Self-adjointness and spectral properties of Dirac operators with magnetic links

0.46 MB·

by  Fabian Portmann

SELF-ADJOINTNESS AND SPECTRAL PROPERTIES OF
DIRAC OPERATORS WITH MAGNETIC LINKS
FABIAN PORTMANN,JE´RE´MY SOK,ANDJAN PHILIPSOLOVEJ
Abstract. We define Dirac operators on S3 (and R3) with magnetic
7
fields supported on smooth, oriented links and prove self-adjointness of
1
certain (natural) extensions. We then analyze their spectral properties
0
and show, among otherthings, that these operators havediscrete spec-
2
trum. Certain examples, such as circles in S3, are investigated in detail

b and we compute thedimension of thezero-energy eigenspace.
e
F

4
1
Contents

]
h 1. Introduction 2
p
1.1. Overview 2
-
h 1.2. Main results 3
at 2. Links and geometrical structure on S3 4
m 2.1. Links as magnetic fields on S3 5
[ 2.2. The Seifert frame 6

2 2.3. Spinc spinor bundles on S3 6
v 2.4. Charge conjugation 8
7 3. The Dirac operator for a magnetic link 8
8
3.1. Self-adjointness for a magnetic knot 8
9
4 3.2. Proof of Theorem 7 12
0 3.3. Generalization to a magnetic link 19
.
1 3.4. Remarks on the Pauli operator 24
0 4. Convergence & compactness properties of Dirac operators 24
7
4.1. Metrics for knots and Seifert surfaces in C2 24
1
: 4.2. Convergence of Dirac operators 25
v
4.3. Proofs of Theorems 22 & 23 27
i
X
5. Hopf links 34
r 5.1. Non-existence of zero modes for circles 34
a
5.2. The Erd˝os-Solovej construction 34
5.3. Proof of Theorem 29 35
Appendix A. 38
A.1. Proof of Proposition 5 38
A.2. Technical Lemmas 40
References 43
Key words and phrases. Dirac operators, knots, links, self-adjointness, discrete spec-
trum,Fredholm.
Theauthors acknowledge support from theERC grant Nr. 321029 “The mathematics
of the structure of matter” and VILLUM FONDEN through the QMATH Centre of Ex-
cellence grant. nr.10059. This work hasbeen donewhen all theauthorswere workingat
theuniversity of Copenhagen.
1
2 F.PORTMANN,J.SOK,ANDJ.P.SOLOVEJ
1. Introduction
In this paper we study Dirac operators on S3 with magnetic fields that
are supported on smooth, oriented links γ S3. These magnetic fields
can be seen as having single field lines on S⊂carrying flux 2πα 0 on
k
≥
each component γ of γ. Their singular nature (they are in some sense
k
measures) makes the definition of the associated Dirac operator a delicate
task, which is fundamentally different from the smooth case. Since the
construction is local, it also works on R3 – we have however chosen to work
on S3 because then the Dirac operators have discrete spectrum. On R3
these singular magnetic fields can be considered as generalizations of the
celebrated Aharonov-Bohm magnetic solenoids allowing for more general
interlinking curves than just one single line.
1.1. Overview. On R3, Dirac operators with singular electric potentials
have been studied before, see for example [16,17] for a studyof Dirac opera-
tors with δ-potentials supported on shells, or [10] for more general measure-
valued electric potentials (which are singular with respect to the Lebesgue
measure). Furthermore, results for the Laplace operator with electric per-
turbations supportedon curves are investigated in e.g. [21–24], as well as on
surfaces in e.g. [12,13]. In [14], the author also investigates Dirac operators
related to links in S3, yet his setup is fundamentally different, as he studies
DiracoperatorsinthecomplementofalinkinS3 endowedwithahyperbolic
metric. Yet, to the best of our knowledge, there exist no results (neither on
R3 nor on S3) concerning Dirac operators with magnetic fields supported on
links.
