish some properties of the scattering data for our operator $H$ with $V\in L_{j+1}^1$, $j\geq 0$. To this end we introduce the Banach algebra ${\mathcal{A}}$ of Fourier transforms of integrable functions,

$$\begin{equation} {\mathcal{A}}= \left\{f(k):\, f(k) = \int _{\mathbb{R}}\mathrm{e}^{\mathrm{i}k p}\hat{f}(p)dp, \,\hat{f}(\cdot )\in L^1({\mathbb{R}}) \right\}, \tag{2.1}{} \end{equation}$$

with the norm $\|f\|_{{\mathcal{A}}}= \|\hat{f}\|_{L^1}$, plus the corresponding unital Banach algebra ${\mathcal{A}}_1$,

$$\begin{equation} {\mathcal{A}}_1 = \left\{f(k):\, f(k) =c+ \int _{\mathbb{R}}\mathrm{e}^{\mathrm{i}k p}\hat{g}(p)dp, \,\hat{g}(\cdot )\in L^1({\mathbb{R}}),\,c\in {\mathbb{C}}\right\}, \tag{2.2}{} \end{equation}$$

with the norm $\|f\|_{{\mathcal{A}}_1}= |c|+\|\hat{g}\|_{L^1}$. We also use the fact, which is known as Wiener’s lemma 17, that if $f\in {\mathcal{A}}_1\setminus {\mathcal{A}}$ and $f(k)\not =0$ for all $k\in {\mathbb{R}}$, then $f^{-1}(k)\in {\mathcal{A}}_1$.

First we recall a few facts from scattering theory 615. If $V\in L^1_1$ there exist Jost solutions $f_\pm (x,k)$ of $H f=k^2 f$, $k\in \overline{{\mathbb{C}}_+}$, which asymptotically behave like $f_\pm (x,k)\sim \mathrm{e}^{\pm \mathrm{i}kx}$ as $x\to \pm \infty$. These solutions are given by

$$\begin{equation} f_\pm (x,k) = \mathrm{e}^{\pm \mathrm{i}kx} h_\pm (x,k), \qquad h_\pm (x,k) = 1 \pm \int _{0}^{\pm \infty } B_\pm (x,y) \mathrm{e}^{\pm 2\mathrm{i}k y}dy, \tag{2.3}{} \end{equation}$$

where $B_\pm (x,y)$ are real-valued and satisfy (see 6, §2 or 15, §3.1)

$$\begin{align} |B_\pm (x,y)|&\le \mathrm{e}^{\gamma _\pm (x)}\eta _\pm (x+y),\tag{2.4}{}\\ |\frac{\partial }{\partial x}B_\pm (x,y)\pm V(x+y)| &\le 2 \mathrm{e}^{\gamma _\pm (x)}\eta _\pm (x+y)\eta _\pm (x), \tag{2.5}{} \end{align}$$

with

$$\begin{equation} \gamma _\pm (x)=\int _{x}^{\pm \infty }(y-x)|V(y)|dy,\quad \eta _\pm (x)=\pm \int _{x}^{\pm \infty }|V(y)|dy. \tag{2.6}{} \end{equation}$$

Since $\eta _\pm \in L^1({\mathbb{R}}_\pm )$ we conclude that

$$\begin{equation} h_\pm (x,\cdot ) - 1, \quad h'_\pm (x,\cdot )\in \mathcal{A}, \quad \forall x\in {\mathbb{R}}. \tag{2.7}{} \end{equation}$$

Here and throughout the rest of this paper a prime will always denote a derivative with respect to the spatial variable $x$. As an immediate consequence of the estimates 2.4 and 2.5 we have the following strengthening of 2.7.

Lemma 2.1.

Let $V \in L_{j+1}^1$. Then $\frac{\partial ^l}{\partial k^l} (h_\pm (x,k)-1)$, $\frac{\partial ^l}{\partial k^l} h'_\pm (x,k)$ are contained in $\mathcal{A}$ for $0\le l \le j$. Moreover, for $\pm x \geq 0$, the $\mathcal{A}$-norms of these expressions do not depend on $x$.

The fact that $f_\pm (x,-k)$ also solve $H f=k^2 f$ for $k\in {\mathbb{R}}$ leads to the scattering relations

$$\begin{equation} T(k)f_\pm (x,k)=R_\mp (k)f_\mp (x,k) + f_\mp (x,-k), \tag{2.8}{} \end{equation}$$

where the transmission coefficient $T$ and the reflection coefficients $R_\pm$ can be expressed in terms of Wronskians. To this end we denote by

$$\begin{equation*} W(f(x),g(x))= f(x)g'(x)-f'(x)g(x) \end{equation*}$$

the usual Wronskian and set

$$\begin{equation*} W(k)=W(f_-(x,k),f_+(x,k)),\qquad W_\pm (k)=W(f_\mp (x,k),f_\pm (x,- k)). \end{equation*}$$

Then

$$\begin{equation} T(k)= \frac{2\mathrm{i}k}{W(k)},\quad R_\pm (k)= \mp \frac{W_\pm (k)}{W(k)}. \tag{2.9}{} \end{equation}$$

The transmission and reflection coefficients are elements of the Wiener algebra, which was established in 9, Theorem 2.1. Here we extend this result to the derivatives of the scattering data.

