arXiv subjects

Kenneth Taliaferro

Publications and source records attributed to Kenneth Taliaferro.

3 recordsLinked to original sources

arXiv2014

Uniqueness of solutions to the 3D quintic Gross-Pitaevskii Hierarchy

In this paper, we study solutions to the three-dimensional quintic Gross-Pitaevskii hierarchy. We prove unconditional uniqueness among all small solutions in the critical space $\mathfrak{H}^1$ (which corresponds to $H^1$ on the NLS level). With slight modifications to the proof, we also prove unconditional uniqueness of solutions to the Hartree hierarchy without smallness condition. Our proof uses the quantum de Finetti theorem, and is an extension of the work by Chen-Hainzl-Pavlović-Seiringer \cite{CHPS}, and our previous work \cite{UniqueLowReg}.

math.AP↗

arXiv2014

Derivation in strong topology and global well-posedness of solutions to the Gross-Pitaevskii hierarchy

We derive the cubic defocusing GP hierarchy in ${\mathbb R}^3$ from a bosonic $N$-particle Schrödinger equation as $N\rightarrow\infty$, in the strong topology corresponding to the space ${\mathcal H}_ξ^1$ introduced in \cite{chpa}. In particular, we thereby eliminate the requirement of regularity ${\mathcal H}_ξ^{1+}$ for the initial data used in \cite{CPBBGKY}. Moreover, the marginal density matrices obtained in this strong limit are allowed to be of infinite rank. This contrasts previous results where weak-* limits were derived, and subsequently enhanced to strong limits based on the condition that the limiting density matrices have finite rank. Furthermore, we prove that positive semidefiniteness of marginal density matrices is preserved over time, which we combine with results in \cite{CPHE}, to obtain the global well-posedness of solutions.

math-ph↗

arXiv2014

Unconditional Uniqueness of the cubic Gross-Pitaevskii Hierarchy with Low Regularity

In this paper, we establish the unconditional uniqueness of solutions to the cubic Gross-Pitaevskii hierarchy on $\mathbb{R}^d$ in a low regularity Sobolev type space. More precisely, we reduce the regularity $s$ down to the currently known regularity requirement for unconditional uniqueness of solutions to the cubic nonlinear Schrödinger equation ($s\ge\frac{d}{6}$ if $d=1,2$ and $s>s_c=\frac{d-2}{2}$ if $d\ge 3$). In such a way, we extend the recent work of Chen-Hainzl-Pavlović-Seiringer.

math.AP↗