Wave operators on Sobolev spaces

TL;DR

This paper establishes a general condition for the existence and completeness of wave operators on Sobolev spaces, with applications to Schrödinger operators and nonlinear Schrödinger equations.

Contribution

It introduces a simple abstract criterion linking wave operators on Sobolev spaces to classical wave operators, expanding scattering theory tools.

Findings

Abstract condition for wave operators on Sobolev spaces

Application to Schrödinger operators with potentials

Extension to nonlinear Schrödinger equations

Abstract

We provide a simple sufficient condition in an abstract framework to deduce the existence and completeness of wave operators (resp. modified wave operators) on Sobolev spaces from the existence and completeness of the usual wave operators (resp. modified wave operators). We then give some examples of Schr\"odinger operators to which our abstract result applies. An application to scattering theory for the nonlinear Schr\"odinger equation with a potential is also given.

Equations

$$W_{\pm ,s}=\underset{t\to \pm \infty }{s-lim}{e}^{itH}{e}^{-it{H}_0}{P}_{\mathrm{ac}}(H_0),{\Omega }_{\pm ,s}=\underset{t\to \pm \infty }{s-lim}{e}^{it{H}_0}{e}^{-itH}{P}_{\mathrm{ac}}(H)\text{in}{\mathcal{H}}^s.$$

$${||e^{-itH}{u}_{\pm }-e^{-it{H}_0}{P}_{\mathrm{ac}}(H_0)u_0||}_s\to 0,t\to \pm \infty ,$$

$${||e^{-itH}{P}_{\mathrm{ac}}(H)u_0-e^{-it{H}_0}{u}_{\pm }||}_s\to 0,t\to \pm \infty .$$

$$(i{\partial }_t+\Delta -V)u=N(u);u{|}_{t=0}=u_0\in H^s({\mathbb{R}}^n).$$

$$||⟨H{⟩}^{s/2}f||\lesssim ||⟨H_0{⟩}^{s/2}f||,||⟨H_0{⟩}^{s/2}g||\lesssim ||⟨H{⟩}^{s/2}g||.$$

$${||u_0-{\phi }_R(H_0)u_0||}_s^2={\int }_{\sigma (H_0)}|1-\phi (\lambda /R){|}^{2}d{||E_{H_0}(\lambda )u_0||}_s^2\to 0,R\to \infty ,$$

$$\underset{n,m}{\sup }{||(W(t_n)-W(t_m))(u_0-{\phi }_R(H_0)u_0)||}_s\lesssim {||u_0-{\phi }_R(H_0)u_0||}_s<\epsilon .$$

$$[W(t_n)-W(t_m)]u_R={\psi }_R(H)[W(t_n)-W(t_m)]u_R+(1-{\psi }_R)(H)[W(t_n)-W(t_m)]u_R.$$

$${||{\psi }_R(H)[W(t_n)-W(t_m)]u_R||}_s\lesssim R^{s/2}||[W(t_n)-W(t_m)]u_R||\to 0,m,n\to \infty .$$

$${||(1-{\psi }_R)(H)[W(t_n)-W(t_m)]u_R||}_s$$

$$\le {||[{\psi }_R(H_0)-{\psi }_R(H)]U_0(t_n)P_{\mathrm{ac}}(H_0)u_R||}_{{}^s}+{||[{\psi }_R(H_0)-{\psi }_R(H)]U_0(t_m)P_{\mathrm{ac}}(H_0)u_R||}_s\to 0$$

$$⟨H_0{⟩}^{s/2}[(H_0-z{)}^{-1}-(H-z{)}^{-1}]$$

$$=⟨H_0{⟩}^{s/2}(H_0-z{)}^{-1}V(H-z{)}^{-1}$$

$$=⟨H_0{⟩}^{s/2+1/2}(H_0-z{)}^{-1}\cdot ⟨H_0{⟩}^{-1/2}|V{|}^{1/2}\cdot \mathrm{sgn}V|V{|}^{1/2}⟨H_0{⟩}^{-1/2}\cdot ⟨H_0{⟩}^{1/2}(H-z{)}^{-1},$$

$$\frac{(n-2{)}^2}{4}\int |x{|}^{-2}|f{|}^2\le \int |\nabla f{|}^2,f\in C_0^{\infty }({\mathbb{R}}^n),$$

