Abstract
We prove uniqueness of ground states for the pseudorelativistic Hartree equation,
in the regime of with sufficiently small -mass. This result shows that a uniqueness conjecture by Lieb and Yau [1987] holds true at least for except for at most countably many .
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.
Citation
Enno Lenzmann. "Uniqueness of ground states for pseudorelativistic Hartree equations." Anal. PDE 2 (1) 1 - 27, 2009. https://doi.org/10.2140/apde.2009.2.1
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