All self-adjoint extensions of the magnetic Laplacian in nonsmooth domains and gauge transformations
aa r X i v : . [ m a t h - ph ] S e p All self-adjoint extensions of the magnetic Laplacian innonsmooth domains and gauge transformations
C´esar R. de Oliveira, Wagner Monteiro
Departamento de Matem´atica, UFSCar, S˜ao Carlos, SP, 13560-970 Brazil
September 25, 2020
Abstract
We use boundary triples to find a parametrization of all self-adjoint extensionsof the magnetic Schr¨odinger operator, in a quasi-convex domain Ω with compactboundary, and magnetic potentials with components in W ∞ (Ω). This gives also a newcharacterization of all self-adjoint extensions of the Laplacian in nonregular domains.Then we discuss gauge transformations for such self-adjoint extensions and generalizea characterization of the gauge equivalence of the Dirichlet magnetic operator for theDirichlet Laplacian; the relation to the Aharonov-Bohm effect, including irregularsolenoids, is also discussed. In particular, in case of (bounded) quasi-convex domainsit is shown that if some extension is unitarily equivalent (through the multiplicationby a smooth unit function) to a realization with zero magnetic potential, then thesame occurs for all self-adjoint realizations. Contents Dirichlet and Neumann traces over the maximal domain 13 H A min and gauge trans-formations 26 H A min in quasi-convex domains . . . . . . . . 275.3 Gauge equivalence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 Let the 1-form A = P nj =1 A j d x j be a magnetic potential in a subset of R n [7].The Schr¨odinger expression H A = ( − i ∇ + A ) = n X j =1 (cid:18) − i ∂∂x j + A j (cid:19) (in appropriate units) is the starting point of the description of the behavior of aquantum nonrelativistic particle in Ω ⊂ R n under the influence of the magnetic field B = d A ; in the two and three dimensional cases, A may be identified with a vectorfield and B = rot A (with just one component, i.e., a function, in two dimensions). To-gether with the Laplacian H = − ∆, which corresponds to a zero magnetic potential,its self-adjoint realizations are among the most prominent operators in Mathematicsand Physics. The self-adjointness is a requirement for the operator to describe anenergy observable in Quantum Mechanics.Usually it is not a trivial step to classify all self-adjoint extensions of a givensymmetric differential operator, especially in domains with boundary irregularities;even without mentioning that both deficiency indices are infinite in this situation.One of the main purpose of this work is to describe all self-adjoint extensions of H A in a quasi-convex [9] open set Ω ⊂ R n , n ≥
2, with compact boundary ∂ Ω, A avector field with components in W ∞ (Ω), and initial operator domain C ∞ (Ω) (seeTheorem 5.4); by now it is enough to mention that the class of quasi-convex domainscontains all convex domains and all domains of class C ,r , for r > /
2. Anotherpurpose is to apply this parametrization to study some properties of all such self-adjoint extensions under gauge transformations (Theorems 5.10 and 5.14). The firstgoal is achieved by following ideas of [9], where the authors deal with the Laplacian,but here such ideas are supplemented with the construction of boundary triples. Byrestricting ourselves to suitable bounded magnetic potentials, we were able to keepthe hypothesis of quasi-convexity, and additionally of considering unbounded Ω withcompact boundaries. articularly attractive is the case in which Ω is not simply connected and themagnetic field is zero in Ω, but A = 0. This opens the possibility for the Aharonov-Bohm effect [2, 17, 13], and our results give the first description of all of its self-adjointextensions as well, and for irregular solenoids (see [1, 6] for the case Ω = R \ { } ).In [13] there is an interesting characterization of the absence of this effect for theDirichlet extension, that is, when the Dirichlet extension is unitarily equivalent tothe case with A = 0 (i.e., the Dirichlet Laplacian); by using gauge transformations,we have then a version of this result in all self-adjoint extensions, but for boundedand connected (although irregular) regions. By Theorem 5.14, one concludes that ifone extension of H A is gauge unitarily equivalent to an extension of the Laplacian,then the same occurs for all self-adjoint realizations, and this is independent of thespectral type of each realization. This seems to be the first proof of such physicallyexpected phenomenon, and here for (bounded regions) and irregular quasi-convexdomains.The characterization of all self-adjoint extensions in the magnetic case, in quasi-convex domains, has some differences with respect to the Laplacian case originallydiscussed in [9]; besides the presence of the magnetic potential A in many estimates,the main differences are related to the Neumann trace map, which influences an inte-gration by parts formula, and the introduction of the space N / A ( ∂ Ω) (Definition 4.6)used in the extension of the modified Neumann trace to the domain of the maximaloperator. It is interesting to note that, differently from [9], we use boundary triplesso that our parametrization of all self-adjoint extensions is in terms of unitary op-erators on the space N ( ∂ Ω), and the set of such unitary operators is independentof the magnetic potential A (see Theorem 5.4). We think this is a more transparentconstruction, particularly we use (bounded) unitary operators on boundary spaces;moreover, we have a natural bijection among such extensions for different magneticpotentials (see Remark 5.5), and this discussion is not restricted to bounded Ω (al-though its boundary is supposed to be compact). Complementary to [9, 3], for thecase A = 0 we have got a new parametrization of all self-adjoint extensions of theLaplacian in (possibly unbounded) quasi-convex domains.In [12], there is, in particular, a characterization of all self-adjoint extensions ofminimal symmetric elliptic differential operators of even-order in L (Ω), for smooth Ω(see also [18] for a general parametrization that reduces to results in [12] ). For aninteresting discussion about the motivations for considering quasi-convex domains,the differences from the approach of [12] and its relations to the Laplacian, see theIntroduction of [9], which is our primary reference for the discussion of extensions.Here we just mention some points. For smooth Ω, the domains of the Dirichlet andNeumann Laplacians are subspaces of H (Ω), whereas for Lipschitz Ω the domainof the Dirichlet Laplacian is a subspace of H (Ω), and this can not be improved ingeneral (one needs an additional effort to get similar results for some quasi-convexdomains). For Lipschitz domains the range of the combined Dirichlet and Neumanntraces, defined on H (Ω), is not a Cartesian product of boundary Sobolev spaces (seeTheorem 4.1). By using the concept of almost boundary triples , in [3] the authorshave found a parametrization of the family of all self-adjoint extensions of the min-imal Laplacian in Lipschitz domains. Although more general than the quasi-convex omain case, the cost for the larger generality is a more abstract construction, andso more difficult to work with in applications; for instance, as already mentioned, thedomain of the Dirichlet and Neumann Laplace operators are not contained in H (Ω),and this regularity is fundamental in some explicit calculations in the quasi-convexcase.The concept of quasi-convexity is a balance that has permitted a characterizationof all self-adjoint extensions of the Laplacian, including the magnetic one, which isnot too abstract; hence we restrict ourselves to the quasi-convex case in this work.In Chapter 2 we recall some basics facts about Sobolev spaces in Lipschitz domainsand Dirichlet trace on Sobolev spaces; many notations are introduced. In Chapter 3we will introduce the maximal and minimal magnetic operators and show how theyare related to each other, and then state some integration by parts formulas relatedto H A , as well as some density results that will be important in the rest of this work.In Chapter 4 we will extend the magnetic Dirichlet and Neumann trace operators tothe domain of the maximal operator; the range of these operators are the importantspaces N / ( ∂ Ω) and N / A ( ∂ Ω), respectively. We also briefly review the conceptof quasi-convex domains and present one regularity result about the domain of theDirichlet extension; this is a slight generalization of a regularity result obtainedin [9]. In Chapter 5 we briefly review the concept of boundary triples and use it toobtain a parametrization of the family of all self-adjoint extensions of H A in a quasi-convex domain; then we use this parametrization to set some results about gaugeequivalence of self-adjoint extensions corresponding to the two operators H A , H B ,where A and B are two gauge equivalent magnetic potentials and have componentsin W ∞ (Ω). Applications to the Aharonov-Bohm setting also appear in this section. Acknowledgments:
CRdO thanks the partial support by CNPq (Brazilian agency,under contract 303503/2018-1), and WM was supported by CAPES (Brazilian agency).
In this section we recall some basics facts and notations about Sobolev spacesnecessary for this work, including the notions of Dirichlet and magnetic Neumanntrace operators. For more details, definitions and proofs see, for example, [15].
An open set Ω of R n , with n ≥
2, is said to be a Lipschitz domain if there exists anopen cover { O i } ≤ i ≤ k , of its boundary ∂ Ω, such that for i ∈ { , ..., k } , O i ∩ Ω is equalto the part of O i below the graph of a Lipschitz function ϕ i : R n − → R (considered,possibly, in a coordinate system obtained by a rigid motion). In a similar way wecan define a domain of class C ,r , the only difference is that the functions ϕ i aresupposed of class C ,r . iven an open set Ω of R n and s ∈ R , we denote by H s (Ω) the correspondingSobolev space [15]. For the same open set we can introduce the space H s (Ω) = (cid:8) u ∈ H s ( R n ) | supp ( u ) ⊆ Ω (cid:9) , s ∈ R , which is equipped with the norm induced by H s ( R n ). We also introduce the followingspaces, ˙ H s (Ω) = C ∞ (Ω) in H s (Ω)and ˜ H s (Ω) = (cid:8) u ∈ H s (Ω) | u = U | Ω with U ∈ H s (Ω) (cid:9) . It is possible to prove that, for any open set Ω, we have ( H s (Ω)) ∗ = H − s (Ω) with s ∈ R (1)and ( H s (Ω)) ∗ = H − s (Ω) with s ∈ R . (2)More precisely, we are identifying an element u ∈ H − s (Ω) with the anti-linear func-tional f u in H s (Ω) given by f u ( v ) = Z R n vU d n x, where u = U | Ω , U ∈ H − s ( R n ) and v ∈ H s (Ω). A similar identification is madein (2).It is also possible to prove that, if Ω is a Lipschitz domain with compact boundary,we have ˙ H s (Ω) = ˜ H s (Ω) for s > − , s ∈ R \ (cid:26)
12 + N (cid:27) , N ∈ N , and ( H s (Ω)) ∗ = H − s (Ω) with − < s < . For a Lipschitz domain with compact boundary it is possible to define the follow-ing Sobolev spaces over the boundary ∂ Ω, H s ( ∂ Ω) with − ≤ s ≤ H s ( ∂ Ω)) ∗ = H − s ( ∂ Ω) with − ≤ s ≤ . Recall that for a Lipschitz domain Ω it is possible to define, in almost all points ofthe boundary ∂ Ω, a unit normal vector field ν = ( ν , ..., ν n ) pointing outward to Ω. In this work the notation X ∗ will be always used to denote the adjoint space of X , that is , the spaceof continuous antilinear functionals of X. et, then, Ω be a Lipschitz domain with compact boundary; for each function g ofclass C in a neighborhood of ∂ Ω we consider ∂g∂τ j,k = ν i ( ∂ k g ) | ∂ Ω − ν k ( ∂ i g ) | ∂ Ω . Given a function f ∈ L ( ∂ Ω), consider the functional ∂f∂τ j,k given by ∂f∂τ j,k : g ∈ C ( ∂ Ω) → Z ∂ Ω d n − ω f ∂g∂τ k,j , where d n − ω is the surface measure on ∂ Ω (it is well defined by Rademacher Theo-rem). It is possible to show that for s ∈ [0 ,
1] the operator ∂∂τ j,k maps H s ( ∂ Ω) into H s − ( ∂ Ω) continuously. Furthermore, in [8] one finds the proofs of the followinglemmas:
Lemma 2.1.
