VECTOR-VALUED SCHR¨ODINGER OPERATORS IN L p -SPACES

. In this paper we consider the vector-valued operator div( Q ∇ u ) − V u of Schr¨odinger type. Here V = ( v ij ) is a nonnegative, locally bounded, matrix-valued function and Q is a symmetric, strictly elliptic matrix whose entries are bounded and continuously diﬀerentiable with bounded derivatives. Concerning the potential V , we assume an that it is pointwise accretive and that its entries are in L ∞ loc ( R d ). Under these assumptions, we prove that a realization of the vector-valued Schr¨odinger operator generates a C 0 -semigroup of contractions in L p ( R d ; C m ). Further properties are also investigated.

semigroup is extrapolated to the L p -scale. We should point out that in the presence of an unbounded drift term the differential operator does not always generate a strongly continuous semigroup on L p -spaces with respect to Lebesgue measure, see [20]. Thus, in some cases appropriate growth conditions need to be imposed on the coefficients to ensure generation of a semigroup on L p with respect to Lebesgue measure.
In this article we will consider systems of parabolic equations which are coupled only through a potential term. To be more precise, consider the differential operator Au = div(Q∇u) − V u = : ∆ Q u − V u acting on vector-valued functions u = (u 1 , . . . , u m ) : R d → C m . Here Q is a bounded, symmetric and strictly elliptic matrix with continuously differentiable entries that have bounded derivatives. The expression div(Q∇u) should be understood componentwise, i.e. div(Q∇u) = (div(Q∇u 1 ), . . . , div(Q∇u m )). The matrixvalued function V : R d → R m×m is assumed to be pointwise accretive and to have locally bounded coefficients. In contrast to the situation where an unbounded drift is present, no additional growth assumptions on the potential V are needed to ensure generation of a strongly continuous semigroup on L 2 (R d ; R m ). Indeed, following Kato [15], who considered the scalar situation, we shall construct a densely defined, m-dissipative realization A of the operator A in L 2 (R d ; R m ). By virtue of the Lumer-Phillips theorem A generates a strongly continuous semigroup. Subsequently, we prove that this semigroup extrapolates to a consistent family of strongly continuous contraction semigroups {T p (t)} t≥0 on L p (R d ; R m ) for 1 < p < ∞. We also give a description of the generator A p of {T p (t)} t≥0 and prove that the test functions form a core for the operator A p .
We should point out that in our recent article [16] we were studying a similar setting. However, in [16] we were interested in proving that the domain of the vector-valued Schrödinger operator is the intersection of the domain of the diffusion part and the potential part. To that end, we had to impose growth conditions on the potential part. In the present article, we allow general potential terms without such a growth condition. The price to pay is that we can only characterize the domain of the L p -realization of our operator as the maximal L p -domain.
This article is organised as follows. In Section 2 we prove a version of Kato's inequality for vector-valued functions which is crucial in all subsequent sections. In Section 3 we construct a realization of the operator A in L 2 (R d ; R m ) which generates a strongly continuous contraction semigroup. In Section 4, we extrapolate the semigroup to L p -spaces, where p ∈ (1, ∞). In the concluding Section 5 we characterize the domain of the generator as maximal domain.
Notation. Let d, m ≥ 1. By | · | we denote the Euclidean norm on C j , j = d, m and by ·, · the Euclidean inner product. By B(r) = {x ∈ R d : |x| ≤ r} we denote the Euclidean ball of radius r > 0 and center 0. For 1 ≤ p ≤ ∞, L p (R d ; C m ) is the C m -valued Lebesgue space on R d . For 1 ≤ p < ∞, the norm is given by whereas in the case p = ∞ we use the essential supremum norm For 1 < p < ∞, p refers to the conjugate index, i.e. 1/p + 1/p = 1. Thus L p (R d ; C m ) is the dual space of L p (R d ; C m ) and the duality pairing ·, · p,p is given by By C ∞ c (R d ; C m ), we denote the space of all test functions, i.e. functions f : R d → C m which have compact support and derivatives of any order. W k,p (R d ; C m ) refers to the classical Sobolev space of order k, that is the space of all functions 2. Preliminaries. Throughout this article we make the following assumptions: 1. The map Q : R d → R d×d is such that q ij = q ji is bounded and continuously differentiable with bounded derivative for all i, j ∈ {1, . . . , d} and there exist positive real numbers η 1 and η 2 such that for all x ∈ R d , ξ ∈ C m .
We define the operator ∆ Q : for any test function ϕ ∈ C ∞ c (R d ), where D(R d ) denotes the space of distributions. As usual, we will say that ∆ In this case we will identify ∆ Q u and the function f . The following lemma, taken from [19,Lemma 2.4], generalizes Stampacchia's result concerning the weak derivative of the absolute value of an W 1,p -function, see [9,Lemma 7.6], to vector-valued functions.. Moreover, We can now prove a vector-valued version of Kato's inequality. Then Thus, the Kato inequality holds in the sense of distributions.
