**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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