Let Ω be a Lipschitz domain with compact boundary. Then for s ∈ [0 , we have H s ( ∂ Ω) = (cid:26) f ∈ L ( ∂ Ω , d n − ω ) (cid:12)(cid:12)(cid:12) ∂f∂τ j,k ∈ H s − ( ∂ Ω) , ≤ j, k ≤ n (cid:27) and k f k H s ( ∂ Ω) ≈k f k L ( ∂ Ω , d n − ω ) + n X j,k =1 k ∂f∂τ j,k k H s − ( ∂ Ω) . With those concepts in mind, we can introduce, ∇ tan : H ( ∂ Ω) → L ( ∂ Ω , d n − ω ) , ∇ tan = n X k =1 ν k ∂∂τ k, , ..., n X k =1 ν k ∂∂τ k,n ! where,L ( ∂ Ω , d n − ω ) = (cid:8) f = ( f , ..., f n ) | f i ∈ L ( ∂ Ω , d n − ω ) , i = 1 , ..., n,ν · f = 0 ω -a . s . in ∂ Ω (cid:9) . We will also make use of the following notation, the duality pairing between H s (Ω)and ( H s (Ω)) ∗ , h· , ·i s : H s (Ω) × H s (Ω) ∗ → C . Let W q ∞ (Ω) denote the usual Sobolev space with q ∈ N , that is, the set of therestrictions u = G | Ω of G ∈ W q ∞ ( R n ) equipped with the norm k u k W q ∞ (Ω) = inf G | Ω = u k G k W q ∞ ( R n ) . It is known that u ∈ W ∞ ( R n ) if, and only if, there is K >
0, that depends on u ,such that u is a bounded locally K -Lipschitz function, in particular it follows that,if u ∈ W ∞ (Ω), then u is the restriction of a bounded locally K -Lipschitz functiondefined on R n . Remark 2.2.
Note that if f ∈ W q ∞ (Ω), then the operator of multiplication by f , M f , maps H q (Ω) into itself, moreover, M f | H q (Ω) : H q (Ω) → H q (Ω) is bounded. Thisresult can be seen as particular case of Theorem 3.2 of [16]. .2 Dirichlet and magnetic Neumann trace operators Let Ω be a Lipschitz domain with compact boundary, then the Dirichlet traceoperator γ : C(Ω) → ∂ Ω , γ u = u | ∂ Ω has the following extensions γ D : H s (Ω) → H s − ( ∂ Ω) , < s < , (3) γ D : H (Ω) → H − ǫ ( ∂ Ω) , ∀ ǫ ∈ (0 , , (4)furthermore γ D : H s (Ω) → H s − ( ∂ Ω) , < s < , has a bounded right inverse, inparticular it is onto. In most of this work we will be concerned about the followingoperators related to the bounded vector field A with components in W ∞ (Ω), ∇ A := ∇ + iA and H A = ( − i ∇ + A ) . (5) Remark 2.3.
For u ∈ L (Ω) we can define H A u as a distribution acting on C ∞ (Ω)by h H A u, ϕ i = ( u, H A ϕ ) L (Ω) , for ϕ ∈ C ∞ (Ω). To see that this definition makessense, note that the following holds in the sense of the distributions H A u = −△ u − iA · ∇ u + ( | A | − i div A ) u , where the term A · ∇ u is well defined as a distribution; indeed, since A i ∈ W ∞ (Ω) ,i = 1 , ..., n and ∇ u ∈ ( H − (Ω)) n , it follows that A · ∇ u ∈ H − (Ω), so it is adistribution. Note that if u ∈ L (Ω) and H A u ∈ L (Ω), for ϕ ∈ C ∞ (Ω) we have( H A u, ϕ ) L (Ω) = h H A u, ϕ i = ( u, H A ϕ ) L (Ω) . We introduce the magnetic Neumann trace operator γ AN := ν · γ D ∇ A : H s +1 (Ω) → L ( ∂ Ω) , < s <
32 ; (6)of course, if A = 0, then this is the usual Neumann trace operator and simply denotedby γ N . In this section we describe the initial, maximal and minimal operators associatedwith the formal operator H A = ( − i ∇ + A ) ; then we set some integration by partsidentities related to these operators in Lipschitz domains. .1 Initial, minimal and maximal operators Let Ω be an open set of R n ; we define the initial operator H A byDom ( H A ) = C ∞ (Ω) and H A u := H A u, which is symmetric and densely defined, and so it is closable and its closure is de-noted by H A min , the so-called minimal (magnetic) operator . The maximal (magnetic)operator H A max is given byDom H A max = (cid:8) u ∈ L (Ω) | H A u ∈ L (Ω) (cid:9) , H A max := H A u. The proof of Lemma 3.1, for the Laplace operator, is due to Prof. J. Behrndt(private communication); however, the magnetic potentials introduce additional dif-ficulties.
Lemma 3.1.
Let Ω be a Lipschitz domain with compact boundary. Then ( H A ) ∗ = ( H A min ) ∗ = H A max . If Ω is also bounded, then Dom H A min = H (Ω) . Proof.
We first show that ( H A ) ∗ = H A max or, equivalently, ( H A ) ∗ = H A max . Pick f ∈ Dom ( H A ) ∗ , then f ∈ L (Ω) and there exists g ∈ L (Ω) such that,( g, u ) L (Ω) = ( f, H A u ) L (Ω) , ∀ u ∈ C ∞ (Ω) . So, by Remark 2.3, we have H A f = g ∈ L (Ω) in the sense of distributions; hence f ∈ Dom H A max and H A f = g = ( H A ) ∗ f , and so ( H A ) ∗ ⊂ H A max . On the other hand,for f ∈ Dom H A max , we have( H A f, u ) L (Ω) = ( f, H A u ) L (Ω) , ∀ u ∈ C ∞ (Ω) , and it follows immediately that H A max ⊂ ( H A ) ∗ ; hence ( H A ) ∗ = H A max .Now assume that Ω is bounded. Note that the norm k · k A = (cid:0) k · k (Ω) + k H A ( · ) k (Ω) (cid:1) / is equivalent to the norm of H (Ω) over H (Ω). Indeed, since A ∈ W ∞ (Ω), it isclear that k u k A ≤ C k u k H (Ω) for u ∈ H (Ω) with an appropriate C >
0. On theother hand, since H A = −△ + L , where L is the linear operator of order 1 given by Lu = − iA · ∇ u + ( | A | − i div A ) u , it follows that, for u ∈ H (Ω), k△ ( u ) k L (Ω) ≤ k H A ( u ) k L (Ω) + k Lu k L (Ω) = k H A ( u ) k L (Ω) + C k u k L (Ω) + C k∇ u k L (Ω) . (7) ow, for u ∈ H (Ω), we have k∇ u k L (Ω) = [( −△ u, u ) L (Ω) ] / ≤ k△ u k / (Ω) k u k / (Ω) ≤ ǫ k△ u k L (Ω) + 12 ǫ k u k L (Ω) , (8)for all ǫ >
0. So, by inequalities (7) and (8), we have k△ u k L (Ω) ≤ − C ǫ / k H A ( u ) k L (Ω) + C + C / (2 ǫ )1 − C ǫ / k u k L (Ω) . (9)By Poincar´e’s Inequality, the norm of H (Ω) is equivalent, in H (Ω), to (cid:16) k · k (Ω) + X | α | =2 k ∂ α ( · ) k (Ω) (cid:17) / ; (10)for all f ∈ H (Ω), we have, X | α | =2 k ∂ α f k (Ω) = X | α | =2 ( ∂ α f, ∂ α f ) L (Ω) = X i,j =1 ( ∂ i ∂ j f, ∂ i ∂ j f ) L (Ω) = X i,j =1 ( ∂ i f, ∂ j f ) L (Ω) = k△ f k (Ω) , where in the last equality we have performed a integration by parts, so, we get k u k H (Ω) ≈ ( k u k (Ω) + k△ u k (Ω) ) / (11)for all u ∈ H (Ω). Therefore, by (9) and (11) we obtain k u k H (Ω) ≤ C ′ k u k A for all u ∈ H (Ω), and so k · k A ≈ k · k H (Ω) in H (Ω). From these facts, we obtainDom H A min = Dom H A = Dom H A k·k A = Dom H A k·k H = C ∞ (Ω) k·k H = H (Ω) . Now we discuss a first version of an integration by parts formula associated withthe operator H A . Lemma 3.2.
Let Ω be a Lipschitz domain with compact boundary, u ∈ H (Ω) and v ∈ H (Ω) ; if Φ A ( u, v ) = ( ∇ A u, ∇ A v ) L (Ω) n , then Φ A ( u, v ) = ( H A u, v ) L (Ω) + ( γ A N u, γ D v ) L ( ∂ Ω) . (12) roof. It is enough to consider u, v ∈ C ∞ (Ω) since the general case follows by adensity argument. Note that, H A u = −△ u − iA · ∇ u + ( | A | − i div A ) u ;on the other hand,Φ A ( u, v ) = Z Ω d n x (cid:0) ∇ u · ∇ v + i ∇ u · Av − iA · u ∇ v + | A | uv (cid:1) = ( ∇ u, ∇ v ) L (Ω) + i Z Ω d n x ( ∇ u · Av − A · u ∇ v ) + Z Ω d n x | A | uv = − ( △ u, v ) L (Ω) + ( γ N u, γ D v ) L ( ∂ Ω) + i Z Ω d n x ( ∇ u · Av − A · u ∇ v ) + Z Ω d n x | A | uv = − ( △ u, v ) L (Ω) + ( γ N u, γ D v ) L ( ∂ Ω) + 2 i Z Ω d n x v ∇ u · A + i Z Ω d n x vu div A − i Z ∂ Ω d n − ω uvA · ν + Z Ω d n x | A | uv = ( H A u, v ) L (Ω) + ( γ A N u, γ D v ) L ( ∂ Ω) , where in the fourth equality we have used that uA · ∇ v = − v ∇ u · A − v u div A + div( v uA )and the Theorem 3.34 of [15], which is a version of the divergence theorem for Lips-chitz Domains.The next lemma will be important to obtain some generalization of this integra-tion by parts formula. It is a particular case of Theorem 6.9 in [16]. A result in thisdirection appears in [11], where the author considers the case s = 1, H A = △ (i.e., A = 0) and Ω bounded; in [5], using an approach similar to [11], the authors havegeneralized to the case s < Lemma 3.3.
Let Ω ⊂ R n be a Lipschitz domain with compact boundary. Then,for ≤ s < , C ∞ (Ω) is dense in H s ( A, Ω) = (cid:8) u ∈ H s (Ω) | H A u ∈ L (Ω) (cid:9) , whenequipped with the norm k · k A,s = k · k H s (Ω) + k H A ( · ) k L (Ω) .Proof. This result is a consequence of the fact that A ∈ (W ∞ (Ω)) n and a directapplication of Theorem 6.9 of [16].By Lemma 3.3 we can further extend the operator γ AN defined in (6) in the fol-lowing way: ˜ γ AN : H ( A, Ω) → H − ( ∂ Ω) , where, for u ∈ H ( A, Ω), we have that ˜ γ AN u ∈ H − ( ∂ Ω) is defined as h g, ˜ γ AN u i = Φ A ( u, G ) − h l ( H A u ) , G i , g ∈ H ( ∂ Ω) , (13) here G ∈ H (Ω) is such that γ D G = g and k G k H (Ω) ≤ c k g k H ( ∂ Ω) and l : L (Ω) → H − (Ω) is the natural inclusion. To see that this definition makes sense, it is enoughto show that it does not depend on the particular choice of G that satisfies the abovecondition. By linearity, this is equivalent to: if G ∈ H (Ω) is such that γ D G = 0,then Φ A ( u, G ) − h l ( H A u ) , G i = 0; however this follows from the fact that this holdstrue if u ∈ C ∞ (Ω) since C ∞ (Ω) is dense in H ( A, Ω).