The situation is somewhat different in the two-dimensional case. After
the discovery of the Aharonov-Bohm effect [5] and the subsequent interest
in point-like magnetic fields, spectral properties of Dirac operators with δ-
type magnetic fields have been investigated in numerous situations, see for
example [7,8,30,33,48], as well as for the Pauli operator [19,38]. It should
benotedthatthespectralpropertiesofDiracoperatorsoneven-dimensional
manifolds are somewhat more accessible, since there are many more tools
available. In the smooth case, the Atiyah-Singer Index Theorem [9] can
provide usefulinformation about the index of an operator. With the help of
vanishing theorems one can reduce the computation of the index to a study
of the zero-energy eigenspace. In some cases the latter can be analyzed
efficiently through the Aharonov-Casher theorem, see [6] for the discussion
on R2 and e.g. [15,20] for the problem on S2.
Describing the zero-energy eigenspace on a three-dimensional manifold
has so far proven to be a rather difficult task. A first step on R3 was
undertaken in [37] (see also subsequent works [2,3,18]), where the authors
exhibitedasquare-integrablemagneticfieldforwhichtheDiracoperatorhad
non-trivial kernel. This discovery turned out to be important in the study
of the stability of matter interacting with magnetic fields [29,36], or more
precisely, that stability may fail depending on the physical parameters, e.g.
thefinestructureconstant[25,26,35]. Thegeometrybehindtheconstruction
in [37] was exploited in [4] and in [20]; the idea is to pull back smooth
magnetic fields from S2 to S3 via Hopf maps to obtain a non-trivial kernel
3
for the Dirac operator on S3. Through this mechanism, given any m N,
one can find a smooth magnetic field on S3 such that the kernel of∈the
associated Dirac operator has exactly dimension m. Combining this result
withtheinvarianceofthekerneloftheDiracoperatorunderconformalmaps
– here the stereographic projection – yields the same answer for magnetic
fields on R3, and includes in particular the result from [37].
Theidea from [20] has somewhat served as a starting point for our analy-
sis. Byextendingthemain resultof[20]tothecase wherethemagnetic field
on S2 contains a collection of point measures, we obtain a rather explicit de-
scription of the spectrum of the Dirac operator on S3 with a magnetic field
supported on oriented interlinking circles, the Hopf links (see Theorem 31).
After this insight one is able to draft up a scheme to define the Dirac oper-
ator with a magnetic field supported on an arbitrary oriented link in S3.
1.2. Main results. Consider a smooth, oriented knot γ S3 with flux
⊂
2πα 0. To define the associated Dirac operator we choose a singular
≥
gauge. For any oriented knot γ there exists a smooth, compact and oriented
surface in S S3 such that ∂S = γ. The singular gauge is then defined as
⊂
the Seifert surface weighted by the flux 2πα. This gauge imposes a phase
jump of e−2iπα in the wave function across S, and the operator acts like the
free Dirac operator away from the surface. This essentially describes the
minimal operator (see Section 3.1.1), whose domain can becharacterized by
ψ H1(S3 S)2 : ψ = e−2iπα ψ ,
{ ∈ \ |S+ |S−}
where S denote the top (resp. bottom) of the Seifert surface. In partic-
±
ular, the Dirac operator exhibits a 2π-periodicity for the flux carried by γ.
One readily observes that the phase jump alone is not sufficient to obtain
a self-adjoint operator, and we are lead to investigate different self-adjoint
extensions described by the behavior of the functions close to γ. We are
interested in two special extensions, namely where the singular part of the
spinor “aligns with or against the magnetic field”. The first main result of
this paper is Theorem 7, where we prove the self-adjointness of these opera-
tors. This result is then generalized to links in Theorem 17. In Section 3.1.3
we discuss the relation between these two self-adjoint extensions; the first
extension for γ with flux 0 < 2πα < 2π is equal to the second extension for
the same knot γ with opposite orientation with flux 0 < 2π(1 α) < 2π. In
−
[40] we will show that the extension where the spinor aligns against the field
is the physical extension, as it can be approximated by smooth magnetic
fields in the strong resolvent sense.
After having proved self-adjointness, we introduce an appropriate topol-
ogy on the space of Seifert surfaces and investigate the convergence proper-
tiesoftheseDiracoperatorswhenvaryingthefluxesandtheSeifertsurfaces.