$$⟨H_0{⟩}^{s/2}(H-z{)}^{-1}|x{|}^{-2}(H_0-z{)}^{-1}$$

$$=⟨H_0{⟩}^{s/2}⟨H{⟩}^{-1/2}\cdot ⟨H{⟩}^{1/2}(H-z{)}^{-1}⟨H{⟩}^{1/2}\times$$

$$\times ⟨H{⟩}^{-1/2}⟨H_0{⟩}^{1/2}\cdot ⟨H_0{⟩}^{-1/2}|x{|}^{-1}\cdot |x{|}^{-1}⟨H_0{⟩}^{-1}\cdot ⟨H_0⟩(H_0-z{)}^{-1}$$

$${\partial }_{t}S(t,\xi )=|\xi {|}^2+V^L({\nabla }_{\xi }S(t,\xi ));$$

$$H_{\alpha }f=-f^{\prime \prime },D(H_{\alpha })=\{f\in H^1(\mathbb{R})\cap H^2(\mathbb{R}\setminus \{0\})|f^{\prime }(0+)-f^{\prime }(0-)=\alpha f(0)\}.$$

$$K_z(x,x^{\prime })=-2\alpha \sqrt{z}(i\alpha +2\sqrt{z}{)}^{-1}{e}^{i\sqrt{z}(|x|+|x^{\prime }|)}$$

$$K_z^{2}f(x):=-2\alpha \sqrt{z}(i\alpha +2\sqrt{z}{)}^{-1}⟨H_0{⟩}^{s/2}\int e^{i\sqrt{z}(|x|+|x^{\prime }|/2)}f(x^{\prime })dx.$$

$$i{\partial }_{t}u+\Delta u-V(x)u=|u{|}^{\sigma }u\text{on}\mathbb{R}\times {\mathbb{R}}^n;u{|}_{t=0}=u_0\in {\mathcal{H}}^1,$$

$$\{\begin{array}{ll}a>-\frac{(n-2{)}^2}{4}& \text{ifn=3 and 4/3<σ≤2},\\ a>-\frac{(n-2{)}^2}{4}+{\left(\frac{n-2}{2}-\frac{1}{\sigma }\right)}^2& \text{if3≤n≤6 and max(n−22,n4)<σ<n−24}.\end{array}$$

$${||V||}_{\mathcal{K}}=\underset{x\in {\mathbb{R}}^3}{\sup }{\int }_{{\mathbb{R}}^3}|x-y{|}^{-1}|V(y)|dy$$

$${||u(t)-e^{-itH}{\tilde{u}}_{\pm }||}_{{\mathcal{H}}^1}\to 0,t\to \pm \infty .$$

$${||u(t)-e^{-it{H}_0}{u}_{\pm }||}_{{\mathcal{H}}^1}\to 0,t\to \pm \infty .$$

$$⟨H_0{⟩}^{s/2}[\psi (H_0)-\psi (H)]\phi (H_0)=\frac{1}{2\pi i}{\int }_{\mathbb{C}}\frac{\partial \tilde{\phi }}{\partial \overline{z}}(z)⟨H_0{⟩}^{s/2}[(H_0-z{)}^{-1}-(H-z{)}^{-1}]\phi (H_0)dz\wedge d\overline{z}$$

$${||wf||}_{L^2({\mathbb{R}}^n)}\le {||w||}_{L^r({\mathbb{R}}^n)}{||f||}_{L^{\frac{2r}{r-2}}({\mathbb{R}}^n)}\lesssim {||w||}_{L^r({\mathbb{R}}^n)}{||f||}_{H^{n/r}({\mathbb{R}}^n)}.$$

$${||wf||}_{L^2(\mathbb{R})}\le {||w||}_{L^2(\mathbb{R})}{||f||}_{L^{\infty }(\mathbb{R})}\lesssim {||w||}_{L^2(\mathbb{R})}{||f||}_{H^1(\mathbb{R})}.$$

