Uniqueness of ground states for pseudorelativistic Hartree equations

Abstract

We prove uniqueness of ground states Q ∈ H 1/2 ‫ޒ(‬ 3 ) for the pseudorelativistic Hartree equation, − + m 2 Q − |x| −1 * |Q| 2 Q = −µQ, in the regime of Q with sufficiently small L 2 -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 Hartreetype 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 socalled 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. IntroductionThe pseudorelativistic Hartree energy functional, given (in appropriate units) byarises in the mean-field limit of a quantum system describing many self-gravitating, relativistic bosons with rest mass m > 0. Such a physical system is often referred to as a boson star, and various models for these -at least theoretical -objects have attracted a great deal of attention in theoretical and numerical astrophysics over the past years. In order to gain some rigorous insight into the theory of boson stars, it is of particular interest to study ground states (that is, minimizers) for the variational problemwhere the parameter N > 0 plays the role of the stellar mass. Provided that problem (1-2) has indeed a ground state Q ∈ H 1/2 ‫ޒ(‬ 3 ), one readily finds that it satisfies the pseudorelativistic Hartree equation, − + m 2 Q − |x| −1 * |Q| 2 Q = −µQ, (1-3) with µ = µ(Q) ∈ ‫ޒ‬ being some Lagrange multiplier.MSC2000: 35Q55.

Existence and Newtonian limit of nonlinear bound states in the Einstein–Dirac system

“…It is proved in [9] that the operator L is self-adjoint on L 2 rad with domain H 2 rad and trivial kernel and a single negative eigenvalue with one dimensional eigenspace. (Going out of the radial sector, L has a three dimensional kernel associated with translation invariance, but this does not concern us in this article.)…”

“…Pseudo-relativistic generalizations of Lieb's solutions have been given in [9], and we will make use of a result from this article on the non-degeneracy of the linearization of (1.4): see lemma 4 and the subsequent remark.…”

Perturbation Method for Particle-like Solutions of the Einstein–Dirac Equations

“…It is well known that Choquard's equation (13) has a unique radial, positive solution u 0 with |u 0 | 2 = N for some N > 0 given. Furthermore, u 0 is infinitely differentiable, goes to zero at infinity and is a radial nondegenerate solution; by this we mean that the linearization of (13) around u 0 has a trivial nullspace in L 2 r (R 3 ) (see [3], [4], [2] for more details). Let φ 0 = (ϕ 0 , χ 0 , τ 0 , ζ 0 ) be the ground state solution of (12).…”

“…Soient e, m, ω tels que e 2 − m 2 < 0, 0 < ω < m et supposons m − ω assez petit ; alors il existe une solution non triviale de (1)(2)(3)(4)(5).…”

Quantitative properties on the steady states to a Schrödinger–Poisson–Slater System

“…But his method is allowed to extend to some higher dimensions, say, 4 and 5 dimension, see [12,24] for example. Quite recently, Lieb's result was proved to be true for positive solutions by Ma and Zhao in [16].…”