In what follows we introduce the Dirichlet H A D and the Neumann H A N magneticself-adjoint realizations, two operators that will play a fundamental role ahead. Lemma 3.4.
Let Ω be a Lipschitz domain with compact boundary. Then there existreal numbers c > and C > such that, for all u ∈ H (Ω) , | Φ A ( u, u ) | ≥ c k u k H (Ω) − C k u k (Ω) . (14) Proof.
We haveΦ A ( u, u ) = k∇ u k (Ω) + 2 Z Ω d n − x Im uA · ∇ u + Z Ω d n − x | A | | u | ≥ k∇ u k (Ω) + Z Ω d n − x | A | | u | − Z Ω d n − x | Im uA · ∇ u |≥ (1 − ǫ ) k∇ u k (Ω) + (1 − ǫ ) Z Ω d n − x | A | | u | ≥ (1 − ǫ ) k∇ u k (Ω) − | (1 − ǫ ) || | A | || L ∞ (Ω) Z Ω d n − x | u | ;in the second inequality we have used that | Im u A · ∇ u | ≤ (cid:16) ǫ | u A | + ǫ |∇ u | (cid:17) , and from this the statement the lemma follows.Consider the operators H A D , H A N given byDom H A D = n u ∈ H (Ω) | H A u ∈ L (Ω) , γ D u = 0 in H ( ∂ Ω) o , H A D u := H A u andDom H A N = n u ∈ H (Ω) | H A u ∈ L (Ω) , ˜ γ A N u = 0 in H − ( ∂ Ω) o , H A N u := H A u . Proposition 3.5. If Ω is a Lipschitz domain with compact boundary, then the op-erators H A D and H A N are self-adjoint. roof. Consider first the operator H A D . Let Φ A, D be the following sesquilinear form,Φ A, D ( u, v ) = Φ A ( u, v ) , Dom Φ A, D = H (Ω);by (14) one can conclude that Φ A, D is closed and so the operator ˜ H A D defined byDom ˜ H A D = (cid:8) u ∈ H (Ω) | ∃ w u ∈ L (Ω) so that Φ A, D ( v, u ) = ( v, w u ) L (Ω) , ∀ v ∈ H (Ω) (cid:9) , ˜ H A D u := w u , is self-adjoint. To conclude the proof we show that H A D = ˜ H A D .Let v ∈ Dom ˜ H A D , then since C ∞ (Ω) ⊂ H (Ω) we have Z Ω d n x u w v = Φ A, D ( u, v ) = Z Ω d n x H A u v, ∀ u ∈ C ∞ (Ω) , where the last equality follows by (12). Thus, w v = H A v in the sense of distributions,in particular H A v ∈ L (Ω), and so ˜ H A D ⊂ H A D .Now, let v ∈ Dom H A D ; by taking w v := H A v ∈ L (Ω) and using (13) we have Z Ω d n x u w v = Z Ω d n x u H A v = Φ A, D ( u, v )for all u ∈ H (Ω), and so H A D ⊂ ˜ H A D , which concludes the proof that H A D is self-adjoint.Now we address H A N . Let Φ A, N be the following sesquilinear formΦ A, N ( u, v ) = Φ A ( u, v ) , Dom Φ A, N = H (Ω) . By using equation (14), and the fact that Φ A, N is nonnegative (so bounded frombelow), we conclude that Φ A, N is closed and so ˜ H A N , given byDom ˜ H A N = (cid:8) u ∈ H (Ω) | ∃ w u ∈ L (Ω) so that Φ A, N ( v, u ) = ( v, w u ) L (Ω) , ∀ v ∈ H (Ω) (cid:9) , ˜ H A N u = w u , is self-adjoint. To conclude we show that H A N = ˜ H A N .Let v ∈ Dom ˜ H A N , then since C ∞ (Ω) ⊂ H (Ω), we have Z Ω d n x u w v = Φ A, N ( u, v ) = Z Ω d n x H A u v for all u ∈ C ∞ (Ω), where the last equality follows by (12). Thus, we have w v = H A v in the sense of distributions, in particular H A v ∈ L (Ω). Furthermore, by (13) wehave, for all u ∈ H (Ω),Φ A, N ( u, v ) = ( u, H A v ) L (Ω) + h ˜ γ D u, γ N v i = ( u, w v ) L (Ω) + h γ D u, ˜ γ N v i = Φ A, N ( u, v ) + h ˜ γ D u, γ N v i and so ˜ γ N v = 0 since γ D : H (Ω) → H (Ω) is onto. Hence ˜ H A N ⊂ H A N .On the other hand, if u ∈ Dom H A N then, analogously to the case of the opera-tor H A D , one can show that u ∈ Dom ˜ H A N , and so H A N = ˜ H A N , that is, the operator H A N is self-adjoint. orollary 3.6. Let Ω be a bounded Lipschitz domain, then H A D is an operator withdiscrete spectrum.Proof. Indeed, by the Theorem of Lax-Milgram and equation (14), we can provethat the problem below has a unique solution u ∈ H (Ω) and we have k u k H (Ω) ≤ C ′ k f k L (Ω) for some C ′ > H A + C ) u = f, f ∈ L (Ω) ,γ D u = 0 , where C is the constant in the inequality (14). So, ( H A D + C ) − : L (Ω) → H (Ω) iscontinuous and since the inclusion H (Ω) ֒ → L (Ω) is compact if Ω is bounded, theoperator ( H A D + C ) − : L (Ω) → L (Ω) is compact, thus, H A D is discrete. We review some facts on traces discussed in [9] and present some generalizationsto the situation with magnetic potential. At the end we recall the concept of quasi-convex domain and we will introduce the concept of magnetic Neumann trace; thiswill be an important step for the construction of boundary triples for the maximalmagnetic operator.
Theorem 4.1.
Let Ω be a Lipschitz domain with compact boundary, and denote by ν the unit vector field normal to its boundary ∂ Ω . Let F := (cid:8) ( g , g ) ∈ H ( ∂ Ω) ˙+L ( ∂ Ω , d n − ω ) | ∇ tan g + g ν ∈ H ( ∂ Ω) n (cid:9) be equipped with the norm k (( g , g ) k ∂ Ω = k g k H ( ∂ Ω) + k g k L ( ∂ Ω , d n − ω ) + k∇ tan g + g ν k H ( ∂ Ω) n . Then, the operator γ : H (Ω) → F , γ u := ( γ D u, γ N u ) , is well defined, linear, bounded and has right inverse that is bounded. Furthermore,the kernel of γ is H (Ω) .Proof. This is an easy adaptation of the proof for bounded domains presented in [9].Fix an open ball B r of radius r > ∂ Ω ⊂ B r/ and a function ϕ ∈ C ∞ ( R n )such that ϕ ( x ) = 1 , for all x ∈ B r/ , and ϕ ( x ) = 0 , for all x ∈ R n \ B r/ . Denoteby γ L , L = Ω or Ω ∩ B r , the application γ with domain H (Ω) or H (Ω ∩ B r ),respectively. Note that, for all u ∈ C ∞ (Ω) one has γ Ω2 ( u ) = γ Ω ∩ B r ( ϕu ). Hence k γ Ω2 ( u ) k ∂ Ω = k γ Ω ∩ B r ( ϕu ) k ∂ (Ω ∩ B r ) ≤ C k ϕu k H (Ω ∩ B r ) ≤ C ′ k u k H (Ω) . ince C ∞ (Ω) is dense in H (Ω), from this inequality it follows that γ , with domainC ∞ (Ω), can be extended continuously, and in a unique way, to an application from H (Ω) to F . To see that this extension has a bounded right inverse, it is enough tonote that if ζ is an inverse of γ Ω ∩ B r , then E ◦ ζ is the required inverse of γ Ω2 , where E is a continuous extension operator from H (Ω ∩ B r ) to H (Ω). The last statementof the theorem is an easy consequence of the above construction and the boundedcase discussed in [9]. Definition 4.2 ([9]) . Let Ω be a Lipschitz domain with compact boundary. Thespace N ( ∂ Ω) is defined by N ( ∂ Ω) := n g ∈ L ( ∂ Ω , d n − ω ) | gν i ∈ H ( ∂ Ω) , ≤ i ≤ n o and equipped with the norm k g k N ( ∂ Ω) = (cid:0) n X i =1 k gν i k H ( ∂ Ω) (cid:1) / . (15)The norm (15) is clearly obtained from the inner product h u, v i N ( ∂ Ω) = n X i =1 h ν i u, ν i v i H ( ∂ Ω) , (16)where h u, v i H ( ∂ Ω) is an inner product in the Hilbert space H ( ∂ Ω). This spaceis a reflexive Banach space that can be continuously embedded in L ( ∂ Ω , d n − ω ).Moreover, if Ω is bounded and ∂ Ω ∈ C ,r with r > /
2, we have N ( ∂ Ω) = H ( ∂ Ω)with equivalent norms; see Lemma 6.2 of [9].The following result is a direct consequence of Lemma 6.3 of [9].
Lemma 4.3.
Let Ω be a Lipschitz domain with compact boundary, then γ A N : H (Ω) ∩ H (Ω) → N ( ∂ Ω) is well defined, linear, bounded, onto and has a bounded right inverse, moreover itskernel is H (Ω) . The next result extends the definition of γ D to the domain of H A max . Theorem 4.4.
