Uniqueness of ground states for pseudorelativistic Hartree equations
By E. Lenzmann. 2008
245 citations recorded by the source corpus.
We prove uniqueness of ground states Q ∈ H 1/2 (R 3 ) for the pseudo-relativistic Hartree equation, p −� + m 2 Q − ` |x| 1 ∗ |Q| 2 ´ Q = −µQ, in the regime of Q with sufficiently smallL 2 -mass. This result shows that a uniqueness conjecture by Lieb and Yau in (CMP 112 (1987), 147-174) holds true at least for N = R |Q| 2 ≪ 1 except for at most countably many N. Our proof combines variational arguments with a nonrelativistic limit, which leads to a certain Hartree-type equation (also known as the Choquard- Pekard or Schrodinger-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 nonrelativis- tic Hartree equations.
Subfields: Atomic and Molecular Physics, and Optics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/328624d1560946b1c94dfa88cfea7a0b518f2783
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In Atomic and Molecular Physics, and Optics
In Mathematical Physics
- Chang-Lin Xiang
- Shuangjie Peng
- Carlos Palenzuela
- Gongbao Li
- Steven L. Liebling
- Xianhua Tang
- Hongyu Ye
- Xiaoyutao Luo
Citations in the corpus
- Dynamical boson stars (2012, background): “Recent work includes a mathematical approach in terms of large and small data [89], followed up by studying singularity formation [153] and uniqueness [90, 152] for a certain boson star equation.”
- A singularly perturbed Kirchhoff problem revisited (2016, background): “[10, 12, 13, 24] and to [24, 34] for mathematical researches on fractional type nonlocal Schrödinger equations and convolution type nonlocal Choquard equations, repectively.”
- The Stability of Matter in Quantum Mechanics (2009, unknown): “For small values of κ ∫ |φ(x)|2dx uniqueness was proved by Lenzmann [104].”