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

We prove that the heat kernel associated to the Schrodinger type operator A := ( 1 + vertical bar x vertical bar(alpha)) Delta- vertical bar x vertical bar (beta) satisfies the estimate k (t,x,y) <= c(1)e (lambda 0t)e(c2t-b) (vertical bar x vertical bar vertical bar y vertical bar)(-N-1)/2-(beta-alpha)/4/1+vertical bar y vertical bar(alpha-root 2/beta-alpha+1 vertical bar)x(vertical bar beta-alpha+2/2)e(-root 2/beta-alpha+2 vertical bar)y vertical bar beta-alpha+2/2 for t > 0; vertical bar x vertical bar; vertical bar y vertical bar >= 1, where c(1); c(2) are positive constants and b = beta-alpha+2/beta+alpha-2 provided that N > 2; alpha >= 2 and beta > alpha-2. We also obtain an estimate of the eigenfunctions of Lambda.