Riv. Mat. Univ. Parma, Vol. 7, No. 2, 2016

Anna Canale[a] and Cristian Tacelli[b]

Kernel estimates for a Schrödinger type operator

Pages: 341-350
Received: 22 January 2016
Accepted in revised form: 17 October 2016
Mathematics Subject Classification (2010): 47D07, 35J10, 35K05, 35K10.
Keywords: Schrödinger operators, unbounded coefficients, kernel estimates.
Author address:
[a], [b]: University of Salerno, Dept. of Information Eng., Electrical Eng. and Applied Mathematics Via Giovanni Paolo II, 132 Fisciano (SA), 84084, Italy

Abstract: In this paper the principal result obtained is the estimate for the heat kernel associated to the Schrödinger type operator \((1+|x|^\alpha)\Delta-|x|^\beta\) \[ k(t,x,y)\leq Ct^{-\frac{\theta}{2}}\frac {\varphi(x)\varphi(y)}{1+|x|^\alpha}, \] where \(\varphi=(1+|x|^\alpha)^{\frac{2-\theta}{4}+\frac{1}{\alpha}\frac{\theta-N}{2}}\), \(\theta\geq N\) and \( 0 < t \leq 1 \), provided that \(N > 2\), \(\alpha > 2\) and \(\beta > \alpha-2\). This estimate improves a similar estimate obtained in [3] with respect to the dependence on spatial component.

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