[Paper Review] $L^p$-mapping properties for Schrödinger operators in open sets of $\mathbb R ^d$

Tsukasa Iwabuchi, Tokio Matsuyama+1 more|arXiv (Cornell University)|Feb 26, 2016

Advanced Mathematical Physics Problems4 references3 citations

TL;DR

This paper establishes $L^p$-boundedness of functional calculus for Schrödinger operators $H_V = -\Delta + V$ on arbitrary open sets $\Omega \subset \mathbb{R}^d$ ($d \geq 3$), where $V$ belongs to the Kato class $K_d(\Omega)$ and satisfies a smallness condition on its negative part. Using spectral theory and pointwise estimates for the heat kernel, it proves that $\varphi(\theta H_V)$ is uniformly bounded on $L^p(\Omega)$ for all $1 \leq p \leq \infty$ and $\theta > 0$, for any rapidly decreasing function $\varphi \in \mathscr{S}(\mathbb{R})$, extending classical multiplier results to Schrödinger operators with singular potentials.

ABSTRACT

Let $H_V=-Δ+V$ be a Schrödinger operator on an arbitrary open set $Ω$ of $\mathbb R^d$, where $d \geq 3$, and $Δ$ is the Dirichlet Laplacian and the potential $V$ belongs to the Kato class on $Ω$. The purpose of this paper is to show $L^p$-boundedness of an operator $φ(H_V)$ for any rapidly decreasing function $φ$ on $\mathbb R$. $φ(H_V)$ is defined by the spectral theorem. As a by-product, $L^p$-$L^q$-estimates for $φ(H_V)$ are also obtained.

Motivation & Objective

• To establish $L^p$-boundedness of the functional calculus $\varphi(H_V)$ for Schrödinger operators on arbitrary open sets $\Omega \subset \mathbb{R}^d$, $d \geq 3$, with $V$ in the Kato class.
• To extend classical $L^p$-multiplier results to Schrödinger operators with singular potentials via spectral theory.
• To prove uniform boundedness of $\varphi(\theta H_V)$ on $L^p(\Omega)$ for all $\theta > 0$ and $1 \leq p \leq \infty$, ensuring stability under scaling.
• To provide a foundation for defining Besov spaces associated with $H_V$ using the functional calculus.

Proposed method

• Utilizes the spectral theorem to define $\varphi(H_V)$ for $\varphi \in \mathscr{S}(\mathbb{R})$, the space of rapidly decreasing functions.
• Imposes Assumption A: $V = V_+ - V_-$ with $V_\pm \in K_d(\Omega)$ and $\|V_- olimits_{K_d(\Omega)} < \frac{\pi^{d/2}}{\Gamma(d/2 - 1)}$ to ensure self-adjointness and non-negativity of $H_V$.
• Employs pointwise estimates for the heat kernel $e^{-tH_V}$ derived from D’Ancona and Pierfelice for $d \geq 3$, which are critical for $L^p$-mapping estimates.
• Applies commutator calculus and the iterated commutator formula $\mathrm{Ad}^{k+1}(e^{-itR_{V,\theta}}) = -i\int_0^t \sum \Gamma(k_1,k_2,k_3) \mathrm{Ad}^{k_1}(e^{-isR_{V,\theta}}) \mathrm{Ad}^{k_2+1}(R_{V,\theta}) \mathrm{Ad}^{k_3}(e^{-i(t-s)R_{V,\theta}}) \, ds$ to control regularity and decay.
• Uses approximation by smooth compactly supported functions and distributional convergence to justify identities involving $\mathrm{Ad}^1(R_{V,\theta})$ and the resolvent $R_{V,\theta}$.

Experimental results

Research questions

• RQ1 Does the functional calculus $\varphi(H_V)$ extend boundedly to $L^p(\Omega)$ for $1 \leq p \leq \infty$ when $H_V = -\Delta + V$ and $V$ is in the Kato class on an open set $\Omega \subset \mathbb{R}^d$, $d \geq 3$?
• RQ2 Can the $L^p$-operator norm of $\varphi(\theta H_V)$ be uniformly bounded in $\theta > 0$ for rapidly decreasing functions $\varphi$?
• RQ3 What is the role of the Kato norm condition on $V_-$ in ensuring the boundedness of $\varphi(H_V)$ on $L^p$?
• RQ4 How do the pointwise estimates of the heat kernel $e^{-tH_V}$ on $\mathbb{R}^d$ ($d \geq 3$) contribute to $L^p$-mapping properties of $\varphi(H_V)$?
• RQ5 Can the spectral functional calculus for $H_V$ be used to define and study Besov spaces generated by $H_V$?

Key findings

• The operator $\varphi(H_V)$ is bounded on $L^p(\Omega)$ for all $1 \leq p \leq \infty$ and $\varphi \in \mathscr{S}(\mathbb{R})$, provided $V$ satisfies Assumption A.
• The $L^p$-operator norm of $\varphi(\theta H_V)$ is uniformly bounded in $\theta > 0$, with the bound depending only on $d$ and $\varphi$, not on $\theta$.
• The $L^p$-boundedness result generalizes classical Fourier multiplier theorems on $\mathbb{R}^d$, extending them to Schrödinger operators with Kato-class potentials.
• The proof relies crucially on pointwise estimates for the heat kernel of $H_V$ in $\mathbb{R}^d$ for $d \geq 3$, which are unavailable in lower dimensions.
• The $L^p$-boundedness of $\varphi(H_V)$ enables the construction of $H_V$-generated Besov spaces, as the functional calculus is well-behaved on $L^p$.
• The result is sharp in the sense that the smallness condition on $\|V_- olimits_{K_d(\Omega)}$ is necessary to ensure self-adjointness and boundedness of the functional calculus.