Strichartz estimates for Critical magnetic Schrödinger operators on flat Euclidean cones

TL;DR

The paper analyzes Strichartz estimates for the magnetic Schrödinger operator $\\mathcal{H}_{\\mathbf{A}}$ with critical potentials on the 2D flat cone $\\Sigma$, introducing a precise resolvent and spectral–measure framework on cone geometry. By separable variables in the angular direction, it derives an explicit Schrödinger kernel in terms of angular data $A_{\\alpha,\\rho}$ and $B_{\\alpha,\\rho}$ and uses this to prove a dispersive bound $|e^{-it\\mathcal{H}_{\\mathbf{A}}}(x,y)| \le C_{\\rho}|t|^{-1}$, enabling Keel–Tao Strichartz estimates for admissible pairs in $\\Lambda_0^S$. The wave equation is treated analogously via Stone’s formula and spectral measure analysis, yielding wave Strichartz bounds for $\\Lambda^w_s$ with data in distorted Besov-type spaces $\\dot{\\mathcal{H}}^s_{\\mathbf{A}}(\\Sigma)$. Introduced distorted Besov spaces $\\dot{\\mathcal{B}}^{s}_{p,r,\\mathbf{A}}(\\Sigma)$ and finite-energy frameworks pave the way for nonlinear applications on conic manifolds with magnetic singularities. Overall, the work extends dispersive PDE theory to metric cones under critical magnetic perturbations and provides tools for nonlinear scattering on these geometries.

Abstract

In this paper, we study Schrödinger operator $\mathcal{H}_{\mathbf{A}}$ perturbed by critical magnetic potentials on the 2D flat cone $Σ= C(\mathbb{S}_ρ^1) = (0, \infty) \times \mathbb{S}_ρ^1$, which is a product cone over the circle $\mathbb{S}_ρ^1 = \mathbb{R}/2πρ\mathbb{Z}$ with radius $ρ> 0$, and equipped with the metric $g = dr^2 + r^2 dθ^2$. The goal of this work is to establish Strichartz estimates for $\mathcal{H}_{\mathbf{A}}$ in this setting. A key aspect of our approach is the construction of the Schwartz kernel of the resolvent and the spectral measure for Schrödinger operator on the flat Euclidean cone $(Σ, g)$. The results presented here generalize previous work in \cite{Ford, BFM, FZZ, Zhang1}.

Theorems & Definitions (15)

• Theorem 1.1
• Definition 1.1: Besov spaces associated with $\mathcal{H}_{{{\bf A}}}$
• Theorem 1.2
• Remark 1.1
• Remark 1.2
• Theorem 2.1: Dispersive for Schrödinger
• Proposition 2.1: Schrödinger kernel
• Remark 2.1
• Remark 2.2
• Theorem 3.1: Dispersive estimates for wave