In Theorem 22 we derive a compactness result and in Theorem 23 we show
that for a converging sequence of Seifert surfaces and fluxes, the associ-
ated sequence of Dirac operators converges in the strong resolvent sense.
An immediate consequence of the compactness result is the discreteness of
the spectrum for the Dirac operator with an arbitrary magnetic link, see
Theorem 24.
4 F.PORTMANN,J.SOK,ANDJ.P.SOLOVEJ
Lastbutnotleast, westudythespectralpropertiesofourDiracoperators
in several examples, such as circles in S3 or Hopf links. In Theorem 29 we
show that the Dirac operator associated to a magnetic circle has always a
trivial kernel. This fact will beheavily used in two forthcomingpublications
[39,40].
Indeed, in this paper we are in some sense also laying the groundwork
for two follow-up works. The aforementioned 2π-periodicity in the flux 2πα
suggests the study of the spectral flow of loops of Dirac operators, as 2πα
runs from 0 to 2π. In [39] we will study the spectral flow for such paths of
operators and obtain as a byproduct information about zero modes for the
associated Dirac operators. The study of these newly discovered zero modes
is then continued in [40], where we provide a regularization procedure to
show that these zero modes persist when the singular magnetic fields are
approximated by smooth versions. From elliptic regularity and the confor-
malinvarianceofthekerneloftheDiracoperatorweobtainnewexamplesof
smooth,compactly supportedmagneticfieldsonR3 forwhichtheassociated
Dirac operator has non-trivial kernel.
Notation: Recall the Pauli matrices:
0 1 0 i 1 0
σ1 = 1 0 , σ2 = i −0 , σ3 = 0 1 .
(cid:18) (cid:19) (cid:18) (cid:19) (cid:18) − (cid:19)
The standard inner product on C2 will always be denoted by , .
For any open set Ω in S3, the space of smooth functions onhΩ· ·iis denoted
by D(Ω). Furthermore, we denote by L2(Ω)2 the space of L2(Ω)-functions
with values in C2. The inner product on L2(Ω)2 is given by
ψ,ϕ := ψ,ϕ ,
L2
h i h i
ZΩ
and the integration is performed with respect to the volume form of the
underlying space. For any m Z, we denote by m := √1+m2 the
∈ h i
Japanese bracket.
2. Links and geometrical structure on S3
We will always view the Riemannian manifold S3 as the unit sphere in C2
endowed with the standard metric g from its ambient space. The tangent
3
bundle of S3 is denoted by TS3, and the class of smooth vector fields by
Γ(TS3). SinceS3 can beseen as aLie group, it admits a global orthonormal
frame, which in turn defines an orientation. We choose to work in the
orientation which is mapped to the standard orientation of R3 under the
stereographic projection (for reasons to become clear during the course of
the exposition).
Convention: For the remainder of this paper, by a link γ S3 we always
mean a smooth, oriented submanifold of S3 which is diffeo⊂morphic to the
disjoint union of finitely many copies of S1 (and we henceforth refrain from
explicitlystatingthatthelinkissmoothandoriented). Furthermore,wewill
often write γ = γ , where the γ are knots (the connected components
k k k
of γ).
S
5
2.1. Links as magnetic fields on S3. Webeginbydefiningwhatwemean
by a magnetic knot. Given a knot γ S3, we would like to see it as a
single field line with flux1 2πα 0. Ob⊂serve that a magnetic field on S3 is
≥
normallyatwo-form, butduetothesingularnatureofourobjects, itismore
convenienttoviewthemagneticknotB asaone-current(see[41,Chapter3]
for an introduction to currents), obtained through a combination of the
Hodge dual and the musical isomorphisms
TS3 T∗S3
♭ : −→
(p,X) (p,g (X, ))
3
7→ ·
and its inverse ♯. B = 2πα[γ] then acts on smooth one-forms ω Ω1(S3) as
∈
(B;ω) := 2πα ω.
Zγ
From this definition it is easy to check the closed-condition; in the language
of currents, the differential of B corresponds to the boundary operator,
hence for ϕ D(S3) we have
∈
(∂B;ϕ) = (B;dϕ) = 2πα dϕ = 0.
