Kernel and eigenfunction estimates for some second order elliptic operators

For a potential V   such that V(x)⩾α|x|V(x)⩾|x|α with α>2α>2 we prove that the heat kernel kt(x,y)kt(x,y) associated to the uniformly elliptic operator A=−∑j,k=1n∂k(ajk∂j)+V satisfies the estimatekt(x,y)⩽Ce−μ0tect−b(e−2θα+2|x|1+α2|x|α4+n−12)(e−2θα+2|y|1+α2|y|α4+n−12) for large x,y∈Rnx,y∈Rn and all t>0t>0. Here 0<θ⩽10<θ⩽1 is an appropriate constant, b>α+2α−2 and μ0μ0 is the first eigenvalue of A  . We also obtain an estimate for large |x||x| of the eigenfunctions of A.