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...

In this paper we characterize completely the existence of a C0-semigroup in L 1 (I) (I real interval) generated by a second-order differential operator when suitable boundary conditions at the endpoints are imposed. In spaces of continuous functions similar characterizations have been obtained by Timmermans [6] on the maximal domain, by Cl'ement and Timmermans [2] in the case of Ventcel's boundary conditions and by Campiti, Metafune and Pallara [1] in the case of Neumann's boundary conditions. 1 Introduction and preliminaries Let I =]r 1 ; r 2 [ (\Gamma1 r 1 ! r 2 +1) be a real interval and consider the second-order differential operator Bu(x) = ff(x)u 00 (x) + fi(x)u 0 (x) ; x 2 I ; (1.1) where ff; fi : I ! R are continuous functions and ff(x) ? 0 for every x 2 I . In [2] Cl'ement and Timmermans gave necessary and sufficient conditions in order for B to be the generator of a C 0 -semigroup in C(I) on the domain DV (B) := ae u 2 C(I) " C 2 (I) j lim x!r1 ;r 2 Bu(x) = 0...

Under suitable conditions on the functions a 2 C N 2 ), F 2 C ), and V : R [0; 1), we show that the operator Au = r(aru) + F ru V u with domain W V (R ) = fu 2 ) : V u 2 L )g generates a positive analytic semigroup on L ), 1 < p < 1.

We show that the domain of the Ornstein-Uhlenbeck operator on ; dx) equals the weighted Sobolev space W ; dx), where dx is the corresponding invariant measure. Our approach relies on the operator sum method, namely the commutative and the non commutative Dore-Venni theorems.

We consider a class of degenerate Ornstein-Uhlenbeck operators in R N, of the kind p0X A aij @ 2 NX xix + j i;j=1 i;j=1

Let E be a real Banach space. We study the Ornstein-Uhlenbeck semigroup P = {P(t)}t≥0 associated with the Ornstein-Uhlenbeck operator Lf(x) = 1 2 Tr QD2 f(x) + 〈Ax, Df(x)〉, x ∈ E. Here Q ∈ L (E ∗ , E) is a positive symmetric operator and A is the generator of a C0-semigroup S = {S(t)}t≥0 on E. Under the assumption that P admits an invariant measure µ ∞ we prove that if S is eventually compact and the spectrum of its generator is nonempty, then ‖P(t) − P(s) ‖ L(L 1 (E,µ∞)) = 2 for all t, s ≥ 0 with t ̸ = s. This result is new even when E = R n. We also study the behaviour of P in the space BUC(E). We show that if A ̸ = 0 there exists t0> 0 such that ‖P(t) − P(s) ‖ L(BUC(E)) = 2 for all 0 ≤ t, s ≤ t0 with t ̸ = s. Moreover, under a nondegeneracy assumption or a strong Feller assumption, the following dichotomy holds: either ‖P(t) − P(s) ‖ L(BUC(E)) = 2 for all t, s ≥ 0, t ̸ = s, or S is the direct sum of a nilpotent semigroup and a finite-dimensional periodic semigroup. Finally we investigate the spectrum of L in the spaces L 1 (E, µ∞) and BUC(E).