arXiv:1902.05010v1 [math.AP] 13 Feb 2019
CORNER DOMAINS
FABIO PIZZICHILLO AND HANNE VAN DEN BOSCH
Abstract. We study the self-adjointenss of the two-dimensional Dirac operator with
Quantum-dotand Lorentz-scalar δ-shell boundary conditions, on piecewise C2
domains with finitely many corners. For both models, we prove the existence of a unique self-adjoint realization whose domain is included in the Sobolev space H1/2, the formal form domain of the free Dirac operator. The main part of our paper consists of a detailed study of the problem on an infinite sector, where explicit computations can be made: we find the self-adjoint extensions for this case. The result is then translated to general domains by a coordinate transformation.
1. Introduction
In this paper we study the self-adjoint realizations of the two-dimensional Dirac operator with boundary conditions. The free massless Dirac operator in R2 is given by the differential
expression (1.1) H := −i 0 ∂x− i∂y ∂x+ i∂y 0 = −iσ · ∇, where σ = (σ1, σ2) and the Pauli matrices are defined as
σ1 = 0 1 1 0 , σ2 = 0 −i i 0 , σ3 = 1 0 0 −1 . The Dirac operator describes the evolution of a relativistic particle with spin 1
2. It also arrises as
an effective description of electronic excitations in materials with a hexagonal lattice structure, such as graphene. The free operator in R2 can be seen to be self-adjoint on D(H) := H1(R2, C2),
since it is equivalent to multiplication by σ · k after Fourier transform. For more details, see for instance [19].
Let Ω ⊂ R2 be a bounded, piecewise C2-domain with finitely many corners and boundary ∂Ω.
The Dirac operators we consider are
• the Quantum-dot operator DQ, acting as H in Ω on spinors satisfying appropriate bound-ary conditions;
• the Lorentz-scalar δ-shell operator DL, where H acts on pairs of spinors defined in Ω and its complement, that satisfy a special type of transmission condition at ∂Ω.
We defer the precise formulation of boundary conditions to Section 1.1. Here, we give a short description of each operator and the physical systems that it describes.
Date: February 14, 2019.
2010 Mathematics Subject Classification. Primary 81Q10; Secondary 47N20, 47N50, 47B25.
Key words and phrases. Dirac operator, Quantum-dot, Lorentz-scalar δ-shell, boundary conditions, self-adjoint operator, conformal map, corner domains.
The Quantum-dot operator arises when electrons move in a hexagonal lattice and are confined by a termination of the lattice or by some type of potential. The best-known example of these boundary conditions is the one known as infinite mass, armchair, MIT-bag or chiral, as intro-duced in [8] for theoretical reasons, or experimentally studied, for instance in [18]. A priori, the boundary value problem is well behaved for the one-parameter family of boundary conditions given in (1.2) or (1.6) below.
The δ-shell interaction arises as a limiting case of the free Dirac operator perturbed by a potential that is strongly localized on the curve ∂Ω. Formally, one can think of this perturbation as a potential that is a coupling constant times the Dirac δ distribution on ∂Ω. In order to make sense of this mathematically, it is better to consider it as a boundary value problem. The action of the δ-shell operator on a function defined on the whole space can be seen as the direct sum of the action of the free Hamiltonian on the restriction of the function on Ω and its complement Ωc. Along the curve ∂Ω, both functions are linked by a special type of transmission condition given in (1.4) or (1.7) below. In [13, 14] it is shown that this type of operator is exactly the limit of the operators with smooth potentials that approximate a delta function on the surface. The case that we will study here, is the case where this potential takes the form of position-dependent mass term, or formally mδ∂Ωσ3. We call this model the Lorentz-scalar delta-shell (as
opposed to an electrostatic delta-shell generated by V δ∂Ω1), since it is invariant under Lorentz
transformations.
When Ω is a C2 domain, both models give rise to self-adjoint operators with domains in
the Sobolev space H1. In other words, the boundary value problem has an elliptic regularity
property. For the Quantum-dot model, this was shown in [7]. The δ-shell interaction has been studied previously in dimension 3, but the 3-dimensional theory also applies in dimension 2. Self-adjointness for 3-dimensional delta-shell interactions has been obtained in [9,2,3,4,16,6,5,10], in increasingly general settings, and we refer to [15] for a review on the topic.
In this paper we are interested in relaxing the smoothness hypothesis on the domain. This is justified by the fact that for numerical approximation, smooth curves are often approximated by polygons. From a mathematical point of view this turns out to be an interesting question, and it goes beyond a mere generalization of the methods in previous works. Recently, in [12], the case of polygons has been treated for the MIT -bag model, a particular case of Quantum-dot boundary conditions. In the case where Ω is a sector, the authors prove that the operator defined on H1 is self-adjoint for opening angles in (0, π) and it is not self-adjoint for opening angles in
(π, 2π). In the latter case, it admits a one-parameter family of self-adjoint extensions and among them, only one has domain included in the Sobolev space H1/2.
In this paper, we generalize this result to more general boundary conditions and to curvilinear polygons. Our main result, Theorem 1.5, states that, for a general bounded and piecewise C2
-regular domain Ω with finitely many corners, the operators DLand DQhave a unique self-adjoint
extension with domain contained in H1/2, the natural form domain of H. Since functions in H1/2
do not necessarily have boundary traces, we need to introduce some definitions before stating this precisely.
The proof of Theorem 1.5 is a consequence of localization near each corner. Locally near a corner, the domain can be mapped to an infinite wedge with the same angular opening. For the Quantum-dot case, we find the analogue of the result in [12]: we prove that, when the sector is convex, the operator DQ is self-adjoint on the usual Sobolev space H1, whereas when the sector
is the unique one whose domain is included in H1/2. For the Lorentz-scalar case, we have the
existence of a family of self-adjoint realizations {DL
α}α∈[0,π), and among all of them, DL0 is the
unique one whose domain is included in H1/2. This is done in Theorem 1.6.
The main advantage of the two-dimensional setting is that we have complex analysis and conformal maps at our disposal. However, we hope to generalize our results in a future work to three-dimensional domains whose boundaries have edges. This should at least work when the irregular part of the boundary has some kind of product structure. The question of generalizing the present result to more general delta-shell potentials remains open. The main technical part of our proof is to use conformal transplantations to map the wedge to a strip in the complex plane and use a separation of variables in a systematic way. The crucial ingredient seems to be an explicit description of harmonic spinors satisfying the boundary conditions, and we where unable to achieve this without separation of variables at hand.
1.1. Some definitions and notations. Throughout this paper, Ω ⊂ R2 will be a bounded
domain with piecewise C2-boundary and finitely many (non-degenerate) corners, and ∂Ω will
be its boundary. The positively oriented tangent t and the outward normal n are piecewise C1-functions on ∂Ω. With a slight abuse of notation, γ will be either the boundary trace of a
function defined in Ω, or the pair of traces on ∂Ω of a pair of functions defined in Ω+:= Ω and
Ω−:= R2\ Ω.
In the Quantum-dot-model, the boundary condition is parametrized by η ∈ (0, π), and its given by PηQγu = 0, where
(1.2) PηQγu := 1