Let Ω be a Lipschitz domain with compact boundary. Then thereexists a unique extension ˆ γ D of γ D , ˆ γ D : Dom H A max = (cid:8) u ∈ L (Ω) | H A u ∈ L (Ω) (cid:9) → ( N ( ∂ Ω)) ∗ , (17) that is compatible with (3) in the following sense: for / ≥ s > / and for all u ∈ H s (Ω) with H A u ∈ L (Ω) , one has ˆ γ D u = γ D u . Furthermore, the range of ˆ γ D isdense in ( N ( ∂ Ω)) ∗ and the following integration by parts formula holds, h γ A N w, ˆ γ D u i N ( ∂ Ω) = − (cid:0) ( H A w, u ) L (Ω) − ( w, H A u ) L (Ω) (cid:1) , (18) here w ∈ H (Ω) ∩ H (Ω) and u ∈ L (Ω) with H A u ∈ L (Ω) , and h· , ·i N ( ∂ Ω) : N ( ∂ Ω) × ( N ( ∂ Ω)) ∗ → C represents the pairing between a vector and a linear functional on N ( ∂ Ω) .Proof. Take u ∈ Dom H A max , and define ˆ γ D u ∈ ( N ( ∂ Ω)) ∗ in the following way: take g ∈ N ( ∂ Ω); by Lemma 4.3 there exists w ∈ H (Ω) ∩ H (Ω) such that γ A N ( w ) = g and k w k H (Ω) ≤ C k g k N ( ∂ Ω) . Define, then, ˆ γ D u ∈ ( N ( ∂ Ω)) ∗ through h g, ˆ γ D u i N ( ∂ Ω) := − (cid:0) ( H A w, u ) L (Ω) − ( w, H A u ) L (Ω) (cid:1) . To see that ˆ γ D u is well defined by this relation, we need to show that theabove definition does not depend on the particular choice of w satisfying the aboveconditions, which is equivalent to: if w ∈ H (Ω) ∩ H (Ω) satisfies γ A N ( w ) = 0,then ( H A w, u ) L (Ω) = ( w, H A u ) L (Ω) . Note that in this case, by the Lemma 4.3, w ∈ ker γ A N = H (Ω). If w ∈ C ∞ (Ω) and u ∈ C ∞ (Ω), this equality holds; the gen-eral case follows from this since, C ∞ (Ω) is dense in H (Ω) and C ∞ (Ω) is dense inDom H A max . It is clear that such ˆ γ D is continuous and does satisfy the required inte-gration by parts formula. The uniqueness also follows from the fact that C ∞ (Ω) isdense in Dom H A max .Now we show that this extension is compatible with (3). Pick u ∈ H s (Ω) with H A u ∈ L (Ω), 3 / ≥ s > /
2, and consider u i ∈ C ∞ (Ω) such that u i → u in H s ( A, Ω). Pick w ∈ H (Ω) ∩ H (Ω) and w j ∈ C ∞ (Ω) such that w j → w in H (Ω).Fix then i and consider the following vector field, G j = u i ∇ A w j ∈ H (Ω) , j ∈ N ;by an easy calculation we havediv G j = ∇ A u i ∇ A w j − u i H A w j ,ν · γ D G j = γ D u i γ A N w j . Thus,( u i , H A w ) L (Ω) = lim j →∞ ( u i , H A w j ) L (Ω) = lim j →∞ n(cid:16) − Z Ω d n x div G j (cid:17) + Φ A ( u i , w j ) o = lim j →∞ (cid:16) − Z ∂ Ω d n − ω γ D u i γ A N w j + Φ A ( u i , w j ) (cid:17) = lim j →∞ (cid:16) ( H A u i , w j ) L (Ω) + ( γ A N u i , γ D w j ) L (Ω) − Z ∂ Ω d n − ω γ D u i γ A N w j (cid:17) = ( H A u i , w ) L (Ω) − Z ∂ Ω d n − ω γ D u i γ A N w here in the last equality we have used that γ D w j → γ D w = 0. Hence( u i , H A w ) L (Ω) = ( H A u i , w ) L (Ω) − Z ∂ Ω d n − ω γ D u i γ A N w . By taking i → ∞ in the last equation we obtain Z ∂ Ω d n − ω γ D u γ A N w = (cid:16) ( H A u, w ) L (Ω) − ( u, H A w ) L (Ω) (cid:17) = h γ A N w, ˆ γ D u i N ( ∂ Ω) , and since γ A N : H (Ω) ∩ H (Ω) → N ( ∂ Ω) is onto, the above equation shows that γ D u and ˆ γ D u coincide as antilinear functionals in N ( ∂ Ω), in other words γ D u = ˆ γ D u .Finally, we only need to show that ˆ γ D has a dense range; to see this we showthat (cid:8) u | ∂ Ω | u ∈ C ∞ (Ω) (cid:9) is dense in ( N ( ∂ Ω)) ∗ . Take then Ψ in (( N ( ∂ Ω)) ∗ ) ∗ = N ( ∂ Ω) ֒ → L ( ∂ Ω) that is zero over (cid:8) u | ∂ Ω | u ∈ C ∞ (Ω) (cid:9) ; to conclude it is enoughto show that Ψ is zero. By Lemma 4.3 there exists w ∈ H (Ω) ∩ H (Ω) such that γ A N w = Ψ, and so h γ A N w, ˆ γ D u i N ( ∂ Ω) = 0 , ∀ u ∈ C ∞ (Ω) . By using this equation with (18) we obtain( H A w, u ) L (Ω) = ( w, H A u ) L (Ω) , ∀ u ∈ C ∞ (Ω) ;on the other hand, by (12) we have( H A w, u ) L (Ω) = ( w, H A u ) L (Ω) − ( γ A N w, γ D u ) L ( ∂ Ω) . By these equations(Ψ , γ D u ) L ( ∂ Ω) = ( γ A N w, γ D u ) L ( ∂ Ω) = 0 , ∀ u ∈ C ∞ (Ω) , and since γ D C ∞ (Ω) is dense in L ( ∂ Ω) (this follows by Lemma 3.1 of [9] and Lemma3.3), it then follows that Ψ = 0, and this proves the statement.As a simple consequence of this result we recover Corollary 6.5 of [9].
Corollary 4.5.
Let Ω be a Lipschitz domain with compact boundary, then each ofthe following inclusions is continuous and has dense range. More over, the pairingbetween ( N / ( ∂ Ω)) ∗ and N / ( ∂ Ω) is compatible with the inner product of L (Ω) . N / ( ∂ Ω) ֒ → L ( ∂ Ω) ֒ → ( N / ( ∂ Ω)) ∗ . Proof.
We will just sketch the proof. Denote the first inclusion by I ; since N / ( ∂ Ω)is a reflexive Banach space the second inclusion is nothing else than I ∗ , the dual of I .It is a fact about reflexive Banach spaces that an application I is continuous if, andonly if, I ∗ is continuous, and further, I is injective and has dense range if, and onlyif, the same holds for I ∗ . With this in mind, it is enough to prove the statement ofthe corollary for one inclusion. The fact that I is continuous follows directly fromthe definition of N / ( ∂ Ω). The compatibility of the pairing and the inner product ollows from the fact that the second inclusion is the dual of the first one. To seethat the second inclusion has dense range, note that, by Lemma 3.3, C ∞ (Ω) is densein Dom H A max , thus by the compatibility and density results of Theorem 4.4, the set L = γ D (C ∞ (Ω)) = ˆ γ D (C ∞ (Ω)) is dense in ( N / ( ∂ Ω)) ∗ ; since L ⊂ L ( ∂ Ω), thestatement follows.In what follows we introduce the space N / A ( ∂ Ω), which is a generalization ofthe space N ( ∂ Ω) introduced in Section 6 of [9]. In Section 5 we will use the space N / ( ∂ Ω) to construct a boundary triple for H A max rather the space N / A ( ∂ Ω), how-ever a similar construction can be done using the space N / A ( ∂ Ω); see the Remark 5.7.
Definition 4.6.
Let Ω be a Lipschitz domain with compact boundary. Define thespace N / A ( ∂ Ω) by N / A ( ∂ Ω) := n g ∈ H ( ∂ Ω) | ∇ A tan g ∈ H ( ∂ Ω) o , where ∇ A tan g := ( ∇ tan − i ( ν · A ) ν ) g (above, we have made the abuse of notation that consists of denoting γ D A also by A ),and equip N / A ( ∂ Ω) with the norm k · k N / A ( ∂ Ω) = k · k H ( ∂ Ω) + k∇ A tan · k ( H ( ∂ Ω)) n . Lemma 4.7.
Let Ω be a Lipschitz domain with compact boundary, then N / A ( ∂ Ω) is a reflexive Banach space that can be continuously embedded in H ( ∂ Ω) .Proof. It is evident that the inclusion of N / A ( ∂ Ω) in H ( ∂ Ω) is continuous. To seethat this is a Banach space, take a Cauchy sequence { g n } n ∈ N in N / A ( ∂ Ω). Fromthe last statement it follows that { g n } n ∈ N is Cauchy in H ( ∂ Ω), and so it convergesin this space to g . Thus, ∇ A tan g n converges in L ( ∂ Ω) to ∇ A tan g and since ∇ A tan g n isalso Cauchy in H ( ∂ Ω), it follows that this sequence converges to ∇ A tan g in H ( ∂ Ω).Therefore, g ∈ N / A ( ∂ Ω) and { g n } n ∈ N converges to g in N / A ( ∂ Ω), so, N / A ( ∂ Ω) isa Banach space. The reflexivity property of N / A ( ∂ Ω) follows from the fact that Ψ : g → ( g, ∇ A tan g ) is an isometry between N / A ( ∂ Ω) and a closed subspace of H ( ∂ Ω) × ( H ( ∂ Ω)) n .The next lemma shows that if Ω is bounded with smooth boundary, then N / A ( ∂ Ω)coincides with the ordinary space H ( ∂ Ω); for the definition of the space H ( ∂ Ω)see [15] (pages 98-99).
Lemma 4.8.
Let Ω be a bounded domain of class C ,r with r > / , then N / A ( ∂ Ω) = N ( ∂ Ω) = H ( ∂ Ω) .Proof. By Lemma 3.4 of [9], M ν : u ∈ H / ( ∂ Ω) −→ νu ∈ ( H / ( ∂ Ω)) n s well defined and bounded; thus, k M − i ( ν · A ) ν u k ( H / ( ∂ Ω)) n = k − i ( ν · A ) νu k ( H / ( ∂ Ω)) n ≤ C ′′′ k ( γ D A ) u k ( H / ( ∂ Ω)) n = k ( γ D ( AU ) k ( H / ( ∂ Ω)) n ≤ C ′′ k ( AU ) k H (Ω) ≤ C ′ k u k H / ( ∂ Ω) ≤ C k u k H ( ∂ Ω) for all u ∈ H ( ∂ Ω) ֒ → H / ( ∂ Ω), where U ∈ H (Ω) is such that γ D U = u and k U k H (Ω) ≤ K k u k H / ( ∂ Ω) . That is, M − i ( ν · A ) ν : H ( ∂ Ω) → ( H / ( ∂ Ω)) n is bounded.Since ∇ A tan u = ∇ tan u + M − i ( ν · A ) ν u, it follows that N / A ( ∂ Ω) = N ( ∂ Ω) and that k · k H ( ∂ Ω) + k∇ A tan · k H ( ∂ Ω) ≈ k · k H ( ∂ Ω) + k∇ tan · k H ( ∂ Ω) ; the rest of the prooffollows easily by Lemma 6.8 of [9].The next lemma presents a relationship between the space N / A ( ∂ Ω) and theoperator γ D . Lemma 4.9.
Let Ω be a Lipschitz domain with compact boundary. Then γ D : (cid:8) u ∈ H (Ω) | γ A N u = 0 (cid:9) → N / A ( ∂ Ω) is well defined, linear, onto and has a bounded right inverse; furthermore, its kernelis H (Ω) .Proof. Take u ∈ H (Ω) with γ A N u = 0; then γ N u = − i ( ν · A ) γ D u , (19)and so γ u = ( γ D u, γ N u ) = ( γ D u, − i ( ν · A ) γ D u ) . Therefore, by Theorem 4.1, ∇ A tan ( γ D u ) = ∇ tan ( γ D u ) − i ( ν · A ) ν ( γ D u ) = ∇ tan ( γ D u ) + ( γ N u ) ν ∈ ( H / ( ∂ Ω)) n and k γ D u k N / A ( ∂ Ω) = k γ D u k H ( ∂ Ω) + k∇ A tan ( γ D u ) k ( H ( ∂ Ω)) n = k γ D u k H ( ∂ Ω) + k∇ A tan ( γ D u ) + ( γ N u ) k ( H ( ∂ Ω)) n ≤ k γ D u k H ( ∂ Ω) + k γ N u k L ( ∂ Ω) + k∇ A tan ( γ D u ) + ( γ N u ) k ( H ( ∂ Ω)) n ≤ C k u k H (Ω) . Thus, the operator is well defined and bounded. Now we show the existence of thebounded right inverse. Let ξ be the right inverse of γ given by Theorem 4.1, andconsider the operator l : N / A ( ∂ Ω) → (cid:8) ( g , g ) ∈ H ( ∂ Ω) ˙+L ( ∂ Ω , d n − ω ) | ∇ tan g + g ν ∈ H ( ∂ Ω) n (cid:9) l ( g ) := ( g, − i ( ν · A ) g ) . he operator l is bounded; in fact, k l ( g ) k = k g k H ( ∂ Ω) + k − i ( ν · A ) g k L ( ∂ Ω) + k∇ A tan g k ( H ( ∂ Ω)) n ≤ C ( k g k H ( ∂ Ω) + k∇ A tan g k ( H ( ∂ Ω)) n ) . Therefore ξ ◦ l is also bounded and, for all u ∈ (cid:8) u ∈ H (Ω) | γ A N u = 0 (cid:9) , by equa-tion (19), γ D ◦ ( ξ ◦ l )( u ) = γ D ( ξ ( γ D u, γ N u )) = u and γ A N ◦ ( ξ ◦ l )( u ) = γ N ( ξ ( γ D u, γ N u )) − i ( ν · A ) γ D ( ξ ( γ D u, γ N u ))= γ N u − i ( ν · A ) γ D u = γ A N u = 0;thus, ξ ◦ l is the continuous right inverse that we were looking for.The next result extends the magnetic Neumann trace to the domain of the max-imal operator. Theorem 4.10.
