Schrödinger Operators on Bounded Domains

For the multi-dimensional Schrodinger operator -∆+V with a bounded real potential V on a bounded domain \( \varOmega \subset {\mathbb{R}}^{n} \) with a C2-smooth boundary a boundary triplet and a Weyl function will be constructed.

For the multi-dimensional Schrödinger operator −Δ + V with a bounded real potential V on a bounded domain Ω ⊂ R n with a C 2 -smooth boundary a boundary triplet and a Weyl function will be constructed. The self-adjoint realizations of −Δ + V in L 2 (Ω) and their spectral properties will be investigated. One of the main difficulties here is to provide trace mappings on the domain of the maximal realization, in such a way that the second Green identity remains valid in an appropriate form. It is necessary to introduce and study Sobolev spaces on the domain Ω and its boundary ∂Ω, which will be done in Section 8.2; in this context also the rigged Hilbert spaces from Section 8.1 arise as Sobolev spaces and their duals. The minimal and maximal operators, and the Dirichlet and Neumann trace maps on the maximal domain will be discussed in Section 8.3, and in Section 8.4 a boundary triplet and Weyl function for the maximal operator associated with −Δ + V is provided. The self-adjoint realizations, their spectral properties, and some natural boundary conditions are also discussed in Section 8.4. The class of semibounded self-adjoint realizations of −Δ + V in L 2 (Ω) and the corresponding semibounded forms are studied in Section 8.5. For this purpose a boundary pair which is compatible with the boundary triplet in Section 8.4 is provided. Orthogonal couplings of Schrödinger operators are treated in Section 8.6 for the model problem in which R n decomposes into a bounded C 2 -domain Ω + and an unbounded component Ω − = R n \ Ω + . Finally, in Section 8.7 the more general setting of Schrödinger operators on bounded Lipschitz domains is briefly discussed.

Rigged Hilbert spaces
In this preparatory section the notion of rigged Hilbert spaces or Gelfand triples is briefly recalled. For this, let G and H be Hilbert spaces and assume that G is densely and continuously embedded in H, that is, one has G ⊂ H and the embedding operator ι : G → H is continuous with dense range and ker ι = {0}. In the following the dual space H is identified with H, but the dual space G of antilinear continuous functionals is not identified with G. Instead, the isometric isomorphism I : G → G, g → Ig , where (Ig , g) G = g (g), g ∈ G, (8.1.1) is written explicitly whenever used. In fact, in this section the continuous (antilinear) functionals on G will be identified via the scalar product in H. In the following usually the notation g , g G ×G := g (g) (8. 1.2) is employed for the (antilinear) dual pairing in (8.1.1), and when no confusion can arise the index is suppressed, that is, one writes g , g = g (g) for (8.1.2).
In the above setting the dual operator of the embedding operator ι : G → H is given by (8.1.3) and in terms of the pairing ·, · this means ι h, g = (h, ιg) H , h∈ H, g ∈ G. (8.1.4) Since the scalar product (·, ·) H is antilinear in the second argument one has ι h, λg = λ ι h, g for λ ∈ C, and hence ι h is indeed antilinear. Observe that the dual operator ι in (8.1.3) is continuous since ι is continuous. Moreover, from the identity ker ι = (ran ι) ⊥H it follows that ι is injective, and the range of ι is dense in G since ker ι = (ran ι ) ⊥ G and ι = ι as G is reflexive. Thus, G ι → H ι → G with ran ι ⊂ H dense and ran ι ⊂ G dense, and since G can be viewed as a subspace of H, and H can be viewed as a subspace of G , instead of (8.1.4) also the notation h, g = (h, g) H , h∈ H, g ∈ G, (8.1.5) will be used. The present situation will appear naturally in the context of Sobolev spaces later in this chapter. First the terminology will be fixed in the next definition.
Definition 8.1.1. Let G and H be Hilbert spaces such that G is densely and continuously embedded in H. Then the triple {G, H, G } is a called a Gelfand triple or a rigged Hilbert space.
Assume now that {G, H, G } is a Gelfand triple. Since the embedding operator ι : G → H is continuous, one has g H ≤ C g G for all g ∈ G with the constant C = ι > 0. Moreover, as G is a Hilbert space it follows from Lemma 5.1.9 that the symmetric form t[g 1 , g 2 ] := (g 1 , g 2 ) G , dom t = G, is densely defined and closed in H with a positive lower bound. Hence, by the first representation theorem (Theorem 5.1.18) there exists a unique self-adjoint operator T with the same positive lower bound in H, such that dom T ⊂ dom t and (g 1 , g 2 ) G = t[g 1 , g 2 ] = (T g 1 , g 2 ) H , g 1 ∈ dom T, g 2 ∈ G.
In the next lemma some more properties of the Gelfand triple {G, H, G } and the operator R are collected. (iv) For all h ∈ H and g ∈ G one has ι + ι − h = h and ι − ι + g = g.
(v) The operator R −2 can be extended by continuity to an isometric operator R −2 : G → G which coincides with the isometric isomorphism I : G → G.
Proof. (i) Consider an element g ∈ G and assume, in addition, that g ∈ H. Then one has where (8.1.2) was used in the second equality, and g ∈ H and (8.1.5) were used in the last step. Since R is uniformly positive, one has R −1 ∈ B(H), and using (8.1.6) one obtains Therefore, g G = R −1 g H for all g ∈ H ⊂ G and as H is dense in G with respect to the norm · G , one concludes that G coincides with the completion of H with respect to the norm R −1 · H .
(ii) Observe that by the definition of ι + and (8.1.6) one has ι + g H = Rg H = g G , g ∈ G = dom ι + = dom R, and hence ι + : G → H is isometric. Moreover, since R is bijective, it follows that ι + is an isometric isomorphism. Similarly, for g ∈ G one has where in the last step it was used that I : G → G is an isometric isomorphism. In order to check the identity (8.1.7), let g ∈ G and g ∈ G. Then (8.1.6) and (8.1.1) imply (ι − g , ι + g) H = (R Ig , Rg) H = (Ig , g) G = g , g .
(iv) By the definition of ι + and (iii) it is clear that (v) For h ∈ H one has R −2 h G = R −1 h H = h G by (8.1.6) and (i), and since H is dense in G , it follows that R −2 admits an extension to an isometric operator R −2 : G → G. Moreover, for h ∈ H it follows from the definition of ι − in (ii) and (iii) that R Ih = ι − h = R −1 h, and hence Ih = R −2 h.
Thus I and the restriction R −2 of R −2 coincide on the dense subspace H ⊂ G . This implies I = R −2 .
Now a different point of view is taken on Gelfand triples. In the next lemma it is shown that the powers R s for s ≥ 0 of a uniformly positive self-adjoint operator R in H give rise to Gelfand triples with certain compatibility properties. Then G t ⊂ G s for all t ≥ s ≥ 0 and the following statements hold: (i) {G s , H, G s } is a Gelfand triple and the assertions in Lemma 8.1.2 hold with R, G, and G replaced by R s , G s , and G s , respectively.
Proof. (i) For s ≥ 0 the self-adjoint operator R s is uniformly positive in H and hence G s = dom R s equipped with the inner product (8.1.9) is a Hilbert space which is dense in H. Moreover, from R −s ∈ B(H) and (8.1.9) one obtains that which shows that the embedding G s → H is continuous. Therefore, if G s denotes the dual of G s , then {G s , H, G s } is a Gelfand triple. Comparing (8.1.9) with (8.1.6) shows that the operator R s plays the same role as the representing operator of the inner product in (8.1.6). Hence, the assertions of Lemma 8.1.2 are valid with R, G, and G replaced by R s , G s , and G s , respectively.
