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On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates

On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates We consider the Schrödinger type operator $${\mathcal{ A}=(1+|x|^{\alpha})\Delta-|x|^{\beta}}$$ A = ( 1 + | x | α ) Δ - | x | β , for $${\alpha\in (0,2)}$$ α ∈ ( 0 , 2 ) and $${\beta\ge 0}$$ β ≥ 0 . We prove that, for any $${p\in (1,\infty)}$$ p ∈ ( 1 , ∞ ) , the minimal realization of operator $${\mathcal{A}}$$ A in $${L^p(\mathbb{R}^{N})}$$ L p ( R N ) generates a strongly continuous analytic semigroup $${(T_p(t))_{t\ge 0}}$$ ( T p ( t ) ) t ≥ 0 . For α ∈ (0,2) and β ≥ 2, we then prove some upper estimates for the heat kernel k associated with the semigroup ( T p ( t )) t ≥0 . As a consequence, we obtain an estimate for large | x | of the eigenfunctions of $${\mathcal{A}}$$ A . Finally, we extend such estimates to a class of divergence type elliptic operators. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Evolution Equations Springer Journals

On Schrödinger type operators with unbounded coefficients: generation and heat kernel estimates

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References (32)

Publisher
Springer Journals
Copyright
Copyright © 2015 by Springer Basel
Subject
Mathematics; Analysis
ISSN
1424-3199
eISSN
1424-3202
DOI
10.1007/s00028-014-0249-z
Publisher site
See Article on Publisher Site

Abstract

We consider the Schrödinger type operator $${\mathcal{ A}=(1+|x|^{\alpha})\Delta-|x|^{\beta}}$$ A = ( 1 + | x | α ) Δ - | x | β , for $${\alpha\in (0,2)}$$ α ∈ ( 0 , 2 ) and $${\beta\ge 0}$$ β ≥ 0 . We prove that, for any $${p\in (1,\infty)}$$ p ∈ ( 1 , ∞ ) , the minimal realization of operator $${\mathcal{A}}$$ A in $${L^p(\mathbb{R}^{N})}$$ L p ( R N ) generates a strongly continuous analytic semigroup $${(T_p(t))_{t\ge 0}}$$ ( T p ( t ) ) t ≥ 0 . For α ∈ (0,2) and β ≥ 2, we then prove some upper estimates for the heat kernel k associated with the semigroup ( T p ( t )) t ≥0 . As a consequence, we obtain an estimate for large | x | of the eigenfunctions of $${\mathcal{A}}$$ A . Finally, we extend such estimates to a class of divergence type elliptic operators.

Journal

Journal of Evolution EquationsSpringer Journals

Published: Mar 1, 2015

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