THE DIRICHLET-TO-NEUMANN MAP FOR SCHRÖDINGER OPERATORS WITH COMPLEX POTENTIALS

Let Ω ⊂ Rd be a bounded open set with Lipschitz boundary and let q : Ω → C be a bounded complex potential. We study the Dirichlet-toNeumann graph associated with the operator −∆ + q and we give an example in which it is not m-sectorial.

There are various extensions of the Dirichlet-to-Neumann operator.The first one is where the operator −∆ in (1) is replaced by a formally symmetric pure secondorder strongly elliptic differential operator in divergence form.Then one again obtains a self-adjoint version of the Dirichlet-to-Neumann operator, which enjoys a description with a form by making the obvious changes in (2).Similarly, if one replaces the operator −∆ in (1) by a pure second-order strongly elliptic differential operator in divergence form (which is possibly not symmetric), then the associated Dirichlet-to-Neumann operator is an m-sectorial operator.
There occurs a significant difference if one replaces the operator −∆ in (1) by a formally symmetric second-order strongly elliptic differential operator in divergence form, this time with lower-order terms.Then it might happen that D is no longer a self-adjoint operator, because it could be multivalued.Nevertheless, it turns out that D is a self-adjoint graph, which is lower bounded (see [6] Theorems 4.5 and 4.15, or [8] Theorem 5.7).
The aim of this note is to consider the case where the operator −∆ in ( 1) is replaced by −∆ + q, where q : Ω → C is a bounded measurable complex valued function; in a similar way a general second-order strongly elliptic operator in divergence form with lower-order terms could be considered.In Section 2 the form method from [3,4,5,6] will be adapted and applied to the present situation in an abstract form, and in Section 3 the Dirichlet-to-Neumann graph D associated with −∆ + q will be studied.Although one may expect that D is an m-sectorial graph it turns out in Example 3.7 that this is not the case in general.