(ii) Let s ≥ 0 and consider l ∈ G s+1 = dom R s+1 . It follows from (8.1.9) that and hence ι + = R : G s+1 → G s is isometric. In order to verify that this mapping is onto let k ∈ G s . Then k ∈ H, and as R is bijective, there exists l ∈ dom R such that Rl = k. Therefore, l = R −1 k and as k ∈ G s = dom R s one concludes l ∈ dom R s+1 = G s+1 . This shows that ι + = R : G s+1 → G s is an isometric isomorphism for s ≥ 0. A similar reasoning shows that ι − = R −1 : G s → G s+1 is an isometric isomorphism for s ≥ 0. The remaining assertions ι + ι − g = g for g ∈ G s and ι − ι + l = l for l ∈ G s+1 follow immediately from (8.1.10).

Sobolev spaces, C 2 -domains, and trace operators
In this section Sobolev spaces on R n , open subsets Ω ⊂ R n , and on the boundaries ∂Ω of C 2 -domains are defined and some of their features are briefly recalled. Furthermore, the mapping properties of the Dirichlet and Neumann trace map on a C 2 -domain Ω are recalled and the first Green identity is established.
For s ≥ 0 the scale of L 2 -based Sobolev spaces H s (R n ) is defined with the help of the (classical) Fourier transform F ∈ B(L 2 (R n )) by and H s (R n ) is equipped with the natural norm and corresponding scalar product Then the space H s (R n ) is a separable Hilbert space for every s ≥ 0 and one has H 0 (R n ) = L 2 (R n ). It is also useful to note that the space Furthermore, for each s ≥ 0 one has for all s ≥ 0. In particular, R plays the same role as the operator R in (8.1.6) and R s plays the same role as the operator R s in (8.1.9). Hence, R s , s ≥ 0, gives rise to a Gelfand triple The space H s (Ω) is a separable Hilbert space; the corresponding scalar product will be denoted by ( In order to define Sobolev spaces on the boundary ∂Ω of some domain Ω ⊂ R n assume first that φ : R n−1 → R is a C 2 -function. The vectors in R n−1 will be denoted by x = (x 1 , . . . , x n−1 ) ∈ R n−1 and the notation (x , x n ) is used for (x 1 , . . . , x n ) ∈ R n . Then the domain is called a C 2 -hypograph and its boundary is given by For a measurable function h : ∂Ω φ → C the surface integral on ∂Ω φ is defined as This surface measure also gives rise to the usual L 2 -space on ∂Ω φ , which will be denoted by L 2 (∂Ω φ ). Furthermore, for s ∈ [0, 2] define the Sobolev space of order s on ∂Ω φ by and equip H s (∂Ω φ ) with the corresponding Hilbert space scalar product In the next step the notion of C 2 -hypograph is replaced by a bounded domain with a C 2 -smooth boundary, that is, the boundary is locally the boundary of a C 2 -hypograph.
Let Ω ⊂ R n be a bounded C 2 -domain as in Definition 8.2.1. Then the boundary ∂Ω ⊂ R n is compact and there exists a partition of unity subordinate to the open cover {U j } of ∂Ω, that is, there exist functions η j ∈ C ∞ 0 (R n ), j = 1, . . . , l, with supp η j ⊂ U j such that 0 ≤ η j (x) ≤ 1 for all x ∈ R n and l j=1 η j (x) = 1 for all x ∈ ∂Ω. For a measurable function h : ∂Ω → C the surface integral on ∂Ω is defined as where the C 2 -functions φ j : R n−1 → R define the C 2 -hypographs Ω j as in (8.2.3) and the possible rotation of coordinates is suppressed. This surface integral induces a surface measure and the notion of an L 2 -space L 2 (∂Ω) in the same way as in (8.2.4) and (8.2.5). In the present setting the Sobolev space H s (∂Ω) for s ∈ [0, 2] is now defined by and is equipped with the corresponding Hilbert space scalar product It follows from the construction that H s (∂Ω) is densely and continuously embed- holds for ϕ ∈ H −1/2 (∂Ω) and ψ ∈ H 1/2 (∂Ω), then for s ∈ [0, 3/2] their restrictions and ; here Q is the uniformly positive self-adjoint operator in (8.2.9).
Assume now that Ω ⊂ R n is a bounded C 2 -domain as in Definition 8.2.1. The weak derivative of order |α| of an L 2 -function f is denoted by D α f in the following; as usual, here α ∈ N n 0 stands for a multiindex and |α| = α 1 + · · · + α n . Then for k ∈ N 0 one has is valid. In particular, for f ∈ C ∞ 0 (Ω) and k = 2, integration by parts and the Schwarz theorem give , and this equality extends to all f ∈ H 2 0 (Ω) by (8.2.2). As a consequence one obtains the following useful fact.
Let again Ω ⊂ R n be a bounded C 2 -domain and denote the unit normal vector field pointing outwards of ∂Ω by ν. The notion of trace operator or trace map and some of their properties are discussed next. Recall first that the mapping extends by continuity to a continuous operator denotes the Neumann trace operator. In particular, for all f ∈ C ∞ (Ω) one has respectively. With the help of the trace operators one has another useful characterization of the space H 2 0 (Ω) in (8.2.2), namely, It will also be used that the Dirichlet trace operator τ D : H 2 (Ω) → H 3/2 (∂Ω) admits a continuous surjective extension Recall next that for f ∈ H 2 (Ω) and g ∈ H 1 (Ω) the first Green identity holds. Note that τ N f, τ D g ∈ H 1/2 (∂Ω) by (8.2.14) and (8.2.16). If, in addition, also g ∈ H 2 (Ω), then one concludes from (8.2.18) the second Green identity which is valid for all f, g ∈ H 2 (Ω).
In the next lemma it will be shown that the Neumann trace operator τ N in (8.2.14) admits an extension to the subspace of H 1 (Ω) consisting of all those functions f ∈ H 1 (Ω) such that Δf ∈ L 2 (Ω), and it turns out that the first Green identity (8.2.18) remains valid in an extended form. Here, and in the following, the expression Δf is understood in the sense of distributions. If, in addition, one has that Δf ∈ L 2 (Ω), then Δf is a regular distribution generated by the function Δf ∈ L 2 (Ω) via holds for all g ∈ H 1 (Ω). In the following the notation τ (1) N f := ϕ will be used.
Remark 8.2.5. The assertion in Lemma 8.2.4 and its proof extend in a natural manner to all f ∈ H 1 (Ω) such that −Δf ∈ H 1 (Ω) * . In this situation there still exists a unique element ϕ ∈ H −1/2 (∂Ω) such that (instead of (8.2.21)) one has the slightly more general first Green identity for all g ∈ H 1 (Ω); cf. [573, Lemma 4.3].

Trace maps for the maximal Schrödinger operator
The differential expression −Δ + V is considered on a bounded domain Ω, where the function V ∈ L ∞ (Ω) is assumed to be real. One then associates with −Δ + V a preminimal, minimal and maximal operator in L 2 (Ω), which are adjoints of each other. Furthermore, the Dirichlet and Neumann operators are defined via the corresponding sesquilinear form and the first representation theorem, and some of their properties are collected. In the case where Ω is a bounded C 2 -domain it is shown in Theorem 8.3.9 and Theorem 8.3.10 that the Dirichlet and Neumann trace operators in the previous section admit continuous extensions to the maximal domain; this is a key ingredient in the construction of a boundary triplet in the next section.
