Kernel and eigenfunction estimates for some second order elliptic operators

For a potential V such that V ( x ) ⩾ | x | α with α > 2 we prove that the heat kernel k t ( x , y ) associated to the uniformly elliptic operator A = − ∑ j , k = 1 n ∂ k ( a j k ∂ j ) + V satisfies the estimate k t ( x , y ) ⩽ C e − μ 0 t e c t − b ( e − 2 θ α + 2 | x | 1 + α 2 | x | α 4 + n − 1 2 ) ( e − 2 θ α + 2 | y | 1 + α 2 | y | α 4 + n − 1 2 ) for large x , y ∈ R n and all t > 0 . Here 0 < θ ⩽ 1 is an appropriate constant, b > α + 2 α − 2 and μ 0 is the first eigenvalue of A. We also obtain an estimate for large | x | of the eigenfunctions of A.