Full text

1. Introduction

Let ${\mathcal{H}}$ be a Hilbert space with norm ${||\cdot||}$ and inner product ${\langle}\cdot,\cdot{\rangle}$, and $H_{0},H$ two self-adjoint operators on ${\mathcal{H}}$. The operator norm on ${\mathcal{H}}$ is also denoted by ${||\cdot||}$. Define the Sobolev space ${\mathcal{H}}^{s}:={\langle}H_{0}{\rangle}^{-s/2}{\mathcal{H}}$ of order $s$ with norm ${||f||}_{s}:={||{\langle}H_{0}{\rangle}^{s/2}f||},$ where ${\langle}\cdot{\rangle}=(1+|\cdot|^{2})^{1/2}$. A typical example is the Schrödinger operators $H_{0}=-\Delta$ and $H=H_{0}+V$ on $L^{2}({\mathbb{R}}^{n})$ with a real-valued potential $V:{\mathbb{R}}^{n}\to{\mathbb{R}}$ in which case ${\mathcal{H}}^{s}=H^{s}({\mathbb{R}}^{n})$ is the standard $L^{2}$-based Sobolev space.

We regard $H_{0}$ as a free operator and study the scattering theory for the pair $(H_{0},H)$, namely the asymptotic behavior of $e^{-itH}P_{\mathrm{ac}}(H)$ in the limit $t\to\infty$ as a perturbation of the free motion $e^{-itH_{0}}P_{\mathrm{ac}}(H_{0})$ (or a suitable modified free motion in the long-range case), where $P_{\mathrm{ac}}(H)$ denotes the projection onto ${\mathcal{H}}_{\mathrm{ac}}(H)$, the absolutely continuous spectral subspace of $H$. In particular, we are interested in the asymptotic behavior of $e^{-itH}P_{\mathrm{ac}}(H)$ in ${\mathcal{H}}^{s}$ for $s\neq 0$. To this end, we consider the existence of the wave and inverse wave operators

[TABLE]

Under the norm equivalence condition ${\mathcal{H}}^{s}\cong D({\langle}H{\rangle}^{s/2})$ (see the condition (H1) below), the existence of $W_{\pm,s}$ implies, for any $u_{0}\in{\mathcal{H}}^{s}$ there exist $u_{\pm}\in{\mathcal{H}}^{s}$ such that

[TABLE]

while the existence of $\Omega_{\pm,s}$ implies, for any $u_{0}\in{\mathcal{H}}^{s}$ there exist $u_{\pm}\in{\mathcal{H}}^{s}$ such that

[TABLE]

The existence of the usual wave operators $W_{\pm,0},\Omega_{\pm,0}$ has developed from almost the beginning of mathematical analysis of Quantum Mechanics and there is a huge literature (see, for instance, monographs [11, 4, 12]). On the other hand, to the best knowledge of the author, the case $s\neq 0$ has attracted less interest. However, the wave operators on Sobolev spaces appear naturally in the study of scattering theory for the nonlinear Schrödinger equation

[TABLE]

Indeed, it often happens that, with a suitable nonlinear term $N(u)$, the nonlinear Schrödinger equation has a global solution for $u_{0}\in H^{s}({\mathbb{R}}^{n})$ with some $s>0$, but not for $u_{0}\in L^{2}({\mathbb{R}}^{n})$. It is then natural to consider the scattering theory in the same topology of the initial data space.

This short note provides a simple sufficient condition in an abstract framework to deduce the existence and completeness of $W_{\pm,s}$ from the existence and completeness of $W_{\pm,0}$. Some applications to Schrödinger operators with potentials are also given. We also give an application to the scattering theory for a nonlinear Schrödinger equation with a linear potential.

The paper is organized as follows. Section 2 is devoted to the main theorem and its proof. In Section 3, we give some applications of the main result to Schrödinger operators and the nonlinear scattering theory. Appendix A consists of some supplementary lemmas.

2. Main result

In what follows, we use the following notation. $\mathbb{B}(X,Y)$ (resp. $\mathbb{B}_{\infty}(X,Y)$) denotes the Banach space of bounded (resp. compact) operators from $X$ to $Y$. Let $\mathbb{B}(X)=\mathbb{B}(X,X)$ and $\mathbb{B}_{\infty}(X)=\mathbb{B}_{\infty}(X,X)$. For positive constants $A,B$, $A\lesssim B$ means $A\leq CB$ with some constant $C>0$.

Let us fix $s\in{\mathbb{R}}$ and consider the following series of assumptions.

