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

On Schroedinger type operators with unbounded coefficients: Generation and heat kernel estimates / Lorenzi, Luca Francesco Giuseppe; Rhandi, A.. - In: JOURNAL OF EVOLUTION EQUATIONS. - ISSN 1424-3199. - 15:1(2015), pp. 53-88. [10.1007/s00028-014-0249-z]

On Schroedinger type operators with unbounded coefficients: Generation and heat kernel estimates

LORENZI, Luca Francesco Giuseppe;
2015-01-01

Abstract

We consider the Schr\"odinger type operator ${\mathcal A}=(1+|x|^{\alpha})\Delta-|x|^{\beta}$, for $\alpha\in [0,2]$ and $\beta\ge 0$. We prove that, for any $p\in (1,\infty)$, the minimal realization of operator ${\mathcal A}$ in $L^p(\R^N)$ generates a strongly continuous analytic semigroup $(T_p(t))_{t\ge 0}$. For $\alpha\in [0,2)$ and $\beta\ge 2$, we then prove some upper estimates for the heat kernel $k$ associated to the semigroup $(T_p(t))_{t\ge 0}$. As a consequence we obtain an estimate for large $|x|$ of the eigenfunctions of ${\mathcal A}$. Finally, we extend such estimates to a class of divergence type elliptic operators.
2015
On Schroedinger type operators with unbounded coefficients: Generation and heat kernel estimates / Lorenzi, Luca Francesco Giuseppe; Rhandi, A.. - In: JOURNAL OF EVOLUTION EQUATIONS. - ISSN 1424-3199. - 15:1(2015), pp. 53-88. [10.1007/s00028-014-0249-z]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11381/2786591
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