Let Ω be a Lipschitz domain with compact boundary. Then thereexists one, and only one, extension ˆ γ A N of γ A N with ˆ γ A N : Dom H A max → ( N / A ( ∂ Ω)) ∗ , (20) that is compatible with (6) in the following sense: for all s ≥ / one has ˆ γ A N u = γ A N u, ∀ u ∈ H s (Ω) so that H A u ∈ L (Ω) . Furthermore, this extension has dense range and h ˆ γ A N u, γ D w i N ( ∂ Ω) = − (cid:16) ( w, H A u ) L (Ω) − ( H A w, u ) L (Ω) (cid:17) , (21) for all u ∈ Dom H A max and w ∈ H (Ω) such that γ A N w = 0 , where h· , ·i N / A ( ∂ Ω) : ( N / A ( ∂ Ω)) ∗ × N / A ( ∂ Ω) → C represents the natural pairing between a functional and a vector in N / A ( ∂ Ω) .Proof. Take u ∈ Dom H A max and define ˆ γ A N u ∈ ( N / A ( ∂ Ω)) ∗ in the following way:consider g ∈ N / A ( ∂ Ω), by Lemma 4.9 there exists w ∈ H (Ω) with γ A N w = 0 suchthat γ D w = g and k w k H (Ω) ≤ C k g k N / A ( ∂ Ω) , define then, h ˆ γ A N u, g i N / A ( ∂ Ω) = − (cid:16) ( w, H A u ) L (Ω) − ( H A w, u ) L (Ω) (cid:17) . Analogously to the proof of Lemma 4.4, one verifies that such ˆ γ A N is well defined,unique and bounded. e will check now that this definition is coherent with (6). For s ≥ /
2, take u ∈ H s (Ω) with H A u ∈ L (Ω). Then, for all w ∈ H (Ω) such that γ A N w = 0, by (12),one can write ( H A w, u ) L (Ω) = Φ A ( w, u ) − ( γ A N w, γ D u ) L (Ω) = Φ A ( w, u ) . On the other hand, by (13),Φ A ( u, w ) = ( H A u, w ) L (Ω) + ( γ A N u, γ D w ) L (Ω) . Thus, by the last two equations( γ A N u, γ D w ) L (Ω) = ( u, H A w ) L (Ω) − ( H A u, w ) L (Ω) = h ˆ γ A N u, γ D w i N ( ∂ Ω) and this shows that ˆ γ A N , as defined above, is coherent with (6).To show that the range of ˆ γ A N is dense, it is enough to check that γ A N (C ∞ (Ω)) isdense in ( N ( ∂ Ω)) ∗ . Let Ψ ∈ (( N / A ( ∂ Ω)) ∗ ) ∗ = N / A ( ∂ Ω) be an antilinear functionalthat is zero on γ A N (C ∞ (Ω)); we are going to show that Ψ is equal to 0. Since Ψ ∈ N / A ( ∂ Ω), there is w ∈ H (Ω) with γ A N w = 0 such that γ D w = Ψ. Thus, h ˆ γ A N u, γ D w i N / A ( ∂ Ω) = 0 , ∀ u ∈ C ∞ (Ω) . Hence, by equation (21), one concludes that( u, H A w ) L (Ω) = ( H A u, w ) L (Ω) , ∀ u ∈ C ∞ (Ω) . On the other hand, by (12) one obtains, for all u ∈ C ∞ (Ω),( H A w, u ) L (Ω) = ( γ D w, γ A N u ) L (Ω) − ( γ A N w, γ D u ) L (Ω) + ( w, H A u ) L (Ω) . Using the fact that γ A N w = 0 and that Ψ = γ D w , it is found that Z Ω d n x Ψ γ A N u = 0 , ∀ u ∈ C ∞ (Ω) . Therefore, to conclude that Ψ is zero it is enough to show that γ A N (C ∞ (Ω)) is densein L ( ∂ Ω). But this follows by Lemma 4.3, Corollary 4.5, and the fact that C ∞ (Ω)is dense in H (Ω) ∩ H (Ω). Corollary 4.11.
Let Ω be a Lipschitz domain with compact boundary. Then, wehave the following inclusions N / A ( ∂ Ω) ֒ → L ( ∂ Ω) ֒ → N / A ( ∂ Ω) ∗ , and both have dense ranges, further the pairing between N / A ( ∂ Ω) and ( N / A ( ∂ Ω)) ∗ .Proof. Denote by I the first inclusion and by J the second one. Clearly the first inclu-sion is bounded since N / A ( ∂ Ω) ֒ → H ( ∂ Ω) ֒ → L ( ∂ Ω), therefore the second inclusionis also bounded since J = I ∗ . The image of J is dense by Theorem 4.10. The densityof the range of I follows by the injectivity of J . In fact, let f ∈ L (Ω) be such that f ( I ( N / A ( ∂ Ω))) = 0 then,
J f ( N / A ( ∂ Ω)) = I ∗ f ( N / A ( ∂ Ω)) = f ( I ( N / A ( ∂ Ω))) = 0and so
J f = 0; since J is injective, it follows that f = 0 and this shows that I hasdense range as well. .2 Quasi-convex domains Now we briefly recall the concept of quasi-convex domains; for more details seethe original source [9], in particular Section 8. Roughly, a quasi-convex domain isa particular class of Lipschitz domain with compact boundary that, either is locallyof class C ,r or its boundary has local convexity properties. In [9] the authors haveshown that, for this class of domains, the functions in Dom △ D and Dom △ N havethe H (Ω) regularity. For a general bounded Lipschitz domain, the functions inDom △ D have only the regularity of H (Ω); see Theorem B.2 in [14]. Using someresults stated in [9] about the regularity of the functions in Dom △ D , we will provethat the functions in the domain of H A D have the regularity of H (Ω) as well. Thisresult will be used to classify all self-adjoint extensions of H A mim .Let Ω be a bounded Lipschitz domain in R n ; Ω is said to be of class M H / δ , whichwe denote by ∂ Ω ∈ M H / δ , if for the functions ϕ i in the definition of the Lipschitzdomain we have ∇ ϕ i ∈ ( M H / δ ( R n − )) n and k∇ ϕ k ( MH / δ ( R n − )) n ≤ δ , where M H / δ ( R n − ) := (cid:8) f ∈ L ( R n ) | M f ∈ B ( H ( R n )) (cid:9) ,M f is the operator of multiplication by f , and B ( H ( R n )) is the set of boundedlinear operators on H ( R n ); furthermore, k f k MH / δ ( R n − ) := k M f k B ( H ( R n ) . Definition 4.12.
A Lipschitz domain with compact boundary Ω in R m is said tobe almost-convex if there is a family { Ω n } n ∈ N of open sets of R m with the followingproperties:i) ∂ Ω n ∈ C and Ω n ⊂ Ω, for all n ∈ N .ii) Ω n ր Ω as n → ∞ , in the following sense: Ω n ⊂ Ω n +1 for all n ∈ N and S n ∈ N Ω n = Ω.iii) there are a neighborhood U of ∂ Ω and a real function ρ n of class C , for each n ∈ N , defined in U such that ρ n < U ∩ Ω n , ρ n > U \ Ω n and ρ n = 0 in ∂ Ω n .Furthermore, there exists C ∈ (1 , ∞ ) such that C − ≤ |∇ ρ n ( x ) | ≤ C , ∀ x ∈ ∂ Ω n and ∀ n ∈ N . iv) there is C ≥ n ∈ N , all x ∈ Ω and all vector ξ tangent to ∂ Ω n in x , one has, h Hess ρ n ( x ) ξ, ξ i ≥ − C | ξ | , where h· , ·i denote the inner product of R n and Hess ρ n = n ∂ ρ n ∂x i ∂x j o ≤ i,j ≤ n is theHessian of ρ n .Given the above considerations, we can now recall the definition of a quasi-convexdomain. efinition 4.13. A Lipschitz domain with compact boundary Ω in R m , with m ≥ δ > x ∈ ∂ Ω, thereis a neighborhood Ω x of x , open in Ω, such that one of the following conditions issatisfied:i) Ω x is of class M H / δ if n ≥
3, and of class C ,r , for some r ∈ (1 / ,
1) if n = 2;ii) Ω x is almost-convex.The next result shows that, for a quasi-convex domain, the functions in Dom H A D and Dom H A N , the elements of the domain of the magnetic Dirichlet and Neumannrealizations, respectively, are in H (Ω); in fact, it is a consequence of Theorem 8.11in [9] and additional arguments; the main difficulties is our extension to unboundeddomains and the presence of the magnetic potential (whose regularity assumptionsare important here). Theorem 4.14.
