Spectral multipliers and wave propagation for Hamiltonians with a scalar potential

TL;DR

This paper extends key spectral multiplier and wave propagation estimates from the free Laplacian to perturbed Hamiltonians with scalar potentials, under optimal conditions, enhancing understanding of wave behavior in potential-influenced systems.

Contribution

It establishes resolvent, dispersive, Mihlin multiplier, fractional integration, and Strichartz estimates for Hamiltonians with scalar potentials, generalizing classical results to more complex operators.

Findings

Proves resolvent estimates for perturbed Hamiltonians.

Establishes dispersive bounds for the wave propagator.

Derives full range of Strichartz estimates under optimal conditions.

Abstract

We extend several fundamental estimates regarding spectral multipliers for the free Laplacian on $\mathbb R^3$ to the case of perturbed Hamiltonians of the form $-\Delta+V$, where $V$ is a scalar real-valued potential. In this paper, we prove resolvent estimates, a dispersive bound for the perturbed wave propagator, Mihlin multiplier and fractional integration bounds, and the full range of wave equation Strichartz estimates, under optimal or almost optimal scaling-invariant conditions on the potential and on the spectral multipliers themselves.