Let n ≥ 2, let Ω ⊂ R n be a bounded domain, and assume that the function V ∈ L ∞ (Ω) is real. The preminimal operator associated to the differential expression −Δ + V is defined as that T 0 is a densely defined symmetric operator in L 2 (Ω) which is bounded from below with v − := essinf V as a lower bound, so that and hence, by the Poincaré inequality (8.2.11), In fact, using Lemma 8.2.3 and the fact that V ∈ L ∞ (Ω) one obtains that the graph norm · L 2 (Ω) + T min · L 2 (Ω) is equivalent to the H 2 -norm on the closed subspace H 2 0 (Ω) of H 2 (Ω). Hence, T min is a closed operator in L 2 (Ω) and it follows from (8.2.2) that T 0 = T min . Therefore, T min is a densely defined closed symmetric operator in L 2 (Ω) and T min − v − is uniformly positive.
Besides the preminimal and minimal operator, also the maximal operator T max associated with −Δ + V in L 2 (Ω) will be important in the sequel; it is defined by Here the expression Δf for f ∈ L 2 (Ω) is understood in the distributional sense.
Since V ∈ L ∞ (Ω), it is clear that f ∈ L 2 (Ω) belongs to dom T max if and only if Δf ∈ L 2 (Ω), that is, the (regular) distribution Δf is generated by an L 2function; cf. (8.2.20). Observe that H 2 (Ω) ⊂ dom T max , and it will also turn out that H 2 (Ω) = dom T max . Proposition 8.3.1. Let T 0 , T min , and T max be the preminimal, minimal, and maximal operator associated with −Δ + V in L 2 (Ω), respectively. Then T 0 = T min , and (T min ) * = T max and T min = (T max ) * .
Proof. It has already been shown above that T 0 = T min holds. In particular, this implies T * 0 = (T min ) * and thus for the first identity in (8.3.4) it suffices to show T * 0 = T max . Furthermore, since multiplication by V ∈ L ∞ (Ω) is a bounded operator in L 2 (Ω), it is no restriction to assume V = 0 in the following. Let f ∈ dom T * 0 and consider T * 0 f ∈ L 2 (Ω) as a distribution. Then one has for all that is, f ∈ dom T * 0 and T * 0 f = −Δf = T max f . Thus, the first identity in (8.3.4) has been shown. The second identity in (8.3.4) follows by taking adjoints.
In the following the self-adjoint Dirichlet realization A D and the self-adjoint Neumann realization A N of −Δ + V in L 2 (Ω) will play an important role. The operators A D and A N will be introduced via the corresponding sesquilinear forms using the first representation theorem. More precisely, consider the densely defined forms in L 2 (Ω). It is easy to see that both forms are semibounded from below and that v − = essinf V is a lower bound. The same argument as in (8.3.1) using the Poincaré inequality (8.2.11) on dom t D = H 1 0 (Ω) implies the stronger statement that the form t D − v − is uniformly positive. Furthermore, it follows from the definitions that the form (∇·, is bounded on L 2 (Ω) and hence it follows from Theorem 5.1.16 that also the forms t D and t N are closed in L 2 (Ω). Therefore, by the first representation theorem (Theorem 5.1.18), there exist unique semibounded self-adjoint operators A D and A N in L 2 (Ω) associated with t D and t N , respectively, such that The self-adjoint operators A D and A N are called the Dirichlet operator and Neumann operator, respectively. In the next propositions some properties of these operators are discussed.

then the Dirichlet operator A D is given by
Therefore, A D is given by (8.3.5). In the case that Ω ⊂ R n has a C 2 -smooth boundary the form of the domain of A D in (8.3.6) follows from (8.3.5) and (8.2.17). Next it will be shown that for λ ∈ ρ(A D ) the resolvent (A D − λ) −1 is a compact operator in L 2 (Ω). For this observe first that is everywhere defined and closed as an operator from Hence, the operator in (8.3.7) is bounded by the closed graph theorem. By Rellich's theorem the embedding It remains to verify that A D is the Friedrichs extension S F of T min = T 0 or, equivalenty, the Friedrichs extension of T 0 ; cf. Lemma 5.3.1 and Definition 5.3.2. For this, consider the form t , and so f ∈ H 1 0 (Ω) by (8.2.2). Therefore, by (5.1.16), the closure of the form t T0 is given by Hence, t T0 = t D , and since by Definition 5.3.2 the Friedrichs extension of T 0 is the unique self-adjoint operator corresponding to the closed form t T0 , the assertion follows.
In order to specify the Neumann operator A N , the first Green identity and the trace operators τ (1) will be used; cf. Lemma 8.2.4. For this reason in the next proposition it is assumed that Ω ⊂ R n is a bounded C 2 -domain. It is also important to note that the Neumann operator A N below differs from the Kreȋn-von Neumann extension and the Kreȋn type extensions in Definition 5.4.2; cf. Section 8.5 for more details.

Proposition 8.3.3. Assume that Ω ⊂ R n is a bounded C 2 -domain. Then the Neumann operator A N is given by
Proof. In a first step it follows for f ∈ dom A N ⊂ H 1 (Ω) and all g ∈ C ∞ 0 (Ω) in the same way as in the proof of Proposition 8.3.2 that In particular, for f ∈ dom A N one has f ∈ H 1 (Ω) and −Δf ∈ L 2 (Ω), so that Lemma 8.2.4 applies and yields To show that the resolvent (A N − λ) −1 is a compact operator in L 2 (Ω) one argues in the same way as in the proof of Proposition 8.3.2. In fact, the operator is everywhere defined and closed, and hence bounded by the closed graph theorem.
It is known that functions f in dom A D or dom A N are locally H 2 -regular, that is, for every compact subset K ⊂ Ω the restriction of f to K is in H 2 (K). The next theorem is an important elliptic regularity result which ensures H 2 -regularity of the functions in dom A D or dom A N in (8.3.6) and (8.3.8), respectively, up to the boundary if the bounded domain Ω is C 2 -smooth in the sense of Definition 8.2.1.
Note that under the assumptions in Theorem 8.

the domain of the Dirichlet operator
The direct sum decompositions in the next corollary follow immediately from Theorem 1.7.1 when considering the operator T = −Δ + V , dom T = H 2 (Ω), and taking into account that A D ⊂ T and A N ⊂ T . Corollary 8.3.5. Assume that Ω ⊂ R n is a bounded C 2 -domain and denote by τ D : H 2 (Ω) → H 3/2 (∂Ω) and τ N : H 2 (Ω) → H 1/2 (∂Ω) the Dirichlet and Neumann trace operator in (8.2.13) and (8.2.14), respectively. Then for λ ∈ ρ(A D ) one has the direct sum decomposition and for λ ∈ ρ(A N ) one has the direct sum decomposition As a consequence of the decomposition (8.3.9) in Corollary 8.3.5 and (8.2.12) one concludes that the so-called Dirichlet-to-Neumann map in the next definition is a well-defined operator from H 3/2 (∂Ω) into H 1/2 (∂Ω). Definition 8.3.6. Let Ω ⊂ R n be a bounded C 2 -domain, let A D be the self-adjoint Dirichlet operator, and let τ D : H 2 (Ω) → H 3/2 (∂Ω) and τ N : be the Dirichlet and Neumann trace operator in (8.2.13) and (8.2.14), respectively. For λ ∈ ρ(A D ) the Dirichlet-to-Neumann map is defined as Note that for λ ∈ ρ(A D ) ∩ ρ(A N ) both decompositions (8.3.9) and (8.3.10) in Corollary 8.3.5 hold and together with (8.2.12) this implies that the Dirichletto-Neumann map D(λ) is a bijective operator from H 3/2 (∂Ω) onto H 1/2 (∂Ω).