2.
Forms.In this section we review and extend the form methods and the theory of self-adjoint graphs.
Let V and H be Hilbert spaces.Let a : V × V → C be a continuous sesquilinear form.Continuous means that there exists an M > 0 such that |a(u, v)| ≤ M u V v V for all u, v ∈ V .Let j ∈ L(V, H) be an operator.Define the graph D in H × H by D = {(ϕ, ψ) ∈ H × H : there exists a u ∈ V such that j(u) = ϕ and a(u, v) = (ψ, j(v)) H for all v ∈ V }.
We call D the graph associated with (a, j).
In general, if A is a graph in H, then the domain of A is dom A = {x ∈ H : (x, y) ∈ A for some y ∈ H} and the multivalued part is mul A = {y ∈ H : (0, y) ∈ A}.
We say that A is single valued, or an operator, if mul A = {0}.In that case one can identify A with a map from dom A into H.
Clearly mul D = {0} if j(V ) is not dense in H.If (ϕ, ψ) ∈ D, then there might be more than one u ∈ V such that j(u) = ϕ and a(u, v) = (ψ, j(v)) H for all v ∈ V .For that reason we introduce the space We say that the form a is j-elliptic if there exist µ, ω > 0 such that Re a(u) for all u ∈ V .Graphs associated with j-elliptic forms behave well.
Theorem 2.1.Suppose that a is j-elliptic and j(V ) is dense in H. Then D is an m-sectorial operator.Also W j (a) = {0}.
If Ω ⊂ R d is a bounded open set with Lipschitz boundary, V = H 1 (Ω), H = L 2 (Γ), j = Tr and a is as in ( 2), then D is the Dirichlet-to-Neumann operator as in the introduction; cf.Section 3 for more details.
In general the form a is not j-elliptic.An example occurs if one replaces a in ( 2) by , where ∆ D is the Laplacian on Ω with Dirichlet boundary conditions.Then (3) fails for every µ, ω > 0 if u is a corresponding eigenfunction and j = Tr .In addition, the graph associated with (a, j) is not single valued any more.We emphasize that we are interested in the graph associated with (a, j).To get around the problem that the form a is not j-elliptic, it is convenient to introduce a different Hilbert space and a different map j.
Throughout the remainder of this paper we adopt the following hypothesis.
Hypothesis 2.2.Let V , H and H be Hilbert spaces and let a : V × V → C be a continuous sesquilinear form.Let j ∈ L(V, H) and let D be the graph associated with (a, j).Furthermore, let j ∈ L(V, H) be a compact map and assume that the form a is j-elliptic, that is, there are μ, ω > 0 such that for all u ∈ V .
As example, if Ω ⊂ R d is a bounded open set with Lipschitz boundary as before, then one can choose V = H 1 (Ω), H = L 2 (Γ), H = L 2 (Ω), j = Tr and j is the inclusion map from H 1 (Ω) into L 2 (Ω).For a one can choose a continuous sesquilinear form on H 1 (Ω) like in (2).We consider this example in more detail in Section 3.
In general, if A is a graph in H, then A is called symmetric if (x, y) H ∈ R for all (x, y) ∈ A. The graph A is called surjective if for all y ∈ H there exists an x ∈ H such that (x, y) ∈ A. The graph A is called self-adjoint if A is symmetric and for all s ∈ R \ {0} the graph A + i s I is surjective, where for all λ ∈ C we define the graph (A + λ I) by A symmetric graph A is called bounded below if there exists an ω > 0 such that (x, y) H + ω x 2 H ≥ 0 for all (x, y) ∈ A. Under the above main assumptions we can state the following theorem for symmetric forms.We next wish to study the case when a is not symmetric.Proposition 2.4.Adopt Hypothesis 2.2.Then the graph D is closed.
for all v ∈ V , where the orthogonal complement is in V .We first show that ( j(u n )) n∈N is bounded in H. Suppose not.Set τ n = j(u n ) H for all n ∈ N. Passing to a subsequence if necessary, we may assume that τ n > 0 for all n ∈ N and lim n→∞ for all n ∈ N. Let μ, ω > 0 be as in (4).Then for all n ∈ N. Since ( ψ n H ) n∈N is bounded and ψn H j τn < 1 for all large n ∈ N, it follows that (Re a(w n )) n∈N is bounded.Together with (4) it then follows that (w n ) n∈N is bounded in V .Passing to a subsequence if necessary there exists a w ∈ W j (a) ⊥ such that lim n→∞ w n = w weakly in V .Then j(w) = lim n→∞ j(w n ) in H since j is compact.So j(w) H = 1 and in particular w = 0.Alternatively, for all v ∈ V it follows from (6) that Moreover, j(w) = lim n→∞ 1 τn j(u n ) = lim n→∞ 1 τn ϕ n = 0, where the limits are in the weak topology on H.So w ∈ W j (a).Therefore w ∈ W j (a) ∩ W j (a) ⊥ = {0} and w = 0.This is a contradiction.So ( j(u n )) n∈N is bounded in H.
Let n ∈ N. Then with v = u n in (5) one deduces that where we used (4) in the last step.Hence (Re a(u n )) n∈N is bounded.Using again (4) one establishes that (u n ) n∈N is bounded in V .Passing to a subsequence if necessary, there exists a u ∈ V such that lim u n = u weakly in V .Then j(u) = lim j(u n ) = lim ϕ n = ϕ weakly in H. Finally let v ∈ V .Then (5) gives So (ϕ, ψ) ∈ D and D is closed.
Proposition 2.5.Adopt Hypothesis 2.2.Suppose j is compact.Then the map where u ∈ W j (a) ⊥ is the unique element such that j(u) = ϕ and a(u, v) = (ψ, j(v)) H for all v ∈ V .We first show that the graph of Z is closed.Let ((ϕ n , ψ n )) n∈N be a sequence in D, let (ϕ, ψ) ∈ H × H and u ∈ V .Suppose that lim ϕ n = ϕ, lim ψ n = ψ in H and lim u n = u in V , where Hence Z(ϕ, ψ) = u and Z has closed graph.
The closed graph theorem, together with Proposition 2.4 implies that Z is continuous.Since j is compact, the composition We say that A has compact resolvent if (A − λ I) −1 is a compact operator for all λ ∈ ρ(A).
For the sequel it is convenient to introduce the space V j (a) = {u ∈ V : a(u, v) = 0 for all v ∈ ker j}.
Theorem 2.7.Adopt Hypothesis 2.2.If V j (a) ∩ ker j = {0} and ran j is dense in H, then D is an m-sectorial operator.
Note that the operator A D in the next lemma is the Dirichlet Laplacian if a is as in (2) and j is the inclusion map from H 1 (Ω) into L 2 (Ω).
Lemma 2.8.Adopt Hypothesis 2.2.Suppose that j(ker j) is dense in H and j is injective.Then the graph A D associated with (a| ker j×ker j , j| ker j ) is an operator and one has the following.

