JournalszaaVol. 36, No. 4pp. 377–392

Kernel Estimates for Schrödinger Type Operators with Unbounded Diffusion and Potential Terms

  • Anna Canale

    Università degli Studi di Salerno, Fisciano, Italy

  • Abdelaziz Rhandi

    Università degli Studi di Salerno, Fisciano, Italy

  • Cristian Tacelli

    Università degli Studi di Salerno, Fisciano, Italy
Kernel Estimates for Schrödinger Type Operators with Unbounded Diffusion and Potential Terms cover
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Abstract

We prove that the heat kernel associated to the Schrödinger type operator A:=(1+∣x∣α)Δ−∣x∣βA:=(1+|x|^\alpha)\Delta-|x|^\beta satisfies the estimate

k(t,x,y)≤c1eλ0tec2t−b(∣x∣∣y∣)−N−12−β−α41+∣y∣αe−2β−α+2∣x∣β−α+22e−2β−α+2∣y∣β−α+22k(t,x,y)\leq c_1e^{\lambda_0t}e^{c_2t^{-b}}\frac{(|x||y|)^{-\frac{N-1}{2}-\frac{\beta-\alpha}{4}}}{1+|y|^\alpha} e^{-\frac{\sqrt{2}}{\beta-\alpha+2}|x|^{\frac{\beta-\alpha+2}{2}}} e^{-\frac{\sqrt{2}}{\beta-\alpha+2}|y|^{\frac{\beta-\alpha+2}{2}}}

for t>0,∣x∣,∣y∣≥1t>0,|x|,|y|\ge 1, where c1,c2c_1,c_2 are positive constants and b=β−α+2β+α−2b=\frac{\beta-\alpha+2}{\beta+\alpha-2} provided that N>2, α≥2N>2,\,\alpha\geq 2 and β>α−2\beta>\alpha-2. We also obtain an estimate of the eigenfunctions of AA.

Cite this article

Anna Canale, Abdelaziz Rhandi, Cristian Tacelli, Kernel Estimates for Schrödinger Type Operators with Unbounded Diffusion and Potential Terms. Z. Anal. Anwend. 36 (2017), no. 4, pp. 377–392

DOI 10.4171/ZAA/1593