A further useful consequence of Theorem 8.3.4 is given by the following a priori estimates.
Corollary 8.3.7. Assume that Ω ⊂ R n is a bounded C 2 -domain and let A D and A N be the Dirichlet and Neumann operator, respectively. Then there exist constants C D > 0 and C N > 0 such that Proof. One verifies in the same way as in the proof of Proposition 8.3.2 that for for some C > 0 and C D > 0, and the first estimate follows. The second estimate is proved in the same way.
The next lemma is an important ingredient in the following.
are equivalent on dom T max , and hence it is no restriction to assume that (8.3.12) Next it will be shown that Δf ∈ H 2 0 (Ω for the restriction onto Ω. Denote by f and Δf the extension of f and Δf by zero to B. Then it follows with the help of (8.3.11) that that is, f = 0 in (8.3.11). Hence, C ∞ (Ω) is dense in dom T max with respect to the graph norm.
The following result on the extension of the Dirichlet trace operator onto dom T max is essential for the construction of a boundary triplet for T max . Theorem 8.3.9. Assume that Ω ⊂ R n is a bounded C 2 -domain. Then the Dirichlet trace operator τ D : H 2 (Ω) → H 3/2 (∂Ω) in (8.2.13) admits a unique extension to a continuous surjective operator where dom T max is equipped with the graph norm. Furthermore, Proof. In the following fix λ ∈ ρ(A D ) and consider the operator is continuous and maps onto dom A D . Hence, it follows from Theorem 8.3.4 and (8.2.12) that Υ ∈ B(L 2 (Ω), H 1/2 (∂Ω)) in (8.3.14) is a surjective operator. Next it will be shown that for some h ∈ L 2 (Ω). Then it follows from Theorem 8.3.4 that which shows that h ∈ ker Υ. This completes the proof of (8.3.15).
From (8.3.14) and (8.3.15) it follows that the restriction of Υ to N λ (T max ) is an isomorphism from N λ (T max ) onto H 1/2 (∂Ω). This implies that the dual operator is bounded and invertible, and by the closed range theorem (see Theorem 1.3.5 for the Hilbert space adjoint) one has from Theorem 1.7.1 or Corollary 1.7.5, and write the elements f ∈ dom T max accordingly, Define the mapping Next it will be shown that τ D is an extension of the Dirichlet trace operator τ D : H 2 (Ω) → H 3/2 (∂Ω). For this, consider ϕ ∈ ran τ D = H 3/2 (∂Ω) ⊂ H −1/2 (∂Ω) and note that by (8.3.9) and (8.2.12) there exists a unique f λ ∈ H 2 (Ω) such that Let h ∈ L 2 (Ω) and set k := (A D − λ) −1 h. Then, by (8.3.14), the fact that τ D k = 0, and the second Green identity (8.2.19), then is an isomorphism and hence, in particular, bounded, one has with some constants C, C , C > 0. Thus, τ D is continuous. The proof of Theorem 8.3.9 is complete.
The following result is parallel to Theorem 8.3.9 and can be proved in a similar way.
.14) admits a unique extension to a continuous surjective operator where dom T max is equipped with the graph norm. Furthermore, As a consequence of Theorem 8.3.9 and Theorem 8.3.10 one can also extend the second Green identity in (8.2.19) to elements f ∈ dom T max and g ∈ H 2 (Ω). Corollary 8.3.11. Assume that Ω ⊂ R n is a bounded C 2 -domain, and let be the unique continuous extensions of the Dirichlet and Neumann trace operators from Theorem 8.3.9 and Theorem 8.3.10, respectively. Then the second Green identity in (8.2.19) extends to Proof. Let f ∈ dom T max and g ∈ H 2 (Ω). Since C ∞ (Ω) is dense in dom T max with respect to the graph norm by Lemma 8.3.8 and because τ D and τ N are continuous with respect to the graph norm. Therefore, with the help of the second Green identity (8.2.19), one concludes that which completes the proof.
Note that, by construction, there exists a bounded right inverse for the extended Dirichlet trace operator τ D (see (8.3.16)-(8.3.17)) and similarly there exists a bounded right inverse for the extended Neumann trace operator τ N . This also implies that the Dirichlet-to-Neumann map in Definition 8.3.6 admits a natural extension to a bounded mapping from from H −1/2 (∂Ω) into H −3/2 (∂Ω). , For later purposes the following fact is provided. Proof. Since A D is a self-adjoint extension of T min with discrete spectrum, it suffices to check that T min has no eigenvalues; cf. Proposition 3.4.8. For this, assume that T min f = λf for some λ ∈ R and some f ∈ dom T min . Since dom T min = H 2 0 (Ω), there exist (f k ) ∈ C ∞ 0 (Ω) such that f k → f in H 2 (Ω). Denote the zero extensions of f and f k to all of R n by f and f k , respectively. Then f k → f in L 2 (R n ) and for all h ∈ C ∞ 0 (R n ) and α ∈ N n 0 such that |α| ≤ 2 one computes where (D α f ) denotes the zero extension of D α f to all of R n . It follows from this computation that

A boundary triplet for the maximal Schrödinger operator
In this section a boundary triplet {L 2 (∂Ω), Γ 0 , Γ 1 } for the maximal operator T max in (8.3.3) is provided under the assumption that Ω ⊂ R n is a bounded C 2 -domain. The corresponding Weyl function is closely connected to the extended Dirichletto-Neumann map in Corollary 8.3.12. As examples, Neumann and Robin type boundary conditions are discussed, and it is also explained that there exist selfadjoint realizations of −Δ + V in L 2 (Ω) which are not semibounded and which may have essential spectrum of rather arbitrary form.
Recall from Corollary 8.2.2 that is a Gelfand triple and there exist isometric isomorphisms ι ± : holds for all ϕ ∈ H −1/2 (∂Ω) and ψ ∈ H 1/2 (∂Ω). For the definition of the boundary mappings in the next proposition recall also the definition and the properties of the Dirichlet operator A D (see Theorem 8.3.4), as well as the direct sum decomposition is a boundary triplet for (T min ) * = T max such that Proof. Let f, g ∈ dom T max and decompose f and g in the form f = f D + f η and and since η is real, one also has Therefore, one obtains Let τ N be the extension of the Neumann trace to dom T max from Theorem 8.3.10.
Then it follows together with Corollary 8.3.11 and τ D f D = τ D g D = 0 that and Hence, and, since f D , g D ∈ ker τ D = ker τ D according to Theorem 8.3.9, one sees that for all f, g ∈ dom T max , that is, the abstract Green identity is satisfied. To verify the surjectivity of the mapping let ϕ, ψ ∈ L 2 (∂Ω) and consider ι −1 − ϕ ∈ H −1/2 (∂Ω) and −ι −1 + ψ ∈ H 1/2 (∂Ω). Observe that by (8.2.12) the Neumann trace operator τ N is a surjective mapping Next recall from Theorem 8.3.9 that the extended Dirichlet trace operator τ D maps dom T max onto H −1/2 (∂Ω) and that ker τ D = ker τ D = dom A D . Hence, it follows from the direct sum decomposition dom From the definition of Γ 0 and ker τ D = ker τ D = dom A D it is clear that dom A D = ker Γ 0 , and hence the self-adjoint extension corresponding to Γ 0 coincides with the Dirichlet operator A D , that is, the first identity in (8.4.2) holds. It remains to check the second identity in (8.4.2). For this let f = f D + f η ∈ ker Γ 1 , which means τ N f D = 0. Thus, f D ∈ dom T min by (8.2.15) and it follows that In fact, one has ker Γ 0 = ker Γ 0 and ker Γ 1 = ker Γ 1 , and hence with some 2 × 2 operator matrix W = (W ij ) 2 i,j=1 as in Theorem 2.5.1, see also Corollary 2.5.5. In the present situation it follows from Theorem 8.3.9 and (8.4.1) that the restriction ι − τ D : N η (T max ) → L 2 (∂Ω) is bijective and one concludes With the help of the extended Dirichlet-to-Neumann map in Corollary 8.3.12 one obtains a more explicit description of the domain of the self-adjoint operator A 1 in (8.4.2).
