Uniqueness of Ground States for Pseudo-Relativistic Hartree EquationsReport as inadecuate



 Uniqueness of Ground States for Pseudo-Relativistic Hartree Equations


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We prove uniqueness of ground states $Q$ in $H^{1-2}$ for pseudo-relativistic Hartree equations in three dimensions, provided that $Q$ has sufficiently small $L^2$-mass. This result shows that a uniqueness conjecture by Lieb and Yau in CMP 112 1987,147-174 holds true at least under a smallness condition. Our proof combines variational arguments with a nonrelativistic limit, which leads to a certain Hartree-type equation also known as the Choquard-Pekard or Schroedinger-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.



Author: Enno Lenzmann

Source: https://archive.org/



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