Theorem 2.2.

If $V\in L^1_{j+1}$, then $\frac{d^l}{dk^l}(T(k)-1)\in {\mathcal{A}}$ and $\frac{d^l}{dk^l}R_\pm (k)\in {\mathcal{A}}$ for $0\le l \le j$.

Proof.

We only focus on the resonant case, since the other case is straightforward.

First of all, we abbreviate $h_\pm (k)=h_\pm (0,k)$, $h'_\pm (k)=h'_\pm (0,k)$. Then 2.3 leads us to

$$\begin{align} W(k) &=2\mathrm{i}k h_+(k)h_-(k)+ h_-(k)h'_+(k)-h'_-(k)h_+(k),\tag{2.10}{}\\ W_\pm (k) &=h_\mp (k)h'_\pm (-k)-h_\pm (-k)h'_\mp (k). \tag{2.11}{} \end{align}$$

Following 9 we introduce

$$\begin{equation} \Phi _\pm (k)=h_\pm (k)h'_\pm (0)-h'_\pm (k)h_\pm (0), \tag{2.12}{} \end{equation}$$

$$\begin{equation} K_\pm (x,y)=\pm \int _y^{\pm \infty }B_\pm (x,z)dz,\quad D_\pm (x,y)=\pm \int _y^{\pm \infty }\frac{\partial }{\partial x}B_\pm (x,z)dz, \tag{2.13}{} \end{equation}$$ where $B_\pm (x,y)$ are given by 2.3. Again $K_\pm (0,y)$ is denoted by $K_\pm (y)$ and similarly for $D_\pm (y)$. In 9, Theorem 2.1, the following equation for $\Phi _\pm (k)$ was obtained:

$$\begin{equation} \Phi _\pm (k)=2\mathrm{i}k\Psi _\pm (k),\quad \Psi _\pm (k)=\int _0^{\pm \infty }H_\pm (y)\mathrm{e}^{\pm 2\mathrm{i}ky}dy, \tag{2.14}{} \end{equation}$$

where

$$\begin{equation} H_\pm (x)=D_\pm (x)h_\pm (0)-K_\pm (x)h'_\pm (0),\quad \pm x\geq 0. \tag{2.15}{} \end{equation}$$

Moreover, $H_\pm$ satisfy an estimate similar to 2.4 as we will show in Lemma 2.3 below. As a consequence, $\Psi _\pm$ and its derivatives up to order $j$ will be in the Wiener algebra.

Next a straightforward computation (cf. 9) shows

$$\begin{equation*} \frac{W(k)}{2\mathrm{i}k}= h_-(k) h_+(k) + \begin{cases} \frac{h_+(k)}{h_-(0)}\Psi _-(k)-\frac{h_-(k)}{h_+(0)}\Psi _+(k), & h_+(0)h_-(0)\not =0,\\[5.0pt] \frac{h'_+(k)}{h'_-(0)}\Psi _-(k)-\frac{h'_-(k)}{h'_+(0)}\Psi _+(k), & h_+(0)=h_-(0) =0,\end{cases} \end{equation*}$$

and since $\frac{W(k)}{2\mathrm{i}k} =T(k)^{-1} \ne 0$ for all $k\in {\mathbb{R}}$, and $T(k)\to 1$ as $k\to \infty$, we conclude that $\frac{d^l}{dk^l}(T(k)-1)\in {\mathcal{A}}$ for $0\le l \le j$. Analogously,

$$\begin{equation*} \frac{W_\pm (k)}{2\mathrm{i}k}= \begin{cases} \frac{h_\pm (-k)}{h_{\mp }(0)}\Psi _{\mp }(k)-\frac{h_\mp (k)}{h_\pm (0)}\Psi _\pm (-k) , & h_+(0)h_-(0)\not =0,\\[5.0pt] \frac{h'_\pm (-k)}{h'_\mp (0)}\Psi _\mp (k)-\frac{h'_\mp (k)}{h'_\pm (0)}\Psi _\pm (-k), & h_+(0)=h_-(0) =0,\end{cases} \end{equation*}$$

and hence $R_\pm (k) = \mp \frac{W_\pm (k)}{2\mathrm{i}k} T(k)$ has the claimed properties.

■

To complete the proof of Theorem 2.2 we need the following result which is an extension of 9, Lemma 2.2.

Lemma 2.3.

Let $H_\pm (x)$ be given by 2.15. For $V\in L^1_1$ the following estimate is valid:

$$\begin{equation} |H_\pm (x)| \leq \hat{C} \eta _\pm (x),\quad \pm x\geq 0,\tag{2.16}{} \end{equation}$$

with some constant $\hat{C}>0$ and $\eta _\pm (x)$ given by 2.6. Moreover, for $\Psi _\pm (k)$ defined by 2.14 and for $V\in L^1_{j+1}$ we have

$$\frac{d^l}{dk^l} \Psi _\pm (k)\in {\mathcal{A}},\quad 0\le l \le j.$$

Proof.

It suffices to prove the estimate for $H_\pm$. The Marchenko equation (§3.5 in 15) states that the kernels $B_\pm (x,y)$ solve the equations