Proof. Let us consider the function a ε (u) = |u| 2 + ε 2 Since Recall that lim ε→0 a ε (u) = |u| in L 2 loc (R d ) and lim ε→0 ∇a ε (u) = ∇|u| in L 2 loc (R d ; R d ). Noting that uj aε(u)+ε is uniformly bounded by 1 we see that we can apply the dominated convergence theorem in the first integral above. For the other two integrals, we apply the monotone convergence theorem, using that Q is strictly elliptic and observing that (a ε (u) + ε) −1 decreases to |u| −1 . Note that in all integrals it is sufficient to integrate over the set {u = 0}. For the first and third integral, this is obvious due to the presence of u j which vanishes on {u = 0}. For the second one we infer from Stampacchia's lemma that ∇u j = 0 on {u = 0}. Thus, by letting ε → 0, we obtain This proves (3). Using (2) and (3), also (4) follows.
3. Generation of a semigroup in L 2 (R d ; C m ). Let us consider the differential In this section we prove that A generates a C 0 -semigroup of contractions on the space To that end, we follow the strategy from [15] and introduce some other realizations of the operator A on the space We define the operator L 0 by setting We letÃu = ∆ Q − V * be the formal adjoint of A, where V * is the conjugate matrix of V . We then define the operatorsL andL 0 in analogy to the operators L and L 0 , using the potential V * instead of V .
We now collect some properties of the operators L 0 and L and the adjoint L * 0 . We denote the duality pairing between H −1 (R d ; C m ) and (1) Let f ∈ D(L) and g ∈ C ∞ c (R d , C m ). Using integration by parts, we see that ThusL = L * 0 and henceL is closed. In a similar way one shows that L =L * 0 and thus L is also closed.
We can now prove the main result of this section.
Theorem 3.2. The operator −L is maximal monotone. Proof.
Step 1. We first show that −L * * 0 is maximal monotone. It is easy to see that −L 0 is monotone.
It follows that also the closure of −L 0 , i.e. the operator −L * * 0 is monotone. As −L * * 0 is monotone, rg(1 − L * * 0 ), the range of (1 − L * * 0 ), is a closed subset of H −1 (R d ; C m ), cf. [7, Proposition II-3.14]. Therefore, to prove that −L * * 0 is maximal, it suffices to show that 1 − L * * 0 has dense range. We prove that Since the coefficients of A are real, it suffices to prove that and hence, for every j ∈ {1, . . . , m} in the sense of distributions. Applying (4), we obtain Thus, ∆ Q |u| ≥ |u| in the sense of distributions. Now, let (φ n ) n ⊂ C ∞ c (R d ) be such that φ n ≥ 0 and φ n → |u| in H 1 (R d ). Then Upon n → ∞, we find − |∇|u|| Q 2 2 − u 2 2 ≥ 0 which implies that u = 0. This proves that the range of I − L * * 0 is dense.
We can now infer that A generates a strongly continuous contraction semigroup. Proof. Since −L is monotone, so is −A, the part of −L in L 2 (R d ; C m ). As −L is maximal monotone, so is −A. Indeed, given f ∈ L 2 (R d ; C m ) ⊂ H −1 (R d ; C m ) we find u ∈ D(L) such that u − Au = f . But then Au = u − f belongs to L 2 (R d ; C m ), proving that u ∈ D(A) and u − Au = f . The claim now follows from the Lumer-Phillips theorem. 4. Extension of the semigroup to L p (R d ; C m ). In this section we extrapolate the semigroup {T (t)} t≥0 to the spaces L p (R d ; C m ), 1 < p < ∞. As a first step, we prove that {T (t)} t≥0 is given by the Trotter-Kato product formula for all t > 0 and f ∈ L 2 (R d ; C m ). Here {e t∆ Q } t≥0 is the semigroup generated by ∆ Q in L 2 (R d ; C m ) and {e −tV } t≥0 is the multiplication semigroup generated by the potential −V , i.e. e −tV is multiplication with the matrix given pointwise by ∞ k=0 To prove that the semigroup {T (t)} t≥0 is given by the Trotter-Kato formula (5) we use the following result which is also of independent interest.
almost everywhere. Here both ∆ Q |u| and |u| are functions in L 2 loc (R d ). Now, let ζ ∈ C ∞ c (R d ) be such that χ B(1) ≤ ζ ≤ χ B(2) and define ζ n (x) = ζ(x/n) for x ∈ R d and n ∈ N. We multiply both two sides of the inequality ∆ Q |u| ≥ |u| by ζ n |u| and integrate by parts. We obtain Here we have used in the fourth line that ∇|u| 2 = 2|u|∇|u|. A straightforward computation shows It follows that ∆ Q ζ n ∞ → 0 as n → ∞. Hence, letting n → ∞ in the above inequality, we obtain u 2 ≤ 0, and thus u = 0. This finishes the proof. Proof.
