Sharp Gaussian bounds and $L^p$-growth of semigroups associated with elliptic and Schrödinger operators
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Abstract:
We prove sharp large time Gaussian estimates for heat kernels of elliptic and Schrödinger operators, including Schrödinger operators with magnetic fields. Our estimates are then used to prove that for general (magnetic) Schrödinger operators $A=-\sum _{k = 1}^d (\tfrac {\partial }{\partial x_k}-i b_k)^2 + V$, we have the $L^\infty$-estimate (for large $t$): \[ \Vert e^{-tA} \Vert _{{\mathcal L}(L^\infty (\mathbb {R}^d))} \le C e^{-s(A)t} ( t\ln t)^{d/4}\] where $s(A) := \inf \sigma (A)$ is the spectral bound of $A.$ The same estimate holds for elliptic and Schrödinger operators on general domains.References
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Additional Information
- El Maati Ouhabaz
- Affiliation: Institut de Mathématiques de Bordeaux, Laboratoire d’Analyse et Géométrie, C.N.R.S. UMR 5467, Université Bordeaux 1-351, Cours de la Libération, 33405 Talence, France
- Email: Elmaati.Ouhabaz@math.u-bordeaux1.fr
- Received by editor(s): March 15, 2005
- Received by editor(s) in revised form: June 24, 2005
- Published electronically: May 31, 2006
- Communicated by: Mikhail Shubin
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 134 (2006), 3567-3575
- MSC (2000): Primary 47D08, 47D06, 35P15
- DOI: https://doi.org/10.1090/S0002-9939-06-08430-9
- MathSciNet review: 2240669