Let Ω be a quasi-convex domain with compact boundary. Then, both Dom H A D , Dom H A N ⊂ H (Ω) .Proof. Assume first that Ω is bounded. In this case, the result for H AD follows directlyby Theorem 8.11 of [9] and the following observation: Since A ∈ (W ∞ (Ω)) n , one has H A = −△ + L with L : H (Ω) → L (Ω). Thus, if u ∈ H (Ω) then H A u ∈ L (Ω) if,and only if, △ u ∈ L (Ω) and the extensions of γ D defined in Dom H A D and Dom △ D coincide, therefore, Dom H A D = Dom △ D .Under the assumption that Ω is bounded, we will prove the result for H AN . Theproof is similar to the proof of Theorem 8.11 of [9] (for the Laplacian). Take u ∈ Dom H A N , by the definition of a (bounded) quasi-convex domain and a partition ofthe unity argument, it is enough to show that, if Ω x is as in the definition of quasi-convex domain and ω ∈ C ∞ ( R n ) is such that ∂ Ω ∩ supp ( ω ) ⊂ ∂ Ω ∩ ∂ Ω x , then v = ( ωu ) | Ω x ∈ H (Ω x ). Since u ∈ H (Ω), ∆ u ∈ L (Ω) (which follows by the sameargument from the paragraph above) we have that∆ v = [(∆ ω ) u + 2 ∇ ω · ∇ u + ω ∆ u ] | Ω x ∈ L (Ω) , and v ∈ H (Ω x ). The conclusion of the proof in this case is divided into two cases.Case I: Assume that Ω x is of class C ,r , r > /
2, or that Ω x is of class N H / δ and n ≥
3. In this case the result is a consequence of the following fact that was statedin the proof of Theorem 8.11 of [9]: let be Ω of class C ,r with r > / N H / δ and n ≥
3, if w ∈ H (Ω) with∆ w ∈ L (Ω) , ˜ γ N w ∈ H / (Ω) , (22)then w ∈ H (Ω). Following the proof, we have, already, v ∈ H (Ω x ) and ∆ v ∈ L (Ω x ), on the other hand,˜ γ AN v | ∂ Ω x ∩ ∂ Ω = ν · γ D ( ∇ A ωu ) | ∂ Ω x ∩ ∂ Ω= [ γ D ( ω )˜ γ AN u + γ D ( u )˜ γ N ( ω ) | ∂ Ω x ∩ ∂ Ω = [ γ D ( u )˜ γ N ( ω )] | ∂ Ω x ∩ ∂ Ω ∈ H / (Ω x ) hus,˜ γ N v | ∂ Ω x ∩ ∂ Ω = ˜ γ AN v | ∂ Ω x ∩ ∂ Ω − [ iν · γ D ( Aωu )] | ∂ Ω x ∩ ∂ Ω = [ γ D ( u )˜ γ N ( ω )] | ∂ Ω x ∩ ∂ Ω − [ iν · γ D ( Au ) γ D ( ω )] | ∂ Ω x ∩ ∂ Ω ∈ H / (Ω x )since, γ D ( Au ) ∈ H / ( ∂ Ω ∩ ∂ Ω x ) and by the fact that, under the hypothesis ofregularity of Ω x , assumed at the start of this case, the operator of multiplicationby the components of ν maps H / (Ω x ) into itself in this case. Then, by the factmentioned above, it follows that v ∈ H (Ω x ) and this finish the proof in this case.Case II: Ω x is almost-convex. This case is a easy consequence of an applicationof the lemma 8.8 of [9] to the vector field ( ω ∇ A u ) | Ω x , thus v ∈ H (Ω x ) also in thiscase, and this finishes the proof of the theorem when Ω is bounded.Consider now that Ω is unbounded. Let us start with the case of the operator H AD , take a ball B r of radius r > ∂ Ω ⊂ B r/ , and let ϕ , ϕ be a partitionof the unity associated with { Ω ∩ B r , R n \ B r/ } . Note that since ∂ Ω ⊂ B r/ , wehave R n \ B r/ ⊂ Ω. Hence, for all u ∈ Dom H A D in Ω, u = ϕ u + ϕ u . Since ϕ ∈ C ∞ ( B r ), it follows directly that ϕ u ∈ H (Ω ∩ B r ); on the other hand, H A ( ϕ u ) = ϕ H A ( u ) + [ H A , ϕ ] u where [ H A , ϕ ], the commutator of H A and the operator ofmultiplication by ϕ , is a differential operator of order 1 that maps H (Ω) in L (Ω).Thus, since H A u ∈ L (Ω) and u ∈ H (Ω), it follows that H A ( ϕ u ) ∈ L (Ω ∩ B r ) andso ϕ u ∈ Dom H A D , , where Dom H A D , is the Dirichlet operator associated with H A in Ω ∩ B r . Therefore, by the case of bounded Ω, we find that ϕ u ∈ H (Ω ∩ B r ). Onthe other hand, we have H A ( ϕ u ) = ϕ H A ( u ) + [ H A , ϕ ] u , and since ϕ as well asits derivative of any order are bounded, it follows that H A ( ϕ u ) ∈ L ( R n \ B r/ ), andsince ϕ u ∈ H ( R n \ B r/ ) we find that ϕ u ∈ Dom △ D, , where △ D, is the Dirichletoperator associated with the Laplacian in R n \ B r/ . But it is a known fact that thedomain of △ D, is contained in H ( R n \ B r/ ); see for example Theorem 8.12 of [10].Therefore u = ϕ u + ϕ u ∈ H (Ω), and this proves this case.The case of the operator H AN is carried by a similar argument, in fact, if u ∈ Dom H A N and ϕ i i = 1 , ϕ u ∈ Dom H A N , ⊂ H ( B r ∩ Ω) where H A N , is the Neumann operator associatedwith H A in Ω \ B r/ , and ϕ u ∈ Dom △ D , ⊂ H (Ω \ B r/ ), thus, u = ϕ u + ϕ u ∈ H (Ω) and this finishes the proof. In the following we recall the concept of regularized Neumann trace map [9] withsuitable adaptions to the magnetic context; it will allow us to construct a boundarytriple for the operator H A max .In Subsection 4.2 we saw that, for quasi-convex domains, Dom H A D ⊂ H (Ω);hence, Dom H A D = H (Ω) ∩ H (Ω). We are ready to introduce the operator τ A N ,called the regularized magnetic Neumann trace , as follows. Definition 4.15.
Let Ω be a quasi-convex domain and z ∈ C \ σ ( H A D ). The opera- or τ A N is defined by τ A,z N : Dom H A max → N ( ∂ Ω) ,τ A,z N u := γ A N ( H A D − z ) − ( H A max − z ) u , u ∈ Dom H A max . Clearly, this operator is well defined and bounded.
Theorem 4.16.
Let Ω be a quasi-convex domain and z ∈ C \ σ ( H A D ) , then theoperator τ A,z N has the following properties:i) τ A,z N ( H (Ω) ∩ H (Ω)) = N ( ∂ Ω) .ii) Ker τ A,z N = H (Ω) ∔ (cid:8) u ∈ L (Ω) | ( H A − z ) u = 0 (cid:9) iii) ( H A max u, v ) L (Ω) − ( u, H A max v ) L (Ω) = −h τ A,z N u, ˆ γ D v i N ( ∂ Ω) + h τ A,z N v, ˆ γ D u i N ( ∂ Ω) ,for all u, v ∈ Dom H A max .Proof. i)Note that, for z ∈ C \ σ ( H A D ), the operator ( H A − z ) : H (Ω) ∩ H (Ω) → L (Ω) is onto, so the result follows by Lemma 4.3.ii) It is clear that H (Ω) ∔ (cid:8) u ∈ L (Ω) | ( H A − z ) u = 0 (cid:9) ⊂ Ker τ A,z N . The fact thatwe have a direct sum follows from the invertibility of H A D − z . Now, let u ∈ Ker τ A,z N ,and set w = ( H A D − z ) − ( H A − z ) u . Then, w ∈ H (Ω) ∩ H (Ω) and ˜ γ A N w = τ A,z N u = 0,thus, w ∈ H (Ω), and u = ( u − w ) + w , with w ∈ H (Ω) and u − w ∈ L (Ω) with( H A − z )( u − w ) = 0, so, Ker τ A,z N ⊂ H (Ω) ∔ (cid:8) u ∈ L (Ω) | ( H A − z ) u = 0 (cid:9) .iii) Take u, v ∈ (cid:8) u ∈ L (Ω) | H A u ∈ L (Ω) (cid:9) and put ˜ u = ( H A D − z ) − ( H A − z ) u and ˜ v = ( H A D − z ) − ( H A − z ) v ; both elements are in H (Ω) ∩ H (Ω). We have( H A − z )˜ u = ( H A − z ) u and ( H A − z )˜ v = ( H A − z ) v , furthermore ˜ γ A N ˜ u = τ A,z N u and˜ γ A N ˜ v = τ A,z N v , thus,( H A u, v ) L (Ω) − ( u, H A v ) L (Ω) = (( H A − z ) u, v ) L (Ω) − ( u, ( H A − z ) v ) L (Ω) = (( H A − z )˜ u, v ) L (Ω) − ( u, ( H A − z )˜ v ) L (Ω) = (˜ u, ( H A − z ) v ) L (Ω) − h ˜ γ A N ˜ u, ˆ γ D v i N ( ∂ Ω) − ( u, ( H A − z )˜ v ) L (Ω) = (˜ u, ( H A − z ) v ) L (Ω) − ( u, ( H A − z )˜ v ) L (Ω) − h τ A,z N u, ˆ γ D v i N ( ∂ Ω) = (˜ u − u, ( H A − z ) v ) L (Ω) − h τ A,z N u, ˆ γ D v i N ( ∂ Ω) = h τ A,z N v, ˆ γ D u i N ( ∂ Ω) − h τ A,z N u, ˆ γ D v i N ( ∂ Ω) , where in the second equation we have employed (18), in the third we have used that˜ γ A N ˜ u = τ A,z N u and, in the fifth (18), ( H A − z )(˜ u − u ) = 0 and ˆ γ D (˜ u − u ) = − ˆ γ D ( u ). Lemma 4.17.
Let Ω be a quasi-convex domain. Then the operator ˆ γ D , which sat-isfies (17) , is onto. More specifically, for any fixed z ∈ C \ σ ( H A D ) , there exists alinear and bounded application ( N ( ∂ Ω)) ∗ ∋ θ → u θ ∈ L (Ω) such that ˆ γ D u θ = θ and ( H A − z ) u θ = 0 . roof. Fix z ∈ C \ σ ( H A D ). By the observation at the beginning of this section, wecan write ( H A D − z ) − : L (Ω) → H (Ω) ∩ H (Ω) , which is a bounded map. Thus, given θ ∈ ( N ( ∂ Ω)) ∗ , using Lemma 4.3, the antilinearfunctional θγ A N ( H A D − z ) − : L (Ω) → C is well defined and bounded. Then, by Riesz Theorem, there exists a unique ˜ u θ ∈ L (Ω) such that θγ A N ( H A D − z ) − ( f ) = ( f, ˜ u θ ) L (Ω) (23)and k ˜ u θ k L (Ω) = k θγ A N ( H A D − z ) − k B (L (Ω ,C )) ≤ C k θ k ( N ( ∂ Ω)) ∗ . Furthermore, theapplication θ → ˜ u θ is linear. By setting f = ( H A − z ) w , with w ∈ H (Ω) ∩ H (Ω),in (23), we have θ ( γ A N ( w )) = (( H A − z ) w, ˜ u θ ) L (Ω) , ∀ w ∈ H (Ω) ∩ H (Ω) . In particular, for w ∈ C ∞ (Ω), we have (( H A − z ) w, ˜ u θ ) L (Ω) = θ ( γ A N ( w )) = θ (0) = 0,so, ( H A − z )˜ u θ = 0 in the sense of distributions. Therefore, for w ∈ H (Ω) ∩ H (Ω)we can write θ ( γ A N ( w )) = (( H A − z ) w, ˜ u θ ) L (Ω) − ( w, ( H A − z )˜ u θ )= −h γ A N w, ˆ γ D ˜ u θ i N ( ∂ Ω) where in the second equation we have employed (18). Since γ A N : H (Ω) ∩ H (Ω) → N ( ∂ Ω) is onto, we conclude that ˆ γ D ( − ˜ u θ ) = θ . Thus, u θ = − ˜ u θ satisfies theproperties of the lemma, and this finishes the proof. Lemma 4.18.