Proposition 8.4.3. Let Ω ⊂ R n be a bounded C 2 -domain, let A D be the self-adjoint Dirichlet realization of −Δ + V in L 2 (Ω), and fix η ∈ ρ(A D ) ∩ R. Moreover, let D(η) be the extended Dirichlet-to-Neumann map in Corollary 8.3.12. Then the self-adjoint extension A 1 of T min in (8.4.2) is defined on Proof. It is clear from Theorem 8.4.1 that Let τ N be the extension of the Neumann trace τ N to the maximal domain in Theorem 8.3.10. Then the boundary condition τ N f D = 0 can be rewritten as With the help of the extended Dirichlet-to-Neumann map If η ∈ R is chosen smaller than the lower bound m(A D ) of A D , then it follows from the second identity in (8.4.2), Lemma 5.4.1, and Definition 5.4.2 that the Kreȋn type extension S K,η = T min + N η (T max ) of T min and A 1 coincide. In the special case m(A D ) > 0 and η = 0 one has A 1 = S K,0 , which is the Kreȋn-von Neumann extension of T min ; cf. Definition 5.4.2.
In the next proposition the γ-field and the Weyl function corresponding to the boundary triplet {L 2 (∂Ω), Γ 0 , Γ 1 } in Theorem 8.4.1 are provided. Note that for f = f D + f η decomposed as in (8.4.1) one has as ker τ D = ker τ D = dom A D by Theorem 8.3.9. It is also clear from (8.4.1) that Γ 0 is a bijective mapping from N η (T max ) onto L 2 (∂Ω).

Proof. Since by definition γ(η) is the inverse of the restriction of Γ
, and hence by the definition of Γ 1 it follows that The assertion M (η)ϕ = 0 for all ϕ ∈ L 2 (∂Ω) is clear from the above.
The Weyl function M in Proposition 8.4.4 is closely connected with the Dirichlet-to-Neumann map D(λ) and its extension D(λ), λ ∈ ρ(A D ), in Definition 8.3.6 and Corollary 8.3.12. This connection will be made explicit in the next lemma. First, consider f = f D + f η ∈ dom T max as in (8.4.1). In the present situation one has Hence, making use of D(η) τ D f η = τ N f η (see Corollary 8.3.12) and the identity ker τ D = dom A D , it follows that  (8.4.11) where (8.4.7) was used in the last step for f = f λ . Since f λ D ∈ dom A D ⊂ H 2 (Ω), the regularization property (8.4.8) follows from (8.2.12). From (8.4.11) one also concludes that and since M (λ)Γ 0 f λ = Γ 1 f λ by the definition of the Weyl function, this shows (8.4.9). It remains to prove the second assertion (8.4.10). For this note that the restriction of ι −1 − : L 2 (∂Ω) → H −1/2 (∂Ω) to H 2 (∂Ω) is an isometric isomorphism from H 2 (∂Ω) onto H 3/2 (∂Ω) by Corollary 8.2.2. Furthermore, it follows from the definition that the extended Dirichlet-to-Neumann map D(λ) coincides with the Dirichlet-to-Neumann map D(λ) on H 3/2 (∂Ω). With these observations it is clear that (8.4.10) follows when restricting (8.4.9) to H 2 (∂Ω).
Remark 8.4.6. The boundary mappings in Theorem 8.4.1 and the corresponding γ-field and Weyl function depend on the choice of η ∈ ρ(A D ) ∩ R and the decom- Suppose now that the boundary mappings are defined with respect to some other η ∈ ρ(A D ) ∩ R and decompose f accordingly as f = f η D + f η . If Γ η 0 , Γ η 1 denote the boundary mappings in Theorem 8.4.1 with respect to η, and Γ η 0 , Γ η 1 denote the boundary mappings in Theorem 8.4.1 with respect to η , then one has Γ η (8.4.12) In fact, that Γ η 0 f = Γ η 0 f for f ∈ dom T max is clear from Theorem 8.4.1, and for the remaining identity in (8.4.12) it follows from Lemma 8.4.5 that Finally, note that the γ-fields and Weyl functions of the boundary triplets in Theorem 8.4.1 for different η and η transform accordingly; cf. Proposition 2.5.3.
Next some classes of extensions of T min and their spectral properties are briefly discussed. Let {L 2 (∂Ω), Γ 0 , Γ 1 } be the boundary triplet in Theorem 8.4.1 with corresponding γ-field γ and Weyl function M in Proposition 8.4.4. According to Corollary 2.1.4, the self-adjoint (maximal dissipative, maximal accumulative) extensions A Θ ⊂ T max of T min are in a one-to-one correspondence to the selfadjoint (maximal dissipative, maximal accumulative) relations Θ in L 2 (∂Ω) via If Θ is an operator in L 2 (∂Ω), then the domain of A Θ is given by (8.4.14) Let Θ be a self-adjoint relation in L 2 (∂Ω) and let A Θ be the corresponding self-adjoint realization of −Δ + V in L 2 (Ω). By Corollary 1.10.9, Θ can be represented in terms of bounded operators A, B ∈ B(L 2 (∂Ω)) satisfying the conditions A * B = B * A, AB * = BA * , and A * A + B * B = I = AA * + BB * such that In this case one has and for λ ∈ ρ(A Θ ) ∩ ρ(A D ) the Kreȋn formula for the corresponding resolvents Although Ω is a bounded C 2 -domain, it will turn out in Example 8.4.9 that the spectrum of A Θ is in general not discrete, and thus continuous spectrum may be present. It then follows from Theorem 2.6.2 and Theorem 2.6.5 that λ ∈ ρ(A D ) belongs to the continuous spectrum σ c (A Θ ) (essential spectrum σ ess (A Θ ) or discrete For a complete description of the spectrum of A Θ recall that the symmetric operator T min is simple according to Proposition 8.3.13 and make use of a transform of the boundary triplet {L 2 (∂Ω), Γ 0 , Γ 1 } as in Chapter 3.8. This reasoning implies that λ is an eigenvalue of A Θ if and only if λ is a pole of the function It is important to note in this context that the multiplicity of the eigenvalues of A Θ is not necessarily finite and that the dimension of the eigenspace ker (A Θ − λ) of an isolated eigenvalue λ of A Θ coincides with the dimension of the range of the residue of M Θ at λ. Furthermore, the continuous and absolutely continuous spectrum of A Θ can be characterized as in Section 3.8, e.g., one has In the special case that the self-adjoint relation Θ in L 2 (∂Ω) is a bounded operator the boundary condition reads as in (8.4.14) and according to Section 3.8 the spectral properties of the self-adjoint operator A Θ can also be described with the help of the function The general boundary conditions in (8.4.13) and (8.4.14) contain also typical classes of boundary conditions that are treated in spectral problems for partial differential operators, as, e.g., Neumann or Robin type boundary conditions. In the following example the standard Neumann boundary conditions are discussed. Note that the Neumann operator does not coincide with the Kreȋn type extension S K,η or the Kreȋn-von Neumann extension S K,0 of T min in Proposition 8.4.3.