We can now extend {T (t)} t≥0 to L p (R d ; C m ). Proof. Let 1 < p < ∞ and f ∈ L 2 (R d ; C m ) ∩ L p (R d ; C m ). Assumption (1) yields |e −tV (x) f (x)| ≤ |f (x)| for all x ∈ R d and t ≥ 0. So e −tV f p ≤ f p , for all t ≥ 0. On the other hand, it is well-known that {e t∆ Q } t≥0 extends to a contractive C 0 -semigroup on L p (R d ; C m ). Consequently, for every t > 0, both e t∆ Q and e −tV leave the set invariant. Since B p is a closed subset of L 2 (R d ; C m ) as a consequence of Fatou's lemma, it follows from the Trotter-Kato formula (5), that T (t)B p ⊂ B p . So, . By density, we can extend T (t) to a contraction T p (t) on L p (R d ; C m ). The semigroup law for {T p (t)} t≥0 can be obtained immediately.
Let us now turn to the generator of {T p (t)} t≥0 . Fix t > 0 and f ∈ C ∞ c (R d ; C m ). Then f ∈ D(A) and where the integral is computed in L 2 (R d ; C m ). However, Af has compact support whence Af ∈ L p (R d ; C m ) and the map t → T p (t)Af is continuous from [0, ∞) into L p (R d ; C m ). Hence, (6) holds true in L p (R d , C m ), i.e.
This implies that t → T p (t)f is differentiable in [0, ∞). It follows that f ∈ D(A p ) and A p f = Af .
Remark 4.4. It is also possible to extend T to a consistent contraction semigroup Upon ε → 0, we find As in the proof of Proposition 4.1, upon n → ∞, we conclude that Therefore, u = 0.
In the case when p > 2, one multiplies in (7) by ζ n |u| p −2 u and argues in a similar way. (ii) This is an immediate consequence of (i) and [7,].
Using a similar strategy as in [8] or [18] we show in the next result that the domain D(A p ) is equal to the L p -maximal domain of A.
Proposition 5.2. Let 1 < p < ∞. Assume Hypotheses 2.1. Then So, by local elliptic regularity, we obtain u ∈ W 2,p loc (R d ; C m ). Hence, Au = A p u belongs to L p (R d ; C m ), which shows that u ∈ D p,max (A).
In order to prove the other inclusion it suffices to show that λ − A is injective on D p,max (A), for some λ > 0. To this end, let u ∈ D p,max (A) be such that (λ − A)u = 0. Assume that p ≥ 2. Multiplying by ζ n |u| p−2 u and integrating (by part) over R d one obtains So, as in the proof of the above proposition, we conclude that u = 0. The case p < 2 can be obtained similarly, by multiplying the equation (λ−A)u = 0 by ζ n (|u| 2 + ε) p−2 2 u, ε > 0, instead of ζ n |u| p−2 u.
We end this article by giving an example which shows that generation of C 0semigroups for scalar-valued Schrödinger operators with complex potentials can be deduced from the vector-valued case developed in the previous sections. where v ∈ L ∞ loc (R d ) and 0 ≤ w ∈ L ∞ loc (R d ). Then Hypotheses 2.1 are satisfied and we can deduce from Theorem 4.3 and Proposition 5.2, that A p , the L p -realization of the operator with domain D(A p ) := {u ∈ L p (R d ; C 2 ) ∩ W 2,p loc (R d ; C 2 ) : Au ∈ L p (R d ; C 2 )} generates a C 0 -semigroup on L p (R d ; C 2 ). Moreover C ∞ c (R d ; R 2 ) is a core for A p . Diagonalizing the matrix 0 −1 1 0 we see that A p is similar to a diagonal operator. More precisely, with P = 1 1 −i i we have It follows that the Schrödinger operators ∆ ± iv − w with domain {f ∈ L p (R d ) ∩ W 2,p loc (R d , ) : ∆f ± ivf − wf ∈ L p (R d )} generate C 0 -semigroups on L p (R d ). Moreover C ∞ c (R d ) is a core for these operators. In general, these semigroups can not be expected to be analytic, see [16,Example 3.5]. However, imposing additional assumptions on the potential V , e.g. that the numerical range is contained in a sector, one can also prove analyticity of the semigroup, see [19,Proposition 4.5]. More precisely, there we find the following result: Proposition 5.4. Assume Hypotheses 2.1 and there is a positive constant C such that Re V (x)ξ, ξ ≥ C|Im V (x)ξ, ξ |, ∀x ∈ R d , ξ ∈ C m .
Then {T p (t)} t≥0 can be extended to an analytic semigroup on L p (R d , C m ).
Using this, we see that these semigroups are analytic provided that there is a constant C > 0 such that |v(x)| ≤ Cw(x) for a.e. x ∈ R d .