Resolvent and spectral measure for Schr\"odinger operators on flat Euclidean cones

TL;DR

This paper constructs the resolvent and spectral measure for Schr"odinger operators on flat Euclidean cones and establishes dispersive estimates for associated propagators, advancing understanding of quantum dynamics on conical geometries.

Contribution

It provides explicit constructions of resolvent and spectral measures on Euclidean cones and proves dispersive estimates for Schr"odinger and half-wave propagators in this setting.

Findings

Explicit Schwartz kernel of resolvent and spectral measure constructed.

Dispersive estimates established for Schr"odinger propagator.

Dispersive estimates established for half-wave propagator.

Abstract

We construct the Schwartz kernel of resolvent and spectral measure for Schr\"odinger operators on the flat Euclidean cone $(X,g)$, where $X=C(\mathbb{S}_\sigma^1)=(0,\infty)\times \mathbb{S}_\sigma^1$ is a product cone over the circle, $\mathbb{S}_\sigma^1=\R/2\pi \sigma\Z$, with radius $\sigma>0$ and the metric $g=dr^2+r^2 d\theta^2$. As products, we prove the dispersive estimates for the Schr\"odinger and half-wave propagators in this setting.