• (H1)

${\mathcal{H}}^{s}\cong D({\langle}H{\rangle}^{s/2})$. In other words, for all $f\in{\mathcal{H}}^{s}$ and $g\in D({\langle}H{\rangle}^{s/2})$,

[TABLE]

• (H2)

$\displaystyle(H_{0}-z)^{-1}-(H-z)^{-1}\in\mathbb{B}_{\infty}({\mathcal{H}}^{s})\cup\mathbb{B}_{\infty}({\mathcal{H}},{\mathcal{H}}^{s})$ for all $z\in{\mathbb{C}}\setminus{\mathbb{R}}$.

• (H3)

There exists a family of unitary operators $\{U_{0}(t)\}_{t\in{\mathbb{R}}}$ on ${\mathcal{H}}$ which commutes with $H_{0}$, that is $U_{0}(t){\mathcal{H}}^{2}\subset{\mathcal{H}}^{2}$ and $[U_{0}(t),H_{0}]=0$ on ${\mathcal{H}}^{2}$ for all $t\in{\mathbb{R}}$.

• (H4)

For any $f\in{\mathcal{H}}_{\mathrm{ac}}(H_{0})$, $U_{0}(t)f\to 0$ weakly in ${\mathcal{H}}$ as $t\to\pm\infty$. Moreover, the wave operators $\displaystyle W_{\pm}:=\operatorname*{s-lim}_{t\to\pm\infty}e^{itH}U_{0}(t)P_{\mathrm{ac}}(H_{0})$ in ${\mathcal{H}}$ exist.

• (H5)

The inverse wave operators $\displaystyle\Omega_{\pm}:=\operatorname*{s-lim}_{t\to\pm\infty}U_{0}(t)^{*}e^{-itH}P_{\mathrm{ac}}(H)$ in ${\mathcal{H}}$ exist.

Remark 2.1**.**

(1) (H1) and (H2) imply $[\psi(H_{0})-\psi(H)]\varphi(H_{0})\in\mathbb{B}_{\infty}({\mathcal{H}},{\mathcal{H}}^{s})$ for any $\varphi,\psi\in C_{0}^{\infty}({\mathbb{R}})$ (see Lemma A.1 in Appendix A below).

(2) By a standard approximation argument, we see that $U_{0}(t)$ also commutes with $\varphi(H_{0})$ for any $\varphi\in L^{\infty}({\mathbb{R}})\cup C({\mathbb{R}})$. In particular, $U_{0}(t)$ is bounded on ${\mathcal{H}}^{s}$ uniformly in $t\in{\mathbb{R}}$.

(3) Under (H4), (H5) is equivalent to $\mathop{\mathrm{Ran}}\nolimits W_{\pm}={\mathcal{H}}_{\mathrm{ac}}(H)$ in which case $\Omega_{\pm}=(W_{\pm})^{*}$ (see [11]).

(4) By the Riemann-Lebesgue lemma, for all $f\in{\mathcal{H}}_{\mathrm{ac}}(H)$, $e^{-itH}f\to 0$ weakly in ${\mathcal{H}}$ as $t\to\pm\infty$.

Theorem 2.2**.**

*Let $s\in{\mathbb{R}}$. Under (H1), (H2) and (H3), the following statements hold:

(1) If (H4) is satisfied then $\displaystyle W_{\pm,s}:=\operatorname*{s-lim}_{t\to\pm\infty}e^{itH}U_{0}(t)P_{\mathrm{ac}}(H_{0})$ in ${\mathcal{H}}^{s}$ exist.

(2) If (H5) is satisfied then $\displaystyle\Omega_{\pm,s}:=\operatorname*{s-lim}_{t\to\pm\infty}U_{0}(t)^{*}e^{-itH}P_{\mathrm{ac}}(H)$ in ${\mathcal{H}}^{s}$ exist.*

Proof.

We prove the existence of $W_{+,s}$ only, the proof of other statements being analogous. Let $u_{0}\in{\mathcal{H}}^{s}$ and $W(t)=e^{itH}U_{0}(t)P_{\mathrm{ac}}(H_{0})$ which is bounded on ${\mathcal{H}}^{s}$ uniformly in $t$ by (H1) and (H3). We shall show that, for any sequence $t_{n}\to\infty$, $\{W(t_{n})u_{0}\}$ is a Cauchy sequence in ${\mathcal{H}}^{s}$. Let us fix ${\varepsilon}>0$ and $\varphi\in C_{0}^{\infty}({\mathbb{R}})$ be such that $\varphi\equiv 1$ near origin and set $\varphi_{R}(\lambda)=\varphi(\lambda/R)$. Then

