• Register
PhysicsOverflow is a next-generation academic platform for physicists and astronomers, including a community peer review system and a postgraduate-level discussion forum analogous to MathOverflow.

Welcome to PhysicsOverflow! PhysicsOverflow is an open platform for community peer review and graduate-level Physics discussion.

Please help promote PhysicsOverflow ads elsewhere if you like it.


New printer friendly PO pages!

Migration to Bielefeld University was successful!

Please vote for this year's PhysicsOverflow ads!

Please do help out in categorising submissions. Submit a paper to PhysicsOverflow!

... see more

Tools for paper authors

Submit paper
Claim Paper Authorship

Tools for SE users

Search User
Reclaim SE Account
Request Account Merger
Nativise imported posts
Claim post (deleted users)
Import SE post

Users whose questions have been imported from Physics Stack Exchange, Theoretical Physics Stack Exchange, or any other Stack Exchange site are kindly requested to reclaim their account and not to register as a new user.

Public \(\beta\) tools

Report a bug with a feature
Request a new functionality
404 page design
Send feedback


(propose a free ad)

Site Statistics

205 submissions , 163 unreviewed
5,037 questions , 2,191 unanswered
5,345 answers , 22,705 comments
1,470 users with positive rep
816 active unimported users
More ...

  Time decay for Hartree equation with Coulomb potential

+ 5 like - 0 dislike

Are there any time-decay results for the solution of the Hartree equation \begin{equation}\frac{1}{i}\partial_t\phi-\Delta\phi=-(|x|^{-1}\ast|\phi|^2)\phi,\quad x\in\mathbb{R}^3\end{equation} in $L^p$-spaces which would ensure e.g. $\phi\in L^2((0,\infty);L^3(\mathbb{R}^3))$ or $\phi\in L^2((0,\infty);L^4(\mathbb{R}^3))$?

According to the information at DispersiveWiki, Coulomb potential is the borderline case and there is no scattering results (in the sense of asymptotic completeness) (??) for the solutions of the above equation. There is a paper by Hayashi and Ozawa on time-decay for Hartree equation with Coulomb or more singular potentials which makes use of pseudo-conformal invariance. This work implies that one can get rates like $\|\phi\|_4\lesssim t^{-3/8}$ or $\|\phi\|_3\lesssim t^{-1/4}$, where $\phi$ is the solution of the above equation. Those are slower rates compared to the rates at which the free solution decays. Are there more recent publications which might imply better rates? Do you know of any $L^\infty$-decay results (for the above equation) which might be interpolated by mass conservation to get faster $L^p$-decay?

This post imported from StackExchange MathOverflow at 2015-10-19 17:10 (UTC), posted by SE-user user49794

asked Apr 20, 2014 in Theoretical Physics by user49794 (25 points) [ revision history ]
edited Jan 14, 2016 by dimension10.admin

Your answer

Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead.
To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL.
Please consult the FAQ for as to how to format your post.
This is the answer box; if you want to write a comment instead, please use the 'add comment' button.
Live preview (may slow down editor)   Preview
Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:
Then drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds).
Please complete the anti-spam verification

user contributions licensed under cc by-sa 3.0 with attribution required

Your rights