arXiv subjects

Xitao Gao

Publications and source records attributed to Xitao Gao.

3 recordsLinked to original sources

arXiv2026

A sharp regularity threshold for Schrödinger maximal estimates on standard tori

We disprove almost everywhere convergence of the Schrödinger evolution on the standard torus $\mathbb{T}^d$ for initial data $f\in H^s(\mathbb{T}^d)$ when \(s d/(d+2)$ with $d\geq2$. By integer dilation and uniform boundedness, we also obtain a single datum in $H^s$ whose evolution is unbounded along a sequence of times tending to zero whenever $s<d/(d+2)$. We also record a logarithmic upper bound in $2D$ at the critical frequency power and a lower bound on shrinking time intervals.

math.AP↗

arXiv2026

Sharp Dispersive Estimates for the Schrödinger Equation with an Attractive Coulomb Potential

We prove sharp dispersive $L^1 \to L^\infty$ estimates for the three-dimensional attractive Coulomb operator $H_Z=-Δ-Z|x|^{-1}$, where $Z>0$. The absolutely continuous part of the Schrödinger evolution decays at the free rate for short times, whereas its leading contribution decays like $|t|^{-1}$ for long times, with amplitude proportional to $Z$. This slower decay is driven by the threshold and is sharp when $Z^2|t|\gg1$.

math.AP↗

arXiv2026

$L^p$ bounds for wave operators with critical electromagnetic potentials

We study the Møller wave operators for scaling critical electromagnetic Hamiltonians in the plane. For smooth transverse magnetic and angular electric potentials, with nonnegative angular operator and magnetic flux outside $\frac12\Z$, we prove that the wave operators relative to $-Δ$ exist, are unitary on $L^2$, and, together with their adjoints, are bounded on every $L^p$, $1<p<\infty$. We then specialize to the the Aharonov--Bohm model and we determine the exact ranges for the boundedness of their wave operators on weighted $L^p(\R^2,|x|^β\,dx)$ spaces, for both the Friedrichs and the Krein realizations. In the Friedrichs case, this gives in particular the already known boundedness on all $L^{p}$ spaces $1<p<\infty$, while boundedness fails at $p=1,\infty$. In the Krein case, both wave operators and adjoints are bounded precisely when $2/(2-η_α)<p<2/η_α$, where $η_α=\max\{α,1-α\}$ (here $α\in(0,1)$). Thus the boundary condition changes the admissible exponents.

math.SP↗