The self-adjoint realization of −Δ + V in L 2 (Ω) corresponding to the selfadjoint operator Θ N in (8.4.16) is denoted by ∂Ω) and hence f ∈ H 2 (Ω) and τ D f = τ D f by (8.2.12) and Theorem 8.3.9. It then follows from (8.4.7) and (8.4.16) that the boundary condition Γ 1 f = Θ N Γ 0 f takes on the form Hence, it has been shown that dom A ΘN ⊂ H 2 (Ω) and that τ N f = 0 for all f ∈ dom A ΘN . Therefore, A ΘN ⊂ A N and since both operators are self-adjoint one concludes that A ΘN = A N .
Note also that by (8.4.16) and Lemma 8.4.5 one has (8.4.15) implies that the resolvents of A D and A N are related via where γ is the γ-field corresponding to the boundary triplet {L 2 (∂Ω), Γ 0 , Γ 1 } in Proposition 8.4.4.
In the next example it is shown that the (essential) spectrum of a self-adjoint realization A Θ of −Δ + V can be very general, depending on the properties of the parameter Θ. In particular, the self-adjoint realization A Θ may not be semibounded.

Semibounded Schrödinger operators
The semibounded self-adjoint realizations of −Δ + V , where V ∈ L ∞ (Ω) is real, and the corresponding densely defined closed semibounded forms in L 2 (Ω) are described in this section. For this purpose it is convenient to construct a boundary pair which is compatible with the boundary triplet in Theorem 8.4.1 and to apply the general results from Section 5.6. Under the additional assumption that V ≥ 0, the nonnegative realizations of −Δ + V and the corresponding nonnegative forms in L 2 (Ω) are discussed as a special case. In this situation the Kreȋn-von Neumann extension appears as the smallest nonnegative extension.
Let Ω ⊂ R n be a bounded C 2 -domain and let A D be the self-adjoint Dirichlet realization of −Δ + V . It is clear from Proposition 8.3.2 that A D coincides with the Friedrichs extension of the minimal operator T min in (8.3.2) and that A D is bounded from below with lower bound m( Furthermore, the resolvent of A D is compact since the domain Ω is bounded. Therefore, the following description of the semibounded self-adjoint extensions of T min is an immediate consequence of Proposition 5.5.6 and Proposition 5.5.8. Proposition 8.5.1. Let Ω ⊂ R n be a bounded C 2 -domain, let {L 2 (∂Ω), Γ 0 , Γ 1 } be the boundary triplet for (T min ) * = T max from Theorem 8.4.1, and let be a self-adjoint extension of T min in L 2 (Ω) corresponding to a self-adjoint relation Θ in L 2 (∂Ω) as in (8.4.13). Then Recall also from Section 8.3 that the densely defined closed semibounded form t AD corresponding to A D is defined on H 1 0 (Ω). Now fix some η < m(A D ), use the direct sum decomposition from (8.4.1) and Proposition 8.3.2, and consider the corresponding boundary triplet {L 2 (∂Ω), Γ 0 , Γ 1 } for (T min ) * = T max in Theorem 8.4.1 given by (8.5.1). It is clear that A 0 = A D coincides with the Friedrichs extension of T min and A 1 = T min + N η (T max ) coincides with the Kreȋn type extension S K,η of T min ; cf. Definition 5.4.2. In order to define a boundary pair for T min corresponding to A 1 = S K,η , consider the densely defined closed semibounded form t SK,η associated with S K,η and recall from Corollary 5.4.16 the direct sum decomposition of dom t SK,η . Comparing (8.5.1) and (8.5.3) one sees that dom T max ⊂ dom t SK,η and that the domain of the Dirichlet operator A D in (8.5.1) is replaced by the corresponding form domain in (8.5.3). The functions f ∈ dom t SK,η will be written in the form f = f η + f F , where f η ∈ N η (T max ) and f F ∈ dom t AD = H 1 0 (Ω). Now define the mapping (8.5.4) It will be shown next that {L 2 (∂Ω), Λ} is a boundary pair that is compatible with the boundary triplet {L 2 (∂Ω), Γ 0 , Γ 1 } in the sense of Definition 5.6.4; although the main part of the proof of Lemma 8.5.2 is similar to Example 5.6.9, the details are provided.
for all f ∈ dom T max and g ∈ dom t SK,η .
Proof. According to Lemma 5.6.5 (ii), it suffices to show that for some a < η the mapping Λ in (8.5.4) is bounded from the Hilbert space to L 2 (∂Ω) and that Λ extends the mapping Γ 0 in (8.5.2). In the present situation it is clear that the compatibility condition A 1 = S K,η is satisfied.
In order to show that Λ is bounded fix some a < η, recall first from (5.1.7) that the Hilbert space norm on H tS K,η −a is given by It follows from Theorem 8.3.9 that the restriction ι − τ D : N η (T max ) → L 2 (∂Ω) is bounded and hence for f = f η + f F ∈ dom t SK,η , decomposed according to (8.5. 3) in f η ∈ N η (T max ) and f F ∈ dom t AD , one has the estimate from Corollary 5.4.15 will be used. To this end, define Then one has f η = I + (η − a)(A D − η) −1 f a and Proposition 1.4.6 leads to the estimate Furthermore, it follows from (5.1.9) and the orthogonal sum decomposition (8.5.7) that From this estimate, (8.5.6), and (8.5.8) one concludes that Λ : From the definition of Λ in (8.5.4) and the decompositions (8.5.1) and (8.5.3) it is clear that Λ is an extension of the mapping Γ 0 in (8.5.2). Moreover, by construction, the condition A 1 = S K,η is satisfied. Therefore, Lemma 5.6.5 (ii) shows that {L 2 (∂Ω), Λ} is a boundary pair for T min corresponding to S K,η which is compatible with the boundary triplet {L 2 (∂Ω), Γ 0 , Γ 1 }. The identity (8.5.5) follows from Corollary 5.6.7.
The next theorem is a variant of Theorem 5.6.13 in the present situation.
Theorem 8.5.3. Let Ω ⊂ R n be a bounded C 2 -domain, let A D be the self-adjoint Dirichlet realization of −Δ + V with lower bound m(A D ), and fix η < m(A D ). Let {L 2 (∂Ω), Γ 0 , Γ 1 } be the boundary triplet for (T min ) * = T max from Theorem 8.4.1 and let {L 2 (∂Ω), Λ} be the compatible boundary pair in Lemma 8.5.2. Furthermore, let Θ be a semibounded self-adjoint relation in L 2 (∂Ω) and let A Θ be the corresponding semibounded self-adjoint extension of T min in Proposition 8.5.1. Then the closed semibounded form ω Θ in L 2 (∂Ω) corresponding to Θ and the densely defined closed semibounded form t AΘ corresponding to A Θ are related by (8.5.9) For completeness, the form t AΘ in Theorem 8.5.3 will be made more explicit using Corollary 5.6.14. First note that, by the definition of the boundary map Λ in (8.5.4) and the decomposition (8.5.3), one can rewrite (8.5.9) as (8.5.10) If m(Θ) denotes the lower bound of the semibounded self-adjoint relation Θ and μ ≤ m(Θ) is fixed, then the closed semibounded form t AΘ in (8.5.9)-(8.5.10) corresponding to A Θ is given by as usual, here Θ op denotes the semibounded self-adjoint operator part of Θ acting in L 2 (∂Ω) op = dom Θ. In the special case where Θ op ∈ B(L 2 (∂Ω) op ) one has and if Θ ∈ B(L 2 (∂Ω)), then Recall also from Corollary 5.4.15 that the form t SK,η can be expressed in terms of the form t AD and the resolvent of A D .
