Annales Henri Poincaré

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

Nonrelativistic Hydrogen Type Stability Problems on Nonparabolic 3-Manifolds

  • Batu GüneysuEmail author


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.


Riemannian Manifold Heat Kernel Dirichlet Form Riemannian Structure Dirac Structure 
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Authors and Affiliations

  1. 1.Institut für MathematikHumboldt-Universität zu BerlinBerlinGermany

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