[Paper Review] The $L^p$ boundedness of wave operators for Schrödinger operators with threshold singularities II. Even dimensional case

Domenico Finco, Kenji Yajima|ArXiv.org|May 10, 2006

Advanced Mathematical Physics Problems15 references16 citations

TL;DR

This paper establishes the $L^p$ boundedness of wave operators $W_{\pm}$ for Schrödinger operators $H = -\Delta + V$ in even dimensions $m \geq 6$, under conditions on the potential $V$ and spectral type (generic or exceptional). It proves that $W_{\pm}$ are bounded on $L^p$ and Sobolev spaces $W^{k,p}$ for $1 \leq p \leq \infty$ under generic conditions, and for a restricted range $m/(m-2) < p < m/2$ under exceptional conditions, using resolvent estimates and integration by parts on oscillatory integrals.

ABSTRACT

In this paper we consider the wave operators $W_{\pm}$ for a Schrödinger operator $H$ in ${\bf{R}}^n$ with $n\geq 4$ even and we discuss the $L^p$ boundedness of $W_{\pm}$ assuming a suitable decay at infinity of the potential $V$. The analysis heavily depends on the singularities of the resolvent for small energy, that is if 0-energy eigenstates exist. If such eigenstates do not exist $W_{\pm}: L^p o L^p$ are bounded for $1 \leq p \leq \infty$ otherwise this is true for $ \frac{n}{n-2} &lt; p &lt; \frac{n}{2} $. The extension to Sobolev space is discussed.

Motivation & Objective

• To establish the $L^p$ boundedness of wave operators $W_{\pm}$ for Schrödinger operators in even-dimensional spaces with threshold singularities.
• To determine the range of $p$ for which $W_{\pm}$ are bounded on $L^p$ and Sobolev spaces $W^{k,p}$, depending on whether the Schrödinger operator is of generic or exceptional type.
• To extend previous results on $L^p$ boundedness to even dimensions $m \geq 6$, particularly under weaker decay and regularity assumptions on the potential $V$.
• To analyze the role of spectral type (generic vs. exceptional) in determining the boundedness range of $W_{\pm}$, especially in relation to dispersive estimates and time decay.

Proposed method

• Uses the intertwining property $f(H)P_{\rm ac} = W_{\pm} f(H_0) W_{\pm}^*$ to transfer boundedness properties from the free resolvent $R_0(\lambda)$ to the wave operators $W_{\pm}$.
• Employs resolvent estimates in $L^p$ and Sobolev spaces, proving that $R(\lambda)^n \in \mathcal{B}(L^p, W^{2n,p})$ and $R(\lambda)^{n+1/2} \in \mathcal{B}(L^p, W^{2n+1,p})$ for large negative $\lambda$.
• Applies integration by parts $s \leq (m+2)/2$ times to oscillatory integrals involving the perturbed kernel $T_\pm(\lambda,x,y)$, using the decay estimate $|\partial_\lambda^s T_\pm(\lambda,x,y)| \leq C_{ns} \lambda^{-3} \langle x\rangle^{-(m-1)/2} \langle y\rangle^{-(m-1)/2}$.
• Establishes the admissibility of the resulting integral kernels $\tilde{\Omega}_{n+1}(x,y)$ via pointwise decay estimates in $\langle |x| \pm |y| \rangle^{-(m+2)/2}$.
• Uses induction on $n$ to prove that $R(\lambda)^n$ maps $L^p$ to $W^{2n,p}$ and $W^{1,p}$ to $W^{2n+1,p}$, relying on the resolvent equation $R(\lambda) = R_0(\lambda) - R_0(\lambda) V R(\lambda)$.
• Leverages the fact that $W_{\pm}$ commute with the resolvent via the intertwining identity to deduce boundedness on Sobolev spaces from boundedness of resolvent powers.

Experimental results

Research questions

• RQ1 For even dimensions $m \geq 6$, what is the range of $p$ for which the wave operators $W_{\pm}$ are bounded on $L^p(\mathbb{R}^m)$ when $H$ is of generic type?
• RQ2 How does the boundedness of $W_{\pm}$ on $W^{k,p}$ spaces depend on the regularity and decay of the potential $V$?
• RQ3 What is the role of the spectral type (generic vs. exceptional) in determining the $L^p$ boundedness of $W_{\pm}$, especially in the context of dispersive estimates?
• RQ4 Can the $L^p$ boundedness of $W_{\pm}$ be extended to Sobolev spaces $W^{k,p}$ under minimal regularity assumptions on $V$?
• RQ5 How do the decay and regularity of $V$ affect the $L^p$ operator norm of $W_{\pm}$, particularly in the exceptional case where $H$ admits zero-energy resonance?

Key findings

• For $m \geq 6$ even, $W_{\pm}$ are bounded on $L^p(\mathbb{R}^m)$ for all $1 \leq p \leq \infty$ if $V$ satisfies $|V(x)| \leq C\langle x\rangle^{-(m+2+\varepsilon)}$, $\mathcal{F}(\langle x\rangle^{2\sigma}V) \in L^{m_*}$ for $\sigma > (m-2)/(m-1)$, and $H$ is of generic type.
• For $1 < p < \infty$, $W_{\pm}$ are bounded on $W^{k,p}(\mathbb{R}^m)$ for $0 \leq k \leq \ell + 2$ if $\partial^\alpha V$ are bounded for $|\alpha| \leq \ell$.
• For $m \geq 6$ even and $H$ of exceptional type, $W_{\pm}$ are bounded on $W^{k,p}(\mathbb{R}^m)$ for $m/(m-2) < p < m/2$ and $0 \leq k \leq \ell + 2$ if $|V(x)| \leq C\langle x\rangle^{-(m+4+\varepsilon)}$ for $m=6$ and $|V(x)| \leq C\langle x\rangle^{-(m+3+\varepsilon)}$ for $m \geq 8$.
• The boundedness of $W_{\pm}$ on $W^{k,p}$ spaces is established via induction on $n$ using the resolvent equation and estimates on $R(\lambda)^n$ in $L^p$ and Sobolev norms.
• The proof relies on $L^p$ estimates for the resolvent $R(\lambda)$ and its powers, showing that $R(\lambda)^n \in \mathcal{B}(L^p, W^{2n,p})$ under regularity and decay conditions on $V$.
• The result improves upon earlier work by [22] for $m \geq 6$, as condition (1.2) and the decay $|V(x)| \leq C\langle x\rangle^{-(m+2+\varepsilon)}$ are weaker than the integral condition (1.5) used in [22].