Finally, the special case V ≥ 0 will be briefly considered. In this situation the minimal operator T min and the Dirichlet operator A D are both uniformly positive and hence in the above construction of a boundary triplet and corresponding boundary pair one may choose η = 0. More precisely, Theorem 8.4.1 has the following form.
Corollary 8.5.4. Let Ω ⊂ R n be a bounded C 2 -domain, let A D be the self-adjoint Dirichlet realization of −Δ+V in L 2 (Ω) with V ≥ 0, and decompose f ∈ dom T max according to (8.4 is a boundary triplet for (T min ) * = T max such that coincide with the Friedrichs extension and the Kreȋn-von Neumann extension of T min , respectively.
It is clear from Proposition 8.4.4 that for all λ ∈ ρ(A D ) the γ-field and Weyl function corresponding to the boundary triplet {L 2 (∂Ω), Γ 0 , Γ 1 } in Corollary 8.5.4 have the form The next proposition is a variant of Proposition 8.5.1 for nonnegative extensions.
Proposition 8.5.5. Let Ω ⊂ R n be a bounded C 2 -domain, assume that V ≥ 0, and let {L 2 (∂Ω), Γ 0 , Γ 1 } be the boundary triplet for (T min ) * = T max from Corollary 8.5.4. Let be a self-adjoint extension of T min in L 2 (Ω) corresponding to a self-adjoint relation Θ in L 2 (∂Ω) as in (8.4.13). Then Proof. Note that the Weyl function M in (8.5.11) satisfies M (0) = 0 and that T min is uniformly positive. Therefore, if A Θ is a nonnegative self-adjoint extension of T min , then Proposition 5.5.6 with x = 0 shows that the self-adjoint relation Θ in L 2 (∂Ω) is nonnegative. Conversely, if Θ is a nonnegative self-adjoint relation in L 2 (∂Ω), then it follows from Corollary 5.5.15 and A 1 = S K,0 ≥ 0 that A Θ is a nonnegative self-adjoint extension of T min .
In the nonnegative case the boundary mapping Λ in (8.5.4) is given by (8.5.12) where one has the direct sum decomposition and according to Lemma 8.5.2 {L 2 (∂Ω), Λ} is a boundary pair that is compatible with the boundary triplet {L 2 (∂Ω), Γ 0 , Γ 1 } in Corollary 8.5.4. In the nonnegative case a description of the nonnegative extensions and their form domains is of special interest. In the present situation Corollary 5.6.18 reads as follows.
Corollary 8.5.6. Let Ω ⊂ R n be a bounded C 2 -domain, let A D be the self-adjoint Dirichlet realization of −Δ + V with V ≥ 0, let {L 2 (∂Ω), Γ 0 , Γ 1 } be the boundary triplet for (T min ) * = T max from Corollary 8.5.4, and let {L 2 (∂Ω), Λ} be the compatible boundary pair in (8.5.12). Then the formula establishes a one-to-one correspondence between all closed nonnegative forms t AΘ corresponding to nonnegative self-adjoint extension A Θ of T min in L 2 (Ω) and all closed nonnegative forms ω Θ corresponding to nonnegative self-adjoint relations Θ in L 2 (∂Ω).

Coupling of Schrödinger operators
The aim of this section is to interpret the natural self-adjoint Schrödinger operator (8.6.1) in L 2 (R n ) with a real potential V ∈ L ∞ (R n ) as a coupling of Schrödinger operators on a bounded C 2 -domain and its complement, that is, A is identified as a selfadjoint extension of the orthogonal sum of the minimal Schrödinger operators on the subdomains and its resolvent is expressed in a Kreȋn type resolvent formula. The present treatment is a multidimensional variant of the discussion in Section 6.5 and is based on the abstract coupling construction in Section 4.6.
Let Ω + ⊂ R n be a bounded C 2 -domain and let Ω − := R n \ Ω + be the corresponding exterior domain. Since C := ∂Ω − = ∂Ω + is C 2 -smooth in the sense of Definition 8.2.1, the term C 2 -domain will be used here for Ω − , although Ω − is unbounded. In the following the common boundary C is sometimes referred to as an interface, linking the two domains Ω + and Ω − . Note that one has the identification L 2 (R n ) = L 2 (Ω + ) ⊕ L 2 (Ω − ). (8.6.2)

Consider the Schrödinger operator
Since the Laplacian −Δ defined on H 2 (R n ) is unitarily equivalent in L 2 (R n ) via the Fourier transform to the maximal multiplication operator with the function x → |x| 2 , it is clear that −Δ, and hence A in (8.6.1), is self-adjoint in L 2 (R n ). Moreover, for f ∈ C ∞ 0 (R n ) integration by parts shows that where v − = essinf V . As C ∞ 0 (R n ) is dense in H 2 (R n ), this estimate extends to H 2 (R n ). Therefore, A is semibounded from below and v − is a lower bound.
The restriction of the real function V ∈ L ∞ (Ω) to Ω ± is denoted by V ± and the same ±-index notation will be used for the restriction f ± ∈ L 2 (Ω ± ) of an element f ∈ L 2 (R n ). The minimal and maximal operator associated with −Δ + V + in L 2 (Ω + ) will be denoted by T + min and T + max , respectively, and the selfadjoint Dirichlet realization in L 2 (Ω + ) will be denoted by A + D ; cf. Proposition 8.3.1, Proposition 8.3.2, and Theorem 8.3.4. For the minimal operator and the maximal operator on the unbounded C 2 -domain one can show in the same way as in the proof of Proposition 8.3.1 that (T − min ) * = T − max and T − min = (T − max ) * . Furthermore, since Ω − has a compact C 2 -smooth boundary, it follows by analogy to Theorem 8.3.4 that the self-adjoint Dirichlet realization A − D corresponding to the densely defined closed semibounded form via the first representation theorem (Theorem 5.1.18) is given by is a self-adjoint operator in L 2 (R n ) with Dirichlet boundary conditions on C. The goal of the following considerations is to identify the self-adjoint Schrödinger operator A in (8.6.1) as a self-adjoint extension of the orthogonal sum of the minimal operators T ± min and to compare A with the orthogonal sum A D in (8.6.3) using a Kreȋn type resolvent formula.
From now on it is assumed that η < essinf V is fixed, so that, in particular, Consider the boundary triplet {L 2 (C), Γ + 0 , Γ + 1 } for T + and hence the self-adjoint operator in (8.6.3) coincides with the self-adjoint extension of T min := T + min ⊕ T − min corresponding to the boundary condition ker Γ 0 . Note also that the corresponding γ-field γ and Weyl function M have the form . In Lemma 8.6.2 it will be shown that a certain relation Θ is self-adjoint in L 2 (C) ⊕ L 2 (C). This relation will turn out to be the boundary parameter that corresponds to the Schrödinger operator A in (8.6.1) via the boundary triplet (8.6.4). The following lemma on the sum of the Dirichlet-to-Neumann maps is preparatory.
Lemma 8.6.1. Let η < essinf V and let D ± (λ) : H 3/2 (C) → H 1/2 (C) be the Dirichlet-to-Neumann maps as in Definition 8.3.6 corresponding to −Δ + V ± . Then for all λ ∈ C \ [η, ∞) the operator Proof. First it will be shown that the operator in (8.6.8) is injective. Assume that (D + (λ) + D − (λ))ϕ = 0 for some ϕ ∈ H 3/2 (C) and some λ ∈ C \ [η, ∞). Then there exist f λ,± ∈ H 2 (Ω ± ) such that (8.6.9) and (8.6.10) In fact, for each h = (h + , h − ) ∈ dom A = H 2 (R n ) one also has τ + D h + = τ − D h − and τ + N h + = −τ − N h − (note that the different signs are due to the fact that the Neumann trace on each domain is taken with respect to the outward normal vector) and hence Lemma 8.6.1 will be used to prove the following lemma on the self-adjointness of a particular relation Θ in L 2 (C) ⊕ L 2 (C).
