Boson Stars as Solitary Waves

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.

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Section: Resultsmentioning

confidence: 99%

Uniqueness of ground states for pseudorelativistic Hartree equations

Lenzmann

2009

APDE

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“…Indeed, referring to the exposition in [3] we see that, for every N > N c , there exists a spherically symmetric initial datum,…”

Section: Blow-upmentioning

confidence: 99%

“…Indeed, existence and spherical symmetry of solutions Q(x), which we define as minimizers of E(u) subject to R 3 |u| 2 dx = N , follows from the discussion in [7]; see also [5] for m = 0. Furthermore, it turns out that the mass parameter, m ≥ 0, plays a decisive role summarized as follows; see also [3] for more details.…”

Section: Introductionmentioning

confidence: 99%

Blowup for nonlinear wave equations describing boson stars

Fröhlich¹,

Lenzmann²

2007

Comm Pure Appl Math

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128

130

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We consider the nonlinear wave equationmodelling the dynamics of (pseudo-relativistic) boson stars. For spherically symmetric initial data,, with negative energy, we prove blowup of u(t, x) in H 1/2 -norm within a finite time. Physically, this phenomenon describes the onset of "gravitational collapse" of a boson star. We also study blow-up in external, spherically symmetric potentials and we consider more general Hartree-type nonlinearities. As an application, we exhibit instability of ground state solitary waves at rest if m = 0.

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“…For the proofs of (i) and (ii)-(iii), we refer to [Lieb and Yau 1987] and [Lenzmann 2006;Fröhlich et al 2007a], respectively. In physical terms, the constant N * > 0 can be regarded as the "Chandrasekhar limit mass" of a pseudorelativistic boson star.…”

Section: Uniqueness Of Ground States For Pseudorelativistic Hartree Ementioning

confidence: 99%

“…For the reader's orientation, we mention that our definition of Ᏹ(ψ) in (1-1) differs from the conventions in [Lieb and Yau 1987;Fröhlich et al 2007a] by an inessential factor of 2 and by the fact that we use √ − + m 2 instead of √ − + m 2 − m. Obviously, these slight alterations in our definition of Ᏹ(ψ) do not affect any results on (1-2) that are derived or quoted in the present paper.…”

Section: It Is Convenient To View the Operator L (Which Is Not ‫-ރ‬Limentioning

confidence: 99%

Untitled

2009

APDE

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UNIQUENESS OF GROUND STATES FOR PSEUDORELATIVISTIC HARTREE EQUATIONS ENNO LENZMANNWe prove uniqueness of ground states Q ∈ H 1/2 ‫ޒ(‬ 3 ) for the pseudorelativistic Hartree equation,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.

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Boson Stars as Solitary Waves

Cited by 160 publications

References 14 publications

Uniqueness of ground states for pseudorelativistic Hartree equations

Uniqueness of ground states for pseudorelativistic Hartree equations

Blowup for nonlinear wave equations describing boson stars

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