[TABLE]

by the dominated convergence theorem. In particular, there exists $R\geq 1$ such that

[TABLE]

We thus may replace $u_{0}$ by $u_{R}:=\varphi_{R}(H_{0})u_{0}$ without loss of generality. Choose $\psi\in C_{0}^{\infty}({\mathbb{R}})$ so that $\psi\varphi\equiv\varphi$. Then

[TABLE]

Since ${||{\langle}H_{0}{\rangle}^{s/2}\psi_{R}(H)||}\lesssim R^{s/2}$ by (H1), the second assumption in (H4) implies

[TABLE]

Moreover, since $(1-\psi_{R})(H)\varphi_{R}(H_{0})=[\psi_{R}(H_{0})-\psi_{R}(H)]\varphi_{R}(H_{0})$, we have

[TABLE]

as $m,n\to\infty$, where we have used (H3) in the first inequality, the first condition in (H4) and Remark 2.1 (1) in the last step, respectively. Hence $\{W(t_{n})u_{R}\}$ is a Cauchy sequence in ${\mathcal{H}}^{s}$. ∎

3. Application to Schrödinger equations

Here we apply the above theorem to the scattering theory for Schrödinger equations on ${\mathbb{R}}^{n}$. Throughout this section, we set ${\mathcal{H}}=L^{2}({\mathbb{R}}^{n})$ and $H_{0}=-\Delta$ with $D(H_{0})=H^{2}({\mathbb{R}}^{n})$ in which case ${\mathcal{H}}^{s}=H^{s}({\mathbb{R}}^{n})$. We first give some typical examples of potentials satisfying (H1) and (H2).

Example 3.1**.**

Let $\max(n/2,1)\leq p<\infty$, and $p>1$ if $n=2$. Suppose that $V:{\mathbb{R}}^{n}\to{\mathbb{R}}$ belongs to $L^{p}({\mathbb{R}}^{n})+L^{\infty}_{0}({\mathbb{R}}^{n})$, where $L^{\infty}_{0}({\mathbb{R}}^{n})$ is the $L^{\infty}$-norm closure of $C_{0}^{\infty}({\mathbb{R}}^{n})$. Then $V$ is $H_{0}$-form compact (see Lemmas A.2 in Appendix A below), having relative bound zero. By the KLMN theorem, we can define a self-adjoint operator $H=H_{0}+V$ as the form sum such that $D({\langle}H{\rangle}^{1/2})\cong{\mathcal{H}}^{1}$. Then (H1) and (H2) hold for $s\in[-1,1]$. Indeed, the complex interpolation and a duality argument show $D({\langle}H{\rangle}^{s/2})\cong{\mathcal{H}}^{s}$ for $s\in[-1,1]$. For the part (H2), we compute

[TABLE]

where ${\langle}H_{0}{\rangle}^{-1/2}|V|^{1/2}$ is compact and the other terms are bounded on ${\mathcal{H}}$ as long as $s\leq 1$.

Example 3.2**.**

Let $n\geq 3$, $a>-(n-2)^{2}/4$ and $V(x)=a|x|^{-2}$. By Hardy’s inequality

[TABLE]

$H=H_{0}+V$ defined as the Friedrichs extension of the quadratic form ${\langle}(H_{0}+V)u,u{\rangle}$ on $C_{0}^{\infty}({\mathbb{R}}^{n})$ satisfies $D({\langle}H{\rangle}^{1/2})\cong{\mathcal{H}}^{1}$. Hence (H1) holds for $s\in[-1,1]$. Moreover, writing

[TABLE]

we see that $|x|^{-1}{\langle}H_{0}{\rangle}^{-1}\in\mathbb{B}_{\infty}({\mathcal{H}})$ since $|x|^{-1}\in L^{p}({\mathbb{R}}^{n})+L_{0}^{\infty}({\mathbb{R}}^{n})$ with some $n/2<p<n$, otherwise are bounded on ${\mathcal{H}}$ if $s\leq 1$. Therefore $(H_{0}-z)^{-1}-(H-z)^{-1}\in\mathbb{B}_{\infty}({\mathcal{H}},{\mathcal{H}}^{s})$ for $s\leq 1$.

We next provide some examples to which our abstract theorem applies.