The following theorem is the main result in this section. It turns out that the self-adjoint operator corresponding to Θ in Lemma 8.6.2 coincides with the Schrödinger operator A.
Theorem 8.6.3. Let {L 2 (C) ⊕ L 2 (C), Γ 0 , Γ 1 } be the boundary triplet for T + max ⊕ T − max from (8.6.4) with γ-field γ, let Θ be the self-adjoint relation in Lemma 8.6.2, and let D ± (λ) be the Dirichlet-to-Neumann maps corresponding to −Δ + V ± . Then the self-adjoint operator A Θ corresponding to the parameter Θ coincides with the Schrödinger operator A in (8.6.1) and for all λ ∈ C \ [η, ∞) one has the resolvent formula where Λ(λ) ∈ B(L 2 (C) ⊕ L 2 (C)) has the form Proof. First it will be shown that the self-adjoint extension A Θ and the self-adjoint Schrödinger operator A in (8.6.1) coincide. Since both operators are self-adjoint, it suffices to verify the inclusion A ⊂ A Θ . For this, consider f ∈ dom A = H 2 (R n ) and note that f By the definition of the boundary mappings Γ 0 and Γ 1 in (8.6.5)-(8.6.6), one has and, as f ∈ H 2 (R n ), it follows that is contained in dom A Θ , and since both operators are self-adjoint, it follows that they coincide, that is, A = A Θ .
Finally, the boundary triplet in (8.6.4) is modified in the same way as in Proposition 4.6.4 to interpret the Schrödinger operator A as the self-adjoint extension corresponding to the boundary mapping Γ 0 . More precisely, the boundary triplets {L 2 (C), Γ + 0 , Γ + 1 } and {L 2 (C), Γ − 0 , Γ − 1 } lead to the boundary triplet It follows from Proposition 4.6.4 that the Schrödinger operator A = −Δ + V in (8.6.1) coincides with the self-adjoint extension defined on ker Γ 0 and that the Weyl function corresponding to the boundary triplet in (8.6.15) is given by where M ± (λ) = ι + ( D ± (η) − D ± (λ))ι −1 − is the Weyl function corresponding to the boundary triplet {L 2 (C), Γ ± 0 , Γ ± 1 }; cf. Proposition 8.4.4 and Lemma 8.4.5. In particular, the results in Section 3.5 and Section 3.6 can be used to describe the isolated and embedded eigenvalues, continuous, and absolutely continuous spectrum of A with the help of the limit properties of the Dirichlet-to-Neumann maps D ± . For this, however, one has to ensure that the underlying minimal operator T min = T + min ⊕ T − min is simple, which follows from Proposition 8.3.13 and [120, Proposition 2.2].

Bounded Lipschitz domains
In this last section Schrödinger operators −Δ+V with a real function V ∈ L ∞ (Ω) on bounded Lipschitz domains are briefly discussed. This situation is more general than the setting of bounded C 2 -domains treated in the previous sections. The main objective here is to highlight the differences to the C 2 -case and to indicate which methods have to be adapted in order to obtain results of similar nature as above.
The notions of a Lipschitz hypograph and a bounded Lipschitz domain are defined in the same way as C 2 -hypographs and bounded C 2 -domains in Section 8.2. More precisely, for a Lipschitz continuous function φ : R n−1 → R the domain is called a Lipschitz hypograph with boundary ∂Ω. The surface integral and surface measure on ∂Ω φ are defined in the same way as in (8.2.4), and this leads to the and equip H s (∂Ω φ ) with the corresponding scalar product (8.2.6). cf. [92,326]. For the present purposes it is particularly useful to note that the mappings = τ D f, τ N g H 1/2 (∂Ω)×H −1/2 (∂Ω) − τ N f, τ D g H −1/2 (∂Ω)×H 1/2 (∂Ω) hold for all f, g ∈ H 1 Δ (Ω). The minimal operator T min and maximal operator T max associated with −Δ + V on a bounded Lipschitz domain are defined in exactly the same way as in the beginning of Section 8.3. The assertions T * min = T max and T min = T * max in Proposition 8.3.1 remain valid in the present situation. Furthermore, the Dirichlet realization A D and Neumann realization A N of −Δ + V are defined as in Section 8.3, and their properties are the same as in Proposition 8.3.2 and Proposition 8.3.3. The first remarkable and substantial difference for Schrödinger operators on a bounded Lipschitz domain appears in connection with the regularity of the domains of A D and A N when comparing with Theorem 8.3.4. In the present case one has the following regularity result from [431,432], see also [92,323].  For a bounded Lipschitz domain and λ ∈ ρ(A D ) the Dirichlet-to-Neumann map is defined as (8.7. 3) where f λ ∈ H 3/2 Δ (Ω) is such that (−Δ + V )f λ = λf λ . This definition is the natural analog of Definition 8.3.6, taking into account the decomposiion in (8.7.3). As before, it follows that for λ ∈ ρ(A D )∩ρ(A N ) the Dirichlet-to-Neumann map (8.7.3) is a bijective operator.
For completeness the following a priori estimates are stated.
Theorem 8.7.5. Assume that Ω ⊂ R n is a bounded Lipschitz domain. Then the Dirichlet and Neumann trace operators in (8.7.2) admit unique extensions to continuous surjective operators where dom T max is equipped with the graph norm. Furthermore, ker τ D = ker τ D = dom A D and ker τ N = ker τ N = dom A N .
By analogy to Corollary 8.3.11, the second Green identity extends to elements f ∈ dom T max and g ∈ dom A D in the form (T max f, g) L 2 (Ω) − (f, T max g) L 2 (Ω) = τ D f, τ N g G 1 ×G1 , and for f ∈ dom T max and g ∈ dom A N the second Green identity reads It will also be used that for λ ∈ ρ(A D ) the Dirichlet-to-Neumann map in (8.7.3) admits an extension to a bounded operator (8.7.6) where f λ ∈ N λ (T max ).
Theorem 8.7.6. Let Ω ⊂ R n be a bounded Lipschitz domain and let A D be the selfadjoint Dirichlet realization of −Δ + V in L 2 (Ω) in Theorem 8.7.2. Fix a number η ∈ ρ(A D ) ∩ R and decompose f ∈ dom T max according to (8.4.1) in the form is a boundary triplet for (T min ) * = T max such that A 0 = A D and A 1 = T min + N η (T max ).
Moreover, the Weyl function M is given by As in the case of bounded C 2 -domains, the Weyl function can be expressed via the Dirichlet-to-Neumann map; here the extended mapping D(λ) in (8.7.6) is used. In the same way as in Lemma 8.4.5 one verifies the relation With the boundary triplet {L 2 (∂Ω), Γ 0 , Γ 1 } in Theorem 8.7.6 and the corresponding γ-field and Weyl function the self-adjoint realizations of −Δ + V on a bounded Lipschitz domain Ω ⊂ R n can be parametrized and the spectral properties can be described in a similar form as in Section 8.4. The discussion of the semibounded extensions and of the corresponding sesquilinear forms with the help of a compatible boundary pair is parallel to the considerations in Section 8.5 and is not provided here. Finally, the coupling technique of Schrödinger operators from Section 8.6 also extends under appropriate modifications to the general situation of Lipschitz domains.
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