Let Ω be a quasi-convex domain. Take z ∈ C \ σ ( H A D ) , then, for all f ∈ L (Ω) and all g ∈ ( N ( ∂ Ω)) ∗ , the following boundary value problem has a uniquesolution u D : ( H A − z ) u = f in Ω ,u ∈ L (Ω) , ˆ γ D u = g in ∂ Ω . Furthermore, there exists C ( z ) > such that k u D k L (Ω) ≤ C ( z ) (cid:8) k f k L (Ω) + k g k ( N ( ∂ Ω)) ∗ (cid:9) . In particular, if g = 0 then u D ∈ H (Ω) ∩ H (Ω) .Proof. By Lemma 4.17, we can take v ∈ L (Ω) such that H A v = 0, ˆ γ D v = g and k v k L (Ω) ≤ C k g k ( N ( ∂ Ω)) ∗ . Define w = ( H A D − z ) − ( f + zv ) ∈ H (Ω) ∩ H (Ω), then k w k L (Ω) ≤ C k f + zv k L (Ω) ≤ C ′ (cid:8) k f k L (Ω) + k g k ( N ( ∂ Ω)) ∗ (cid:9) . t is easy to see that u = v + w is a solution of the problem that satisfies the inequalityin the statement of the lemma. To see that the solution is unique we need to showthat, if u ∈ L (Ω) satisfy ( H A − z ) u = 0 and ˆ γ D u = 0, then u = 0. Fix f ∈ L (Ω)and let u f = ( H AD (Ω) − z ) − f ∈ H (Ω) ∩ H (Ω), then,( f, u ) L (Ω) = (( H AD (Ω) − z ) u f , u ) L (Ω) = ( u f , ( H AD (Ω) − z ) u ) L (Ω) − h γ A N u f , ˆ γ D u i N ( ∂ Ω) = 0 , where the second equality follow from 18, thus, it follows that u = 0.The following corollary is an immediate consequence of the above two lemmas. Corollary 4.19.
Let Ω be quasi-convex and z ∈ C \ σ ( H A D ) . Then the operator ˆ γ D : (cid:8) u ∈ L (Ω) | ( H A − z ) u = 0 (cid:9) → ( N ( ∂ Ω)) ∗ is a continuous isomorphism with a continuous inverse. Let Ω be a quasi-convex domain; due to the above results, we can introduce theoperator M A D , N ( z ) : ( N ( ∂ Ω)) ∗ → ( N ( ∂ Ω)) ∗ for z ∈ C \ σ ( H A D ), defined by M A D , N ( z ) f := − ˆ γ A N u D , where u D is the unique solutionof ( H A − z ) u = 0 , u ∈ L (Ω) , ˆ γ D u = f. This operator is clearly bounded and τ A,z N u = ˆ γ A N u + M A D , N ( z )(ˆ γ D u ) , ∀ u ∈ Dom H A max . H A min and gauge transformations In this section, first we briefly review the concept of boundary triples [4]. Thenwe construct a boundary triple for the operator H A max in the case of quasi-convex Ω,and so we obtain a parametrization of all self-adjoint extensions of H A min throughunitary operators on N ( ∂ Ω). For the case A = 0, this result gives us anotherparametrization to the family of all self-adjoint extension of the minimal Laplacianthat is different from the one obtained in [9], where the parametrization is given interms of self-adjoint operators defined in closed subspaces of ( N / ( ∂ Ω)) ∗ , and isobtained via Theorem II.2.1 of [12]. We think that our parametrization of the familyof all self-adjoint extensions of H A min is more direct and easier to apply.In this section the space N ( ∂ Ω) is considered a Hilbert space equipped with theinner product ( · , · ) N ( ∂ Ω) and the associated norm will be denoted by k · k N ( ∂ Ω) .Furthermore, the inner product ( · , · ) N ( ∂ Ω) will be selected in such way that it hasthe following property: let F be a measurable function such that | F ( x ) | = 1 a.s., then F , the operator of multiplication by F in N ( ∂ Ω), is unitary and ( M F ) ∗ = M F − ;an inner product of this kind can always be constructed in N ( ∂ Ω), see, for example,Remark 6.14 in [9], where the authors assume that Ω is bounded, but (by the sameproof) the result holds true for unbounded domains with compact boundary. Wenote that Theorem 5.4 does not depend of this particular choice of inner product in N ( ∂ Ω), however, Theorem 5.10 does use this particular kind of inner product (aswell as all results that follow from this theorem).Denote by I the surjective isometry I : ( N ( ∂ Ω)) ∗ → N ( ∂ Ω) defined by h f, g i N ( ∂ Ω) = ( f, Ig ) N ( ∂ Ω) , ∀ f ∈ N ( ∂ Ω) and ∀ g ∈ ( N ( ∂ Ω)) ∗ . Definition 5.1.
Let A be a linear, densely defined and closed operator in a Hilbertspace H . A boundary triple for A is a triple ( G, Γ , Γ ), where G is a Hilbert spaceand Γ , Γ : Dom A → G are linear operators such that:i) h f, Ag i − h Af, g i = h Γ f, Γ g i − h Γ f, Γ g i for all f, g ∈ Dom A. ii) The operator (Γ , Γ ) : Dom A → G × G is onto.iii) The set Ker(Γ , Γ ) is dense in H .By Theorems 1.2 and 1.12 of [4], we have Theorem 5.2.
Let A be a linear, symmetric densely defined and closed operator ina Hilbert space H and ( G, Γ , Γ ) a boundary triple for A ∗ . Then, the applicationthat maps to each unitary operator U of G the operator A U defined by Dom A U = { u ∈ Dom A ∗ | i ( + U )Γ u = ( − U )Γ u } ,A U u := A ∗ u , u ∈ Dom A U , sets a bijection between the set of unitary applications of G and the set of all self-adjoint extensions of A . H A min in quasi-convex do-mains The next theorem establishes that ( N ( ∂ Ω) , τ A,z N , ˆΓ D ), with z ∈ R \ σ ( H A D ) andˆΓ D = I ◦ ˆ γ D , is a boundary triple for the operator H A max = ( H A ) ∗ ; this result will bea tool for our description of the family of all self-adjoint extensions of H A min . Theorem 5.3.
Let Ω be a quasi-convex domain with compact boundary, a vectorfield A with components in W ∞ (Ω) and z ∈ R \ σ ( H A D ) , then ( N ( ∂ Ω) , τ A,z N , ˆΓ D ) isa boundary triple for H A max .Proof. By item iii) of Theorem 4.16, for all u, v ∈ Dom H A max we have( u, H A max v ) L (Ω) − ( H A max u, v ) L (Ω) = ( τ A,z N u, ˆΓ D v ) N ( ∂ Ω) − (ˆΓ D u, τ A,z N v ) N ( ∂ Ω) , nd so item i) of Definition 5.1 is satisfied.Item ii) of Definition 5.1 follows since τ A,z N (Ker ˆ γ D ) = N ( ∂ Ω) and ˆΓ D (Ker τ A,z N ) = N ( ∂ Ω), where the first equation is a consequence of item i) of Theorem 4.16, whereasthe second one is a consequence of item ii) of Theorem 4.16, together with the factthat, by Lemma 4.18, ˆ γ D ( (cid:8) u ∈ L (Ω) | ( H A max − z ) u = 0 (cid:9) ) = ( N ( ∂ Ω)) ∗ .Item iii) of Definition 5.1 is also satisfied since C ∞ (Ω) is contained in Ker( τ A,z N , ˆΓ D ).Next a direct consequence of Theorems 5.2 and 5.3 combined with Lemma 3.1. Theorem 5.4.
Let Ω be a quasi-convex domain with compact boundary, a vectorfield A with components in W ∞ (Ω) and z ∈ R \ σ ( H A D ) . Then the application thatassociates the operator H A,zU , defined by
Dom H A,zU = (cid:8) u ∈ Dom H A max | i ( + U ) τ A,z N u = ( − U )ˆΓ D u (cid:9) ,H A,zU u := H A u , to the unitary transformation U on N ( ∂ Ω) establishes a bijection between the set ofsuch unitary transformations and the set of all self-adjoint extensions of H A min . Remark 5.5.
Fix, for instance, z = −
1, since H AD is nonnegative, − ∈ R \ σ ( H A D ), forall admissible A . It is interesting to note that Theorem 5.4 gives a parametrizationof all self-adjoint extensions of H A min in terms of U , which is independent of themagnetic potential A . This establishes a natural bijection between the set of self-adjoint extensions H A, − U of H A min and those H B, − U of H B min , for each pair of admissiblemagnetic potentials A and B through H A, − U ←→ H B, − U . Remark 5.6.
A particular situation is for unbounded connected quasi-convex Ω ⊂ R ,with ∂ Ω a simple closed curve, and A is such that there is no magnetic field in Ω,i.e., rot A = 0 there. This is a typical Aharonov-Bohm setting and, to the best of ourknowledge, Theorem 5.4 gives the first description of all self-adjoint extensions forthis case and, furthermore, also for irregular solenoids ∂ Ω. An important questionin this context is to know when and which self-adjoint extensions H A,zU are unitarilyequivalent to some realization with zero magnetic potential, that is, the presence of A would be physically immaterial. We have something to say in the comments afterTheorem 5.14. Remark 5.7.
A boundary triple for H A max can be constructed in such way that thespace N / A ( ∂ Ω) plays the role of N / ( ∂ Ω) and a regularization τ A,z D of ˆ γ D , analogousto the operator τ A,z N , defined by τ A,z D : Dom H A max → N A ( ∂ Ω) ,τ A,z D u := ˆ γ D ( H A N − z ) − ( H A max − z ) u , u ∈ Dom H A max and z ∈ C \ σ ( H AN ) , plays the hole of τ A,z N in the construction of the boundary triple (from Theorem 5.3).The construction is done in a very similar way to the previous one. The advantageof the boundary triple using the space N / ( ∂ Ω) is that this space does not dependexplicitly on the magnetic potential A (Remark 5.5 makes use of this fact). .3 Gauge equivalence In what follows we introduce the concept of gauge equivalence of vector field in(W ∞ (Ω)) n , with connected domain Ω, and discuss how the self-adjoint extensionsof H A mim , given by Theorem 5.4, behave under such gauge transformations. If A =( A , ..., A n ) is a such vector field on Ω ⊂ R n , n ≥
2, let ω A denote the differential1-form associated with A , that is, in Cartesian coordinates ω A = P ni =1 A i d x i . Inthis subsection we suppose that Ω is open and connected. Definition 5.8.
Let Ω be a connected Lipschitz domain and
A, B vector fields withcomponents in W ∞ (Ω). We say that A is (quantum) gauge equivalent to B if thefollowing conditions hold:i) d( ω B − ω A ) = 0.ii) For each (smooth by parts) closed path γ in Ω, there exists an integer n γ suchthat Z γ ( ω A − ω B ) = 2 πn γ . Let
A, B be two gauge equivalent vector fields in Ω, fix x ∈ Ω and consider thefunction F Ω = F Ω A,B : Ω → C given by F Ω ( x ) := e i R γx ( ω B − ω A ) , (24)where γ x is a path in Ω connecting x to x ∈ Ω; note that this function is well definedby item ii) in the above definition and | F Ω | = 1. We have: Lemma 5.9.
