Spectral Multipliers II: Elliptic and Parabolic Operators and Bochner-Riesz Means
Marius Beceanu, Michael Goldberg
TL;DR
This work analyzes spectral multipliers for the Schrödinger operator $H=-\Delta+V$ in $\mathbb{R}^3$ with rough potentials $V$ in the Kato class, focusing on the regime where zero is a regular point and no zero or positive energy bound states are assumed (with a broader treatment allowing large, possibly negative, potentials in $\mathcal{K}_0$). By leveraging a detailed Birman–Schwinger/operator framework and Wiener's-type perturbation results, the authors derive sharp Poisson and heat kernel bounds, decompose kernels into continuous and bound-state contributions when negative eigenvalues are present, and establish robust $L^p$–$L^q$ bounds for Bochner–Riesz means via spectral multipliers. They demonstrate that $H$ has finitely many negative eigenvalues and that positive-energy obstructions disappear under additional regularity (e.g., $V\in L^{3/2,\infty}$ or $\dot W^{-1/4,4/3}$), with exponential decay of eigenfunctions (Agmon bounds). Moreover, the Bochner–Riesz and Mihlin-type multiplier bounds extend classical results to large rough potentials, unifying and broadening prior work. The framework supports both dispersive-type estimates and spectral multiplier theory in a broad, scalable setting, enabling applications to PDEs with rough potentials and bound states.
Abstract
We establish estimates for the Poisson kernel, the heat kernel, and Bochner--Riesz means defined in terms of $H=-Δ+V$, where $V$ is a possibly large rough real-valued scalar potential and $H$ can have negative eigenvalues. All results are in three space dimensions. We eliminate several unnecessary conditions on $V$, leaving just $V \in \mathcal K_0$, meaning that $V$ is locally integrable and $(-Δ)^{-1}|V|$ is bounded. For the spectral multiplier bounds, we assume that $H$ has no zero or positive energy bound states. For $V \in \mathcal K_0$, we prove that $H$ has at most a finite number of negative bound states. If in addition $V \in \dot W^{-1/4, 4/3}$, then by [GoSc] and [KoTa] there are no positive energy bound states.