Example 3.3** (Short range potential).**

Assume that $V:{\mathbb{R}}^{n}\to{\mathbb{R}}$ satisfies one of the following:

• •

$n=1$ and $V\in L^{1}({\mathbb{R}})$;

• •

$n\geq 2$ and $V=V_{0}+V_{1}+\cdots+V_{N}$, where

• $\bullet$

$|V_{0}(x)|\lesssim{\langle}x{\rangle}^{-\rho}$ for some $\rho>1$ and,

• $\bullet$

$V_{j}\in L^{p_{j}}({\mathbb{R}}^{n})$ for $p_{j}$ satisfying $n/2\leq p_{j}\leq(n+1)/2$ and $p_{j}>1$ for $j=1,...,N$;

• •

$n\geq 3$ and $V(x)=a|x|^{-2}$ with $a>-(n-2)^{2}/4$.

Then (H1) and (H2) hold for $s\in[-1,1]$ by the above examples. Moreover, the usual wave operators $\displaystyle W_{\pm}=\operatorname*{s-lim}_{t\to\pm\infty}e^{itH}e^{-itH_{0}}$ in ${\mathcal{H}}$ exist and are complete. We refer to Reed-Simon [11, Theorem XI.30] for the first case, Ionescu-Schlag [7] for the second case, respectively. For the last case, the existence and completeness of $W_{\pm}$ follow from the fact that $|x|^{-1}$ is both $H_{0}$-smooth and $H$-smooth in the sense that $|x|^{-1}e^{-itH_{0}},|x|^{-1}e^{-itH}\in\mathbb{B}(L^{2}({\mathbb{R}}^{n}),L^{2}({\mathbb{R}}^{1+n}))$ (see [3]) and the smooth perturbation theory by Kato [8]. Hence, for all $s\in[-1,1]$, Theorem 2.2 with $U_{0}(t)=e^{-itH_{0}}$ applies.

Example 3.4** (Long-range potential).**

Let $n\geq 1$ and $V=V^{S}+V^{L}:{\mathbb{R}}^{n}\to{\mathbb{R}}$ be such that

• •

$V^{S}$ is $H_{0}$-compact and ${\langle}H_{0}{\rangle}^{-1}V(x)\mathds{1}_{[1,\infty)}(|x|/r){\langle}H_{0}{\rangle}^{-1/2}\in L^{2}([0,\infty)_{r};L^{2}({\mathbb{R}}^{n}_{x}))$;

• •

$V^{L}\in C^{\infty}({\mathbb{R}}^{n})$ and, with some $\rho>0$, $|\partial_{x}^{\alpha}V(x)|\leq C_{\alpha}{\langle}x{\rangle}^{-\rho-|\alpha|}$ on ${\mathbb{R}}^{n}$ for all $\alpha\in{\mathbb{Z}}_{+}^{n}$.

Then (H1) and (H2) hold for $s\in[-2,2]$ since $V$ is $H_{0}$-compact. Moreover, there exists $S\in C^{\infty}({\mathbb{R}}\times{\mathbb{R}}^{n};{\mathbb{R}})$ such that

• •

$\partial_{\xi}^{\alpha}(S(t,\xi)-t|\xi|^{2})=o(t)$ as $|t|\to\infty$ for $|\alpha|\leq 2$;

• •

for any ${\varepsilon}>0$, there exists $T_{\varepsilon}>0$ such that for $|\xi|\geq{\varepsilon}$, $|t|\geq T_{\varepsilon}$, $S(t,\xi)$ solves

[TABLE]

• •

the modified wave operators $\displaystyle W_{\pm}^{S}=\operatorname*{s-lim}_{t\to\pm\infty}e^{itH}e^{-iS(t,D)}$ in ${\mathcal{H}}$ exist and are complete.

We refer to [4, Theorem 4.7.1]. Moreover, it follows from the above asymptotics of $S$ that for any $f\in{\mathcal{H}}$, $e^{-iS(t,D)}f\to 0$ weakly in ${\mathcal{H}}$ as $t\to\infty$ by the stationary phase theorem. Theorem 2.2 with $U_{0}(t)=e^{-iS(t,D)}$ thus applies for $s\in[-2,2]$.

Remark 3.5**.**

As a typical example, $V(x)=-Z|x|^{-\mu}$ with $Z\in{\mathbb{R}}$ and $\mu\in(0,1]$ satisfies the above condition in Example 3.4 if $n\geq 3$.