Let
A, B ∈ (W ∞ (Ω))) n be gauge equivalent vector fields in Ω . Then, ∇ F Ω = i ( B − A ) F Ω ; moreover F Ω ∈ W ∞ (Ω) ∩ H (Ω) .Proof. It is enough to prove the first statement, since the rest is an easy consequenceof it. Fix x ∈ Ω and an open ball, B x ⊂ Ω, with center x such that A − B is K -Lipschitz in B x , K >
0; the statement will be concluded if we show thatit holds in B x . Note that for x ∈ B x , we have F Ω ( x ) = F Ω ( x ) e iφ ( x ) , where φ ( x ) = R [ x ,x ] ( ω A − ω B ) and [ x , x ] is the line segment connecting x to x ; we have ∇ φ ( x ) = ( B − A )( x ) for x ∈ B x , indeed, if ( B − A )( x ) = ( f ( x ) , ..., f n ( x )) ∂ j φ ( x ) = ∂ j Z [ n X i =1 f i ( x + t ( x − x ))( x i − x i )] dt = Z ∂ j [ n X i =1 f i ( x + t ( x − x ))( x i − x i )] dt = Z [ f j ( x + t ( x − x )) + n X i =1 ∂ j f i ( x + t ( x − x )) t ( x i − x i )] dt = Z [ f j ( x + t ( x − x )) + n X i =1 ∂ i f j ( x + t ( x − x )) t ( x i − x i )] dt = Z ddt ( tf j ( x + t ( x − x ))= f j ( x ) , here the second equality is justified by an application of the dominated convergencetheorem to the limit of the definition of the differential, which can be applied becausethe integrand is bounded since ( B − A ) is K -Lipschitz in B x ; the fourth equalityis a consequence of the fact that ∂ i f j = ∂ j f i , for i, j = 1 , ..., n , and this proves theabove statement.From this lemma the following function is also well defined, F ∂ Ω := γ D F Ω , (25)and for a unitary transformation U on N ( ∂ Ω), we define F U := ( F ∂ Ω ) − U F ∂ Ω . (26) Theorem 5.10.
Let Ω be a quasi-convex domain with compact boundary. Let A, B be gauge equivalent vector fields in (W ∞ (Ω)) n . Then, with the same hypotheses andnotations of Theorem 5.4, the self-adjoint extension H A,zU of H A min is unitarily equiv-alent to H B,z F U (extension of H B min ). More precisely, H A,zU ( F Ω u ) = F Ω H B,z F U u, for all u ∈ Dom H B,z F U = ( F Ω ) − Dom H A,zU or, equivalently, H A,zU F Ω = F Ω H B,z F U .Proof. First note that H BD = ( F Ω ) − H A D F Ω (by an abuse of notation, the multiplica-tion operator by F Ω is also denoted by F Ω ). Indeed, in the proof of Proposition 3.5we have verified that H A D is the self-adjoint operator associated with the form Φ A, D ,the statement then follows by Φ B,D ( u, v ) = Φ A, D ( F Ω u, F Ω v ), for all u, v ∈ H (Ω).Note now that Dom H A max = F Ω Dom H B max and H A max ( F Ω u ) = F Ω H B max u . In fact,it is clear that if u ∈ C ∞ (Ω), then H A max ( F Ω u ) = F Ω H B max u . Fix u ∈ Dom H B max , andtake a sequence { u j } j ∈ N in C ∞ (Ω) converging to u in Dom H B max , with the graphnorm k · k B = k · k L (Ω) + k H B ( · ) k L (Ω) , then, from what we said above, k H A ( F Ω u j ) − F Ω H B u k L (Ω) = k H B ( u j ) − H B ( u ) k → j → ∞ ; in particular, { F Ω u j } i ∈ N is a Cauchy sequence in the graph norm of H A max and, therefore, convergent in this space. On the other hand, since ( F Ω u i , H A ( F Ω u i ))converges in L (Ω) × L (Ω) to ( F Ω u, F Ω H B ( u )), from the fact that H A max is closed, itfollows that H A ( F Ω u ) = F Ω H B ( u ) ∈ L (Ω). Thus F Ω Dom H B max ⊂ Dom H A max and H A ( F Ω u ) = F Ω H B ( u ) ∈ L (Ω) for all u ∈ Dom H B max . Exchanging the roles of A and B in the above arguments, we see that the converse inclusion holds, and thisproves the statement.Note now that, for u ∈ C ∞ (Ω), τ A,z N ( F Ω u ) = γ A N ( H A D − z ) − ( H A − z )( F Ω u )= γ A N ( H A D − z ) − ( F Ω ( H B ( u ) − z ))= γ A N F Ω (( H BD − z ) − ( H B ( u ) − z ))= ν · γ D ∇ A ( F Ω (( H BD − z ) − ( H B ( u ) − z ))= ν · γ D F Ω ∇ B (( H BD − z ) − ( H B ( u ) − z ))= ν · F ∂ Ω γ D ∇ B (( H BD − z ) − ( H B ( u ) − z ))= F ∂ Ω τ B,z N ( u ) , hen, using the last observation and the fact that C ∞ (Ω) is dense in Dom H B max , weconclude that τ A,z N ( F Ω u ) = F ∂ Ω τ B,z N ( u ) for all u ∈ Dom H B max . Note, also, that forall u ∈ Dom H B max one has ˆΓ D ( F Ω u ) = F ∂ Ω ˆΓ D u . Indeed, it is easy to see that thisholds for u ∈ C ∞ (Ω), and the general case is a consequence of the fact that C ∞ (Ω)is dense in Dom H B max .With these facts, to conclude the proof it is enough to show thatDom H B,z F U = ( F Ω ) − Dom H A,zU , or, equivalently, F Ω Dom H B,z F U = Dom H A,zU . Pick then u ∈ Dom H B,z F U ; it follows that F Ω u ∈ Dom H A max and τ A,z N ( F Ω u ) = F ∂ Ω τ B,z N ( u ), so that i ( + U ) τ A,z N ( F Ω ) = i ( + U ) F ∂ Ω τ B,z N u = F ∂ Ω i h ( + F U ) i τ B,z N u = F ∂ Ω ( − F U )ˆΓ D u = ( − U ) F ∂ Ω ˆΓ D u = ( − U )ˆΓ D ( F Ω u ) , and F Ω u ∈ Dom H AU ; then, F Ω Dom H B F U ⊂ Dom H AU . Exchanging the roles of A and B in the arguments, we obtain the converse inclusion, and this finishes theproof.Next, a direct consequence of Theorem 5.10. Corollary 5.11.
Let
A, B be vector fields,
A, B ∈ (W ∞ (Ω)) n and satisfying B = A + ∇ Λ with Λ ∈ W ∞ (Ω) . Then, under the same hypotheses of Theorem 5.4, theself-adjoint extension H A,zU of H A min is unitarily equivalent to the extension H B,z ( e − iλ Ue iλ ) of H B min , with λ = γ D Λ . In fact, H A,zU ( e i Λ u ) = e i Λ H B,z ( e − iλ Ue iλ ) u, for all u ∈ Dom H B,z ( e − iλ Ue iλ ) = e − i Λ Dom H A,zU . Remark 5.12.
One says that two magnetic potentials A and B are classical gaugeequivalent if for each x ∈ Ω there is a smooth function Λ x , defined in a neighborhoodof x , so that B = A + ∇ Λ x holds in that neighborhood. This implies that A and B generate the same magnetic field, but it does not guarantee that such A and B aregauge equivalent in the sense of Definition 5.8. This distinction is at the heart of theAharonov-Bohm effect. Theorem 5.13.
Let Ω be a bounded connected Lipschitz domain and a vector field A with components in W ∞ (Ω) . Recall that H D = −△ D (the Dirichlet Laplacian), andlet λ and λ A, be the first (lowest) eigenvalues of −△ D and H A D , respectively. Then,the following statements are equivalent:i) λ = λ A, .ii) A is gauge equivalent to .iii) H A D is unitarily equivalent to −△ D . he proof of Theorem 5.13 is identical to the proof of Proposition 1.1 in [13], al-though our statement is slightly different. The differences are: 1) Here Ω is supposedto be a bounded Lipschitz domain, not necessarily with a smooth boundary. 2) Sincewe suppose that Ω is bounded, we do not need to add a scalar potential to H A ,that diverges at infinity, to ensure that the resulting Dirichlet extension has discretespectrum. 3) The potential A is in (W ∞ (Ω)) n and is not smooth, as (in principle)assumed in [13]. The main ingredient in this proof is the following identity (cid:13)(cid:13)(cid:13)(cid:16) ∇ − iA − ∇ u u (cid:17) ϕ (cid:13)(cid:13)(cid:13) (Ω) = (( H A − λ ) ϕ, ϕ ) L (Ω) , ∀ ϕ ∈ C ∞ (Ω) , where u ( x ) >
0, for all x ∈ Ω, is an eigenvector associated with the first eigenvalue λ of − ∆ D , which is not degenerate. This identity remains valid under the hypothesisof Theorem 5.13. By applying Theorems 5.10 and 5.13, we will conclude: Theorem 5.14.
Let Ω be a bounded connected and quasi-convex domain, A ∈ (W ∞ (Ω)) n . Let λ and λ A, be the first eigenvalue of −△ D and H A D , respectively.Fix z ∈ R \ σ ( H A D ) . Then the following statements are equivalent:i) λ = λ A, .ii) A is gauge equivalent to .iii) Let F Ω = F Ω A, be given by (24) . Then for all unitary applications U on N ( ∂ Ω) one has, by using (26) , ( F Ω ) − H A,zU F Ω = H ,z F U . (27) iv) There exist unitary applications U and V on N ( ∂ Ω) such that J − H A,zU J = H ,zV , for some J : Ω → C , J ∈ W ∞ (Ω) , and with |J ( x ) | = 1 for all x ∈ Ω .Proof. The equivalence of i) and ii) is a consequence of Theorem 5.13. By Theo-rem 5.10, ii) implies iii), and the fact that iii) implies iv) is obvious. To see that iv)implies ii), note that by Remark 2.2, the operator of multiplication by J maps H (Ω)into itself, since J ∈ W ∞ (Ω). By item iv), J − H A,zU J = H ,zV and we have H u = H ,zV u = J − H A,zU J u = J − H A,z mim J u, ∀ u ∈ H (Ω); (28)therefore, J − H A mim J = H , since, by Lemma 3.1, Dom H A mim = Dom H = H (Ω). Denote G = J − ; so G : Ω → S ⊂ R , and G maps H (Ω) into itself andrelation (28) is equivalent to −△ ( Gu ) = GH A u, ∀ u ∈ H (Ω);in particular, − ( G △ u + u △ G + 2 ∇ G · ∇ u ) = G {−△ u − iA · ∇ u + ( | A | − i div A ) u } , ∀ u ∈ C ∞ (Ω) , hus, u {△ G − ( | A | − i div A ) G } = ∇ u · ( − ∇ G − iGA ) , ∀ u ∈ C ∞ (Ω) , in particular, it follows that ∇ GiG = − A , or, equivalently, ω A = − G ∗ ( dziz ) (here G ∗ isthe pullback operation). In particular, ω A is closed, and for each closed path γ in Ω,we have Z γ ω A = Z γ − G ∗ (cid:0) dziz (cid:1) = Z Gγ − dziz = 2 πn for some n ∈ Z , and ii) follows.By Theorem 5.14 iv), if just one extension of H A min is unitarily equivalent (throughthe multiplication by a function J ) to a realization with zero magnetic potential, thenthe same occurs for all self-adjoint realizations of H A min , that is, the unitarily equiv-alences are always implemented by gauge transformations. And this is independentof the spectral type of each realization. Although these remarks are physically ex-pected, it seems there was no mathematical proof in the literature yet; and here for(bounded) quasi-convex domains.For the Aharonov-Bohm effect we need a multiply-connected domain Ω ⊂ R androt A = 0 in Ω (for simplicity we restrict the discussion to the plane); for instance,a quasi-convex annulus (the inner and outer border given by simple closed curves;the inner border represents the solenoid and the outer one could be a circle); thenTheorem 5.14 says that if A gives no contribution in case of some boundary condition,or simply if the first Dirichlet eigenvalue coincides with that of the Laplacian (item i)in the theorem), then A gives no contribution for all self-adjoint extensions of H A min . Remark 5.15.
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