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

Abstract

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

k(t,x,y)c1eλ0tec2tb(xy)N12βα41+yαe2βα+2xβα+22e2βα+2yβα+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,y1t>0,|x|,|y|\ge 1, where c1,c2c_1,c_2 are positive constants and b=βα+2β+α2b=\frac{\beta-\alpha+2}{\beta+\alpha-2} provided that N>2,α2N>2,\,\alpha\geq 2 and β>α2\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