Example 3.6** (Point interaction).**

Let $\alpha\in{\mathbb{R}}$ and $H_{\alpha}=-\partial_{x}^{2}+\alpha\delta$ be the Schrödinger operator with a delta potential in ${\mathbb{R}}$. More precisely, $H_{\alpha}$ is defined as follows:

[TABLE]

Note that $H_{\alpha}|_{\alpha=0}$ coincides with $H_{0}f=-f^{\prime\prime}$ with $D(H_{0})={\mathcal{H}}^{2}$. Then the form domain of $H_{\alpha}$ is ${\mathcal{H}}^{1}$ and $K_{z}:=(H_{\alpha}-z)^{-1}-(H_{0}-z)^{-1}$ is a rank one operator with the kernel

[TABLE]

where we take a branch of $\sqrt{z}$ so that $\mathop{\mathrm{Im}}\nolimits\sqrt{z}>0$ (see [1, Chapter 1.3]). In particular, $K_{z}$ is in the trace class. We decompose ${\langle}H_{0}{\rangle}^{s/2}K_{z}{\langle}H_{0}{\rangle}^{-s/2}=K_{z}^{2}K_{z}^{1}$, where $K_{z}^{1}:=e^{i\sqrt{z}|x|/2}{\langle}H_{0}{\rangle}^{-s/2}$ and

[TABLE]

Then $K^{1}_{z}$ is compact on ${\mathcal{H}}$ if $s>0$ and $K_{z}^{2}$ is bounded on ${\mathcal{H}}$ if $s<3/2$ since ${\mathcal{F}}(e^{-|x|})(\xi)=O({\langle}\xi{\rangle}^{-2})$. Hence $K_{z}\in\mathbb{B}_{\infty}({\mathcal{H}}^{s})$ if $0<s<3/2$. By the duality, $K_{z}\in\mathbb{B}_{\infty}({\mathcal{H}}^{s})$ for $-3/2<s<0$. Finally, by the scattering theory for trace class operators (see [12]), the usual wave operators on ${\mathcal{H}}$ exist and are complete. Hence Theorem 2.2 with $U_{0}(t)=e^{-itH_{0}}$ applies for $s\in[-1,1]$.

We conclude this section with a simple application of the above examples to the nonlinear scattering theory. For the sake of simplicity we only consider the following defocusing nonlinear Schrödinger equation with a potential $V$:

[TABLE]

where we suppose one of the following assumptions (A1)–(A4):

• (A1)

$n=1$, $\alpha>0$, $V=\alpha\delta$ is the delta potential as in Example 3.6 and $\sigma>4$;

• (A2)

$n=1$, ${\langle}x{\rangle}V,{\langle}x{\rangle}V^{\prime}\in L^{1}({\mathbb{R}})$, $V\geq 0$, $xV^{\prime}\leq 0$ and $\sigma>4$;

• (A3)

$n=3$, $V\in\mathcal{K}_{0}\cap L^{3/2}({\mathbb{R}}^{3};{\mathbb{R}})$, ${||V_{-}||}_{\mathcal{K}}<4\pi$, ${||(x\cdot\nabla V)_{+}||}_{\mathcal{K}}<4\pi$ and $\sigma=2$;

• (A4)

$3\leq n\leq 6$, $V=a|x|^{-2}$ and $(n,a,\sigma)$ satisfies

[TABLE]

Here $\mathcal{K}$ is the so-called global Kato class with norm

[TABLE]

and $\mathcal{K}_{0}$ is the norm closure of bounded compactly supported functions with respect to ${||\cdot||}_{\mathcal{K}}$. Note that under one of these conditions (A1)–(A4) the spectrum of $H=-\Delta+V$ is purely absolutely continuous $\sigma(H)=\sigma_{\mathrm{ac}}(H)=[0,\infty)$ and $P_{\mathrm{ac}}(H)=\mathop{\mathrm{Id}}\nolimits_{L^{2}({\mathbb{R}}^{n})}$.