Zγ
As in the classical electromagnetic setting, we would like to construct a
magnetic gauge potential A to B such that ∂A = B (in the language of
currents). This can be doneas follows; by Seifert’s theorem [28,45] we know
that given any (oriented) knot γ S3, there exists a smooth, connected and
oriented surface S S3 such tha⊂t ∂S = γ (as oriented submanifolds). On
ω Ω2(S3), A = 2π⊂α[S] acts as
∈
(A;ω):= 2πα ω. (2.1)
ZS
For any ω Ω1(S3) we then have by Stokes theorem that
∈
(∂A;ω) = (A;dω)= 2πα dω = 2πα ω =(B;ω). (2.2)
ZS Zγ
Note that if A′ is another magnetic gauge potential for B (meaning ∂S′ = γ
with the correct orientation and α′ = α), then they should be related by
a boundary term. In the case when S and S′ only intersect on γ, a quick
computation shows that
A A′ = 2πα[∂V], (2.3)
−
where [V] is the (signed) volume enclosed by the two surfaces S and S′,
seen as a three-current. Should the surfaces intersect on more than γ,
one can use the result of [44] to construct a sequence of Seifert surfaces
S = S ,S ,...,S = S′ such that S S = γ, 1 i n. The gauge
0 1 n i i−1
∩ ≤ ≤
transformationisthenperformedstepwisefromS toS usingtheprevious
i−1 i
result.
1In thecase of negative flux,we simply studytheknot with opposite orientation with
flux2π|α|.
6 F.PORTMANN,J.SOK,ANDJ.P.SOLOVEJ
The generalization to a magnetic link is now straight forward; given a
link γ = γ S3, the magnetic link B is defined as
k k ⊂
S B = 2πα [γ ], α 0.
k k k
≥
k
X
A natural choice for the magnetic gauge potential is A = 2πα [S ],
k k k
where S S3 is any Seifert surface with ∂S = γ . Note that if the fluxes
k k k
⊂ P
on two components of the link coincide, one may choose a common Seifert
surface for them.
2.2. The Seifert frame. Given any Seifert surface S S3, there exists a
⊂
natural construction to endow the boundary ∂S = γ with a natural moving
frame, the Seifert frame. It is given by the triple (T,S,N), where T is the
tangent vector to γ, N is the normal of the oriented Seifert surface and S
is the unit vector such that (T,S,N) is positively oriented (thus along the
link it points towards the inside of the Seifert surface). On each component
of γ, the Seifert frame coincides with the Darboux frame and satisfies the
Darboux equations
T 0 κ κ T
D g n
S = κ 0 τ S ,
g r
ds   −  
N κ τ 0 N
n r
− −
where κ ,κ are the geodesic and normal curvature respectively, and τ is
g n r
the relative torsion, see [46, Chapter 7] for more details.
For the coming chapters it will be necessary to extend the Seifert frame
(which is apriori only defined on γ) to a tubular neighborhood
B [γ] := p S3 :dist (p,γ) < ε
ε { ∈ g3 }
of the curve. The parameter ε> 0 is chosen such that the map
B [σ ; γ] B [γ]
ε 0 ε
exp : N −→ (2.4)
(γ(s),v(s)) exp (v(s))
7→ γ(s)
is a diffeomorphism, where γ is the normal fiber to γ in S3 and σ its null-
0
N
section. We define the frame at a point p = exp(t v (s))γ(s), v = 1, by
0 0 0
k k
parallel transporting (T,S,N) along the geodesic
t [0,t ] exp (tv (s)) = cos(t)γ(s)+sin(t)v (s), (2.5)
∈ 0 7→ γ(s) 0 0
which is a great circle in S3. Furthermore, the geodesic distance t will
0
henceforth be denoted as ρ (p) = ρ(p).
γ
2.3. Spinc spinor bundles on S3. We now have to definea Dirac operator
with a singular magnetic potential A = 2πα [S ]. Such an operator is
k k k
defined on the L2-sections of a vector bundle, called a Spinc spinor bundle.
P
We review in this part its definition and its properties.
