We give sufficient conditions for the discreteness of the spectrum of differential operators of the form Au = \Gamma\Deltau +hrF;rui in L 2 (R n ) where d(x) = e \GammaF (x) dx and for Schrodinger operators in L 2 (R n ). Our conditions are also necessary in the case of polynomial coefficients. Mathematics subject classification (1991): 35P05, 35J10, 35J70 1 Introduction In this paper we study the discreteness of the spectrum of two strictly related second order elliptic differential operators with unbounded coefficients on R n . These operators are A = \Gamma\Delta + n X i=1 @F @x i @ @x i ; B = \Gamma\Delta + V; with F 2 C 2 (R n ) and V 2 C(R n ). B is the classical Schrodinger operator, whereas A is a special case of second order operators with (possibly) unbounded coefficients of the first order terms. These operators are of interest when dealing with diffusion processes on all of R n in presence of a drift represented by the first order terms. Unlike...