It was proved by Banica-Visciglia [2] for (A1), Lafontaine [9] for (A2), Hong [6] for (A3) and Lu-Miao-Murphy [10] for (A4), respectively that (3.1) is globally well-posed in ${\mathcal{H}}^{1}$ and the solution $u$ scatters to a linear solution in ${\mathcal{H}}^{1}$ in the sense that there exist ${\widetilde{u}}_{\pm}\in{\mathcal{H}}^{1}$ such that

[TABLE]

On the other hand, it follows from Examples 3.3 and 3.6 that Theorem 2.2 with $U_{0}(t)=e^{-itH_{0}}$ and $s=1$ holds. Hence, we have the following

Corollary 3.7**.**

Let $u_{0}\in{\mathcal{H}}^{1}$. Then the solution $u$ obtained by the above previous works scatters to a free solution in ${\mathcal{H}}^{1}$, namely there exist $u_{\pm}\in{\mathcal{H}}^{1}$ such that

[TABLE]

Remark 3.8**.**

(1) With the additional condition $V\in W^{1,3/2}({\mathbb{R}}^{n})$, Hong [6] has proved in case of (A3) that the solution $u$ scatters to a free solution in ${\mathcal{H}}^{1}$. We here do not need such an additional regularity.

(2) It was claimed in [9] that $u$ scatters to a free solution in ${\mathcal{H}}^{1}$ under the condition (A2). However, the proof in [9] used the same argument as in [2, Propositions 3.1] in which it was shown that $\{e^{itH}u(t)\}$ is Cauchy in ${\mathcal{H}}^{1}$ as $t\to\infty$. This implies the scattering to a linear solution in ${\mathcal{H}}^{1}$, but the scattering to a free solution in ${\mathcal{H}}^{1}$ seems to be not an obvious consequence.

Appendix A Some supplementary lemmas

Lemma A.1**.**

Under (H1) and (H2), $[\psi(H_{0})-\psi(H)]\varphi(H_{0})\in\mathbb{B}_{\infty}({\mathcal{H}},{\mathcal{H}}^{s})$ for any $\varphi,\psi\in C_{0}^{\infty}({\mathbb{R}})$.

Proof.

We shall show ${\langle}H_{0}{\rangle}^{s/2}[\psi(H_{0})-\psi(H)]\varphi(H_{0})\in\mathbb{B}_{\infty}({\mathcal{H}})$. Helffer-Sjöstrand’s formula implies

[TABLE]

where $\widetilde{\varphi}\in C_{0}^{\infty}({\mathbb{R}}^{2})$ is an almost analytic extension of $\varphi$ satisfying $\partial_{\overline{z}}\widetilde{\varphi}(z)=O({\langle}z{\rangle}^{-N}|\mathop{\mathrm{Im}}\nolimits z|^{N})$ for any $N\geq 1$ ([5]). By (H1), (H2), the operator in the integrand is compact and its operator norm is $O(|\mathop{\mathrm{Im}}\nolimits z|^{-1})$. Hence, the integral converges in norm, being compact on ${\mathcal{H}}$. ∎

Lemma A.2**.**

Let $\max(n,2)\leq r<\infty$, $r>2$ if $n=2$, and $w\in L^{r}({\mathbb{R}}^{n})$. Then $w{\langle}D{\rangle}^{-1}$ is compact on $L^{2}({\mathbb{R}}^{n})$.

Proof.

We recall Sobolev’s inequality ${||f||}_{L^{q}({\mathbb{R}}^{n})}\lesssim{||f||}_{H^{s}({\mathbb{R}}^{n})}$, where $s\geq n(1/2-1/q)$ if $2\leq q<\infty$ or $s>n/2$ if $q=\infty$. If $r>2$ and $n\geq 2$ then

[TABLE]

When $r=2$ and $n=1$, we similarly have

[TABLE]

Since $n/r\leq 1$ by assumption, these estimates show ${||w{\langle}D{\rangle}^{-1}||}\lesssim{||w||}_{L^{r}({\mathbb{R}}^{n})}.$ Let $w_{n}\in C_{0}^{\infty}({\mathbb{R}}^{n})$ be such that ${||w-w_{n}||}_{L^{r}({\mathbb{R}}^{n})}\to 0$ as $n\to\infty$. Then $w_{n}{\langle}D{\rangle}^{-1}\to w{\langle}D{\rangle}^{-1}$ in norm by the above computation. Since $w_{n}{\langle}D{\rangle}^{-1}$ is compact, $w{\langle}D{\rangle}^{-1}$ is also compact. ∎