[Paper Review] Remarks on $L^p$-limiting absorption principle of Schrödinger operators and applications to spectral multiplier theorems

Shanlin Huang, Xiaohua Yao+1 more|arXiv (Cornell University)|Jul 10, 2016

Advanced Mathematical Physics Problems38 references3 citations

TL;DR

This paper establishes an $L^p$-type limiting absorption principle for Schrödinger operators $H = -\Delta + V$ on $\mathbb{R}^n$ ($n \geq 3$), proving uniform $L^{\frac{2(n+1)}{n+3}}$-$L^{\frac{2(n+1)}{n-1}}$ resolvent estimates for $\lambda > 0$ under integrability and spectral regularity conditions on $V$. The result enables sharp spectral multiplier theorems and extends uniform Hardy-Littlewood-Sobolev inequalities to fractional Schrödinger operators.

ABSTRACT

This paper comprises two parts. We first investigate a $L^p$ type of limiting absorption principle for Schrödinger operators $H=-Δ+V$, i.e., In $\mathbb{R}^n$ ($n\ge 3$) we prove the $ε-$uniform $L^{\frac{2(n+1)}{n+3}}$-$L^{\frac{2(n+1)}{n-1}}$ estimates of the resolvent $(H-λ\pm iε)^{-1}$ for all $λ>0$ when the potential $V$ belongs to some integrable spaces and a spectral condition of $H$ at zero is assumed. As an application, we establish a sharp spectral multiplier theorem and $L^p$ bound of Bochner-Riesz means associated with Schrödinger operators $H$. Next, we consider the fractional Schrödinger operator $H=(-Δ)^α+V$ ($0<2α

Motivation & Objective

• To extend the $L^p$-type limiting absorption principle to the case $\lambda_0 = 0$ for Schrödinger operators $H = -\Delta + V$.
• To establish a sharp Hörmander-type spectral multiplier theorem for $H$ using uniform resolvent estimates.
• To generalize the uniform Hardy-Littlewood-Sobolev inequality to fractional Schrödinger operators $(-\Delta)^\alpha + V$ with $0 < 2\alpha < n$.
• To characterize the spectral behavior of $H$ via the absence of embedded eigenvalues and regularity at zero.

Proposed method

• Prove uniform $L^{\frac{2(n+1)}{n+3}}$-$L^{\frac{2(n+1)}{n-1}}$ resolvent estimates for $(H - \lambda \pm i\epsilon)^{-1}$ with $\lambda > 0$, under $V \in L^{\frac{n}{2}+\sigma} \cap L^{\frac{n}{2}}$ and zero regularity in $L^{\frac{2(n+1)}{n-1}}$.
• Use the Stone formula to relate resolvent bounds to spectral measure estimates, enabling spectral multiplier results.
• Apply the limiting absorption principle and compactness arguments to analyze the resolvent of $(-\Delta)^\alpha + V$ in the fractional case.
• Leverage the theory of Lorentz spaces and weighted estimates, particularly extending results of Kenig-Ruiz-Sogge to the perturbed setting.
• Use Hölder's inequality and perturbation theory to control the resolvent of $H - z$ in $L^p$-operator norms.
• Apply Fredholm theory and Rellich's compactness theorem to prove the existence of a spectral set $E$ of Lebesgue measure zero where singular spectrum may lie.

Experimental results

Research questions

• RQ1 Can the $L^p$-type limiting absorption principle be extended to the case $\lambda_0 = 0$ for Schrödinger operators with $V \in L^{\frac{n}{2}+\sigma} \cap L^{\frac{n}{2}}$?
• RQ2 What is the sharp range of $p$ for which uniform $L^p$ resolvent estimates hold for $H = -\Delta + V$ when zero is a regular spectral point?
• RQ3 How do uniform resolvent estimates for $H$ lead to sharp spectral multiplier theorems and Bochner-Riesz bounds?
• RQ4 Can the uniform Hardy-Littlewood-Sobolev inequality for $(-\Delta)^\alpha$ be extended to the perturbed case $(-\Delta)^\alpha + V$?
• RQ5 What conditions ensure the absence of embedded eigenvalues and the regularity of the spectral measure at zero for Schrödinger operators?

Key findings

• The uniform $L^{\frac{2(n+1)}{n+3}}$-$L^{\frac{2(n+1)}{n-1}}$ resolvent estimate $\sup_{0<\epsilon<1}\|(-\Delta + V - (\lambda + i\epsilon))^{-1}\|_{L^{\frac{2(n+1)}{n+3}}-L^{\frac{2(n+1)}{n-1}}}} \leq C\lambda^{-\frac{1}{n+1}}$ holds for $\lambda > 0$ under $V \in L^{\frac{n}{2}+\sigma} \cap L^{\frac{n}{2}}$ and zero regularity in $L^{\frac{2(n+1)}{n-1}}$.
• A sharp Hörmander-type spectral multiplier theorem is established for $H = -\Delta + V$, with the spectral measure norm bounded by $C\lambda^{-\frac{1}{n+1}}$.
• The Bochner-Riesz operator associated with $H$ satisfies $L^p$ bounds for $\frac{2(n+1)}{n+3} \leq p \leq \frac{2(n+1)}{n-1}$, matching the resolvent estimate range.
• For the fractional Schrödinger operator $H = (-\Delta)^\alpha + V$ with $0 < 2\alpha < n$, a uniform Hardy-Littlewood-Sobolev inequality is proven in the form $\|(H - z)^{-1}\|_{L^p - L^{p'}} \leq C|z|^{\frac{n}{2\alpha}(\frac{1}{p} - \frac{1}{p'}) - 1}$ for $\max(\frac{2\alpha}{n}, \frac{n+3}{2(n+1)}) < \frac{1}{p} \leq \frac{n+2\alpha}{2n}$.
• When $\|V\|_{L^{\frac{n}{2\alpha}}} \leq c_0$ for a small enough $c_0 > 0$, the resolvent of $H = (-\Delta)^\alpha + V$ satisfies uniform $L^p$-operator bounds.
• The singular spectrum of $(-\Delta)^m + V$ is contained in a Lebesgue null set $E \subset \mathbb{R}$, and the resolvent is continuous on $\mathbb{C}_+ \setminus \{0\} \setminus E$, with uniform bounds on compact subsets of $\mathbb{R} \setminus E \setminus \{0\}$.