(a)
ker If ker A D = {0} and ran j is dense in H, then mul D = {0}.
Proof.The graph A D in H × H associated with (a| ker j×ker j , j| ker j ) is given by Now suppose that k ∈ mul A D .Let u ∈ ker j be such that j(u) = 0 and a(u, v) = (k, j(v)) H for all v ∈ ker j.The assumption that j is injective yields u = 0 and hence 0 = a(u, v) = (k, j(v)) H for all v ∈ ker j.Since j(ker j) is dense in H it follows that k = 0. Therefore mul A D = {0} and A D is an operator.'(a)'.'⊃'.Let u ∈ V j (a) ∩ ker j.Then u ∈ ker j.Moreover, a(u, v) = 0 for all v ∈ ker j.So j(u) ∈ dom A D and A D j(u) = 0. Therefore j(u) ∈ ker A D .
The converse inclusion can be proved similarly.'(b)'.Since A D has compact resolvent, this statement follows from part (a) and the injectivity of j.
In Corollary 3.4 we give a class of forms such that the converse of Lemma 2.8(c) is valid.
We conclude this section with some facts on graphs.In general, let A be a graph in H.In the following definitions we use the conventions as in the book [22] of Kato.The numerical range of A is the set 2 ) such that (x, y) H ∈ Σ θ for all (x, y) ∈ A − γ I and A − (γ − 1)I is invertible.The graph A is called quasi-accretive if there exists a γ ∈ R such that Re(x, y) H ≥ 0 for all (x, y) ∈ A − γ I.The graph A is called quasi m-accretive if there exists a γ ∈ R such that Re(x, y) H ≥ 0 for all (x, y) ∈ A − γ I and A − (γ − 1)I is invertible.Clearly every m-sectorial graph is sectorial and quasi m-accretive.Moreover, every sectorial graph is quasi-accretive.Lemma 2.9.Let A be a graph.

(a)
If not dom A ⊥ mul A, then the numerical range of A is the full complex plane.

(b)
If A is a quasi-accretive graph, then dom A ⊥ mul A. 3. Complex potentials.In this section we consider the Dirichlet-to-Neumann map with respect to the operator −∆ + q, where q is a bounded complex valued potential on a Lipschitz domain.
Throughout this section fix a bounded open set Ω ⊂ R d with Lipschitz boundary Γ.Let q : Ω → C be a bounded measurable function.Choose V = H 1 (Ω), H = L 2 (Γ), j = Tr : H 1 (Ω) → L 2 (Γ), H = L 2 (Ω) and j the inclusion of V into H.Then j and j are compact.Moreover, ran j is dense in H by the Stone-Weierstraß theorem.Define a : H Then a is a sesquilinear form and it is j-elliptic.Let D be the graph associated with (a, j).Note that all assumptions in Hypothesis 2.2 are satisfied.In order to describe D, we need the notion of a weak normal derivative.
Let u ∈ H 1 (Ω) and suppose that there exists an f ∈ L 2 (Ω) such that ∆u = f as distribution.Let ψ ∈ L 2 (Γ).Then we say that u has weak normal derivative ψ if for all v ∈ H 1 (Ω).Since ran j is dense in H it follows that ψ is unique and we write ∂ ν u = ψ.
The alluded description of the graph D is as follows.
Proof.The easy proof is left to the reader.
Let A D = −∆ D + q, where ∆ D is the Laplacian on Ω with Dirichlet boundary conditions.Then A D is as in Lemma 2.8.Moreover, (A D ) * = −∆ D + q.Proposition 3.2.Let u ∈ ker A D .Then u has a weak normal derivative, that is, ∂ ν u ∈ L 2 (Γ) is defined.Similarly, if u ∈ ker(A D ) * , then u has a weak normal derivative.
The claim for (A D ) * follows by replacing q by q.
Note that the right hand side is indeed defined and it is a subspace of L 2 (Γ) by Proposition 3.2.

1 .
Introduction.The classical Dirichlet-to-Neumann operator D is a positive selfadjoint operator acting on functions defined on the boundary Γ = ∂Ω of a bounded open set Ω ⊂ R d with Lipschitz boundary.The operator D is defined as follows.Let ϕ, ψ ∈ L 2 (Γ).Then ϕ ∈ dom D and Dϕ = ψ if and only if there exists a u ∈ H 1 (Ω) such that    Tr u = ϕ, −∆u = 0 weakly on Ω,
(c)If A is a quasi m-accretive graph, then mul A = (dom A) ⊥ .Proof.'(a)'.There are x ∈ dom A and y ∈ mul A such that (x, y ) H = 0. Without loss of generality we may assume that x H = 1.There exists a y ∈ H such that (x, y) ∈ A.Then (x, y + τ y ) ∈ A for all τ ∈ C.So (x, y + τ y ) H ∈ W (A) for all τ ∈ C.'(b)'.This follows from Statement (a).'(c)'.By Statement (b) it remains to show that (dom A) ⊥ ⊂ mul A. By assumption there exists a γ ∈ R such that Re(x, y) H ≥ 0 for all (x, y) ∈ A − γ I and A − (γ − 1)I is invertible.Without loss of generality we may assume that γ = 0. Let y ∈ (dom A) ⊥ .Define x = (A + I) −1 y.Then x ∈ dom A and (x, y − x) ∈ A. So − x 2 H = Re(x, y − x) H ≥ 0 and x = 0. Then (0, y) ∈ A and y ∈ mul A as required.