Vol. 2, No. 1, 2009

Download this article
Download this article For screen
For printing
Recent Issues

Volume 16
Issue 2, 309–612
Issue 1, 1–308

Volume 15, 8 issues

Volume 14, 8 issues

Volume 13, 8 issues

Volume 12, 8 issues

Volume 11, 8 issues

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Editorial Board
Editors’ Interests
Subscriptions
 
Submission Guidelines
Submission Form
Policies for Authors
Ethics Statement
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Author Index
To Appear
 
Other MSP Journals
Uniqueness of ground states for pseudorelativistic Hartree equations

Enno Lenzmann

Vol. 2 (2009), No. 1, 1–27
Abstract

We prove uniqueness of ground states Q H12(3) for the pseudorelativistic Hartree equation,

Δ + m2Q ( x1 Q2)Q = μQ,

in the regime of Q with sufficiently small L2-mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for N =|Q|2 1 except for at most countably many N.

Our proof combines variational arguments with a nonrelativistic limit, leading to a certain Hartree-type equation (also known as the Choquard–Pekard or Schrödinger–Newton equation). Uniqueness of ground states for this limiting Hartree equation is well-known. Here, as a key ingredient, we prove the so-called nondegeneracy of its linearization. This nondegeneracy result is also of independent interest, for it proves a key spectral assumption in a series of papers on effective solitary wave motion and classical limits for nonrelativistic Hartree equations.

Keywords
pseudorelativistic Hartree equation, ground state, uniqueness, boson stars
Mathematical Subject Classification 2000
Primary: 35Q55
Milestones
Received: 25 January 2008
Revised: 17 September 2008
Accepted: 11 January 2009
Published: 1 February 2009
Authors
Enno Lenzmann
Massachusetts Institute of Technology
Department of Mathematics
Room 2-230
Cambridge, MA 02139
United States