A Spinc spinor bundle Ψ over a 3-manifold is a two-dimensional com-
M
plex vector bundle over together with an inner product , and an
M h· ·i
isometry σ :T∗ Ψ(2), called the Clifford map, where
M →
Ψ(2) := M End(Ψ) :M = M∗,tr(M) =0 .
{ ∈ }
A connection Ψ on Ψ is a Spinc connection if for any vector field X
∇ ∈
Γ(T ) it satisfies
M
7
(1) X ξ,η = Ψξ,η + ξ, Ψη , ξ,η Γ(Ψ).
h i h∇X i h ∇X i ∀ ∈
(2) [ Ψ,σ(ω)] = σ( ω), ω Ω1( ).
∇X ∇X ∀ ∈ M
Here, is the Levi-Civita connection on S3. It is well known that, up to
X
isomor∇phism, the Spinc spinor bundle on S3 is unique and it is the trivial
bundleΨ = S3 C2. For completeness wequickly sketch aproof of this fact.
Let Ψ be a S×pinc spinor bundle over S3 with Clifford map σ. Since S3
is a Lie group, it is parallelizable and we can choose a global orthonormal
frame e ,e ,e . For each p S3, the fiber Ψ can be decomposed into
1 2 3 p
{ } ∈
two complex lines
⊥ ⊥
Ψ = ker(σ(e3(p)) 1) ker(σ(e3(p))+1) =:Λ+(e3) Λ−(e3).
p p p
− ⊕ ⊕
This induces a decomposition of Ψ into two complex line bundles
Ψ= Λ+(e3) Λ−(e3).
⊕
As the second homotopy group π (S1) is trivial, any complex line bundle
2
over S3 is trivial. In particular, Λ+(e3) admits a global section λ , which
+
we can choose to have unit length at any point p S3. Similarly, we can
∈
defineaglobalsectionλ onΛ−(e3),andfixtheirrelativephasebyrequiring
−
that
ω(e )+iω(e ) = λ ,σ(ω)λ , ω Ω1(S3).
1 2 − +
h i ∀ ∈
This then yields the desired trivialization.
If Ψ is a Spinc connection, its connection form is denoted by
∇
λ , Ψλ λ , Ψλ
Mλ(X) := hλ+,∇XΨλ+i hλ+,∇XΨλ−i ,
(cid:18)h − ∇X +i h − ∇X −i(cid:19)
where X Γ(TB [γ]). We then have (as in [20, Prop. 2.9])
ε
∈
M (X)
λ
i e , e e , e i e , e
= h 1 ∇X 2i −h 3 ∇X 2i− h 3 ∇X 1i
2 e3, Xe2 +i e3, Xe1 e1, Xe2
(cid:18)−h ∇ i h ∇ i −h ∇ i (cid:19)
iωλ(X)IdC2, (2.6)
−
where ω is the (local) real one form
λ
i
ω (X) := ( λ , λ + λ , λ ).
λ + X + − X −
2 h ∇ i h ∇ i
By [20, Prop. 2.11], the connection form of the induced connection from the
Levi-Civita connection on S3 is given by (2.6) with ω = 0 (the connection
λ
form is then denoted by Mcan).
λ
Remark 1. From now on we will use the convention that whenever acts
∇
on a vector field, we mean the Levi-Civita connection, whereas if acts on
∇
aspinor,thenwemean thetrivial (induced)connection ontheSpinc bundle
on S3. In particular, iσ( ) denotes the free Dirac operator on S3.
− ∇
8 F.PORTMANN,J.SOK,ANDJ.P.SOLOVEJ
2.4. Charge conjugation. Observe that Ψ is in fact a Spin spinor bun-
dle, meaning it is a Spinc spinor bundle together with an antilinear bundle
isometry C : Ψ Ψ such that η,Cη = 0 and C2η = η for all η Ψ. In
→ h i − ∈
the previous trivialization, the map C is given by
S3 C2 S3 C2
C : × −→ ×
(p,ψ) (p,iσ ψ).
2
7→
Furthermore, for any unital ω Ω1(S3) we can again define Λ (ω), and one
±
∈
can show that C is a (fiber to fiber anti-linear) isomorphism between these
two complex line bundles,
C: Λ (ω) Λ (ω). (2.7)
± ∓
→
Thetrivialconnection commutes with C,henceitis aSpinconnection.
