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Uniqueness of ground states for pseudorelativistic Hartree equations

Enno Lenzmann

Vol. 2 (2009), No. 1, 1–27

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.

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