Optimal kernel estimates for elliptic operators with second order discontinuous coefficients

Abstract We consider the family of second order elliptic operators L = Δ + ( a − 1 ) ∑ i , j = 1 N x i x j | x | 2 D i j + c x | x | 2 ⋅ ∇ − b | x | 2 , a > 0 , b , c ∈ R , which includes the Schrodinger operator with inverse square potential, and we prove optimal upper bounds for the heat kernel.