∇
Similarly, one can show that the free Dirac operator iσ( ) also commutes
− ∇
with C.
3. The Dirac operator for a magnetic link
For a smooth magnetic potential α Ω1(S3), the associated Spinc con-
∈
nection on Ψ in a trivialization (λ ,λ ) is given by the connection form 2
+ −
[20, Proposition 2.11]:
Mλcan(X)+iα(X)IdC2.
In this section, we extend the definition to the singular magnetic potential
introduced in the previous section.
3.1. Self-adjointness for a magnetic knot.
3.1.1. The minimal Dirac operator. Consider a magnetic knot B. To define
the minimal Dirac operator, we choose the singular gauge
A = 2πα[S], α 0,
≥
from (2.1). Due to the singular nature of the magnetic gauge potential it is
not possible to write the Dirac operator simply as σ( i +A), and a more
− ∇
involved definition is necessary.
It is straight forward to see that away from the Seifert surface on S3 S,
\
theDiracoperatoractsasthefreeDirac iσ( ). Todeterminethebehavior
− ∇
close to the surface, we see (with the help of a gauge transformation (2.3))
that the elements of the domain need to have a phase jump of e−2πiα across
S.
Denote by S (resp. S ) the top (resp. bottom) of the Seifert surface S
+ −
with respect to N. To simplify the notation we define Ω := S3 S. We
S
\
define the minimal Dirac operator via
(min)ψ := iσ( )ψ
DA − ∇
ΩS
with domain (cid:12)
(cid:12)
dom (min) := clos ψ D(Ω )2 : suppψ S3 γ,
DA G ∈ S ⊂ \
(cid:0) (cid:1) ψ(cid:8) exist & are in C0(S)2, ψ = e−2iπα ψ ,
|S± |S+ |S−
2The sign is different from [20], as ourconvention for theDirac operator is differe(cid:9)nt.
9
where clos denotes the closure in the graph norm. The derivative is taken
G
in [D′(Ω )]2, but we have (min)ψ L2(Ω )2 and we canonically inject
S DA ∈ S
(min)ψ into L2(S3)2.
DA
Remark 2. Note that when given two fluxes 2πα,2πα′ 0, the domains of
the minimal Dirac operators agree if 2π(α α′) 2πZ.≥We may therefore
− ∈
take α to be in [0,1] := [0,1] : 0 1 . A natural distance on [0,1] is
per per
{ ∼ }
given by
dist[0,1]per(α,α′) := distR(α+N,α′+N). (3.1)
Next, we define the domain V := S3 S B [γ] .
S,ε ε
\{ ∪ }
Proposition 3 (Generalized Stokes’ formula). Let e ,e ,e be an or-
1 2 3
thonormal frame for S3. Given ψ,ϕ H1(V(S,ε))2, w{e have }
∈
σ( i )ψ,ϕ ψ,σ( i )ϕ
h − ∇ i− h − ∇ i
ZVS,ε ZVS,ε
3
= ι iσ(ej)ψ,ϕ vol (3.2)
ej h i g3
j=1Z∂Bε[γ]
X (cid:2) (cid:3)
3
+ ι iσ(ej) ψ ,ϕ iσ(ej)ψ ,ϕ vol .
ej h |S+ |S+i−h |S− |S−i g3
Xj=1ZS∩Bε[γ]c h(cid:16) (cid:17) i
If ψ where
S
∈ A
:= ψ [D′(Ω )]2 : iσ( )ψ L2(Ω )2 L2(Ω )2, (3.3)
S S S S
A ∈ − ∇ ∈ ∩
n o
then the boundary terms ψ exist as elements in H−1/2(S γ)2. Further-
|S± loc \
more (3.2) still applies for ϕ H1(Ω )2 with 0 < ε < dist(suppϕ,γ), in
S
∈
which case the boundary term on ∂B [γ] vanishes.
ε
Proof. Assume first ψ,ϕ H1(Ω )2.
