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Annales Henri Poincaré

, Volume 13, Issue 7, pp 1557–1573 | Cite as

Nonrelativistic Hydrogen Type Stability Problems on Nonparabolic 3-Manifolds

  • Batu GüneysuEmail author
Article

Abstract

Let M be a noncompact Riemannian 3-manifold, satisfying some assumptions that guarantee the existence of a minimal positive Green’s function G : M × M → (0, ∞]. We prove the following two stability results. First, we show that there is a C > 0 such that for all κ ≥ 0, all generalized Laplacians \({\fancyscript{P}}\) on M with Lichnerowicz potential term \({V_{\fancyscript{P}}=0}\) , and all \({y \in M}\) one has
$$\fancyscript{P} - \kappa G(\bullet, y)\geq -C\kappa^2. $$
Secondly, we prove that there are C, κ 0 > 0 such that for all generalized Laplacians \({\fancyscript{P}}\) on M with \({\fancyscript{P} \geq 0, |V_{\fancyscript{P}}| \in {\rm L}^2(M)}\) and all \({\Lambda > 0, 0\leq \kappa\leq \kappa_0\Lambda^2, y \in M}\) one has
$$\fancyscript{P} -\kappa G(\bullet,y)+\Lambda \int\limits_{M}\left\|V_{\fancyscript{P}}(x)\right\|^2_x{\rm vol}({\rm d}x) \geq -C\kappa^2. $$
The first inequality corresponds to the nonrelativistic stability of Hydrogen type problems on M with magnetic fields when spin is neglected, whereas the second inequality corresponds to a restricted nonrelativistic stability of Hydrogen type problems when spin is taken into consideration.

Keywords

Riemannian Manifold Heat Kernel Dirichlet Form Riemannian Structure Dirac Structure 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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© Springer Basel AG 2012

Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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