S
∈
We write iσ( ) = i 3 σ(ej) and set f := iσ(ej)ψ,ϕ . The
− ∇ − j=1 ∇ej j h− i
Lie derivative along e is denoted by , and by Cartan’s formula we have
j P Lej
= d ι +ι d.
e e e
L j ◦ j j ◦
Using the above representation of the Lie derivative in combination with
Stokes’ theorem yields
3 3
(f vol ) = ι iσ(ej)ψ,ϕ vol ,
Lej j g3 ej h− i g3
Xj=1ZVS,ε+,ε− Xj=1Z∂WS,ε+,ε− (cid:2) (cid:3)
which after expansion gives the terms on the right-hand side of (3.2). On
the other hand,
3 3
(f vol ) = df (e )vol +f div (e )vol .
Lej j g3 j j g3 j g3 j g3
j=1 j=1
X X
Metric compatibility gives
3 3
df (e )= iσ( )ψ,ϕ ψ, iσ( )ϕ + iσ( ej)ψ,ϕ .
j j h− ∇ i−h − ∇ i h− ∇ej i
j=1 j=1
X X
10 F.PORTMANN,J.SOK,ANDJ.P.SOLOVEJ
However, i 3 σ( ej)+div (e )σ(ej) ψ,ϕ = 0, and the desired
h− j=1 ∇ej g3 j i
formula follows.
P (cid:2) (cid:3)
Now assume ψ and ϕ H1(Ω )2 with 0 < ε < dist(suppϕ,γ). For
S S
∈ A ∈
0 < ε ,ε < εletW V bethesubsetofV obtainedbyremoving
+ − S,ε1,ε2 ⊂ S,ε S,ε
the points p that are either:
(1) above S with dist(p,S) ε ,
+
≤
(2) or below S with dist(p,S) ε .
−
≤
Asabove, weuseStokes’formula, butonW instead. Theboundary
S,ε+,ε−
term on ∂B [γ] vanishes leaving only two corresponding to the boundary
ε
above and below S.
By taking the limits ε 0 with ε fixed, the boundary terms corre-
+ −
→
sponding to S has a well-defined limit. For any ε > 0, this is true for any
+
ϕ H1(Ω )2 withdist(suppϕ,γ) > ε. Thisshowsthat ψ H−1/2(S γ).
∈ S |S+ ∈ loc \
By symmetrythesameholds for ψ . We endtheproofbytaking thelimit
|S−
ε ,ε 0. (cid:3)
+ −
→
(min)
Lemma 4. The operator is symmetric on its domain.
DA
(min)
Proof. Given any two ψ,ϕ dom , we have
∈ DA
(cid:0) (cid:1)
(min)ψ,ϕ = lim iσ( )ψ,ϕ .
hDA iL2 ε→0ZVS,εh− ∇ i
We now use (3.2); the last boundary term vanishes since the functions in
(min)
dom only differ by a phase on S and the boundaries have opposite
DA
orientation. We therefore have
(cid:0) (cid:1)
(min)ψ,ϕ = lim iσ( )ψ,ϕ
hDA iL2 ε→0ZVS,εh− ∇ i
3
= lim ψ, iσ( )ϕ + ι iσ(ej)ψ,ϕ vol
ε→0(cid:16)ZVS,εh − ∇ i Z{ργ=ε}Xj=1 ej (cid:2)h− i g3(cid:3)(cid:17)
(min)
= ψ, ϕ ,
h DA iL2
where the remaining boundary term vanishes when both ψ,ϕ are supported
away from the curve. The general case follows by density. (cid:3)
The Lichnerowicz formula
1 3
[σ(−i∇)]2 = −∆S3 + 4RsSc3 = −∆S3 + 2 (3.4)
suggests a relation between the minimal domain and the Sobolev functions
with the phase jump across S. Indeed, we have the following proposition,
whose proof can be found in the Appendix.
Proposition 5. Let H1(S3) L2(S3) be defined as
A ⊂
H1(S3) := f L2(S3) : ( f) L2(Ω ), X Γ(TS3);
A { ∈ ∇X |ΩS ∈ S ∀ ∈
ψ =e−2iπα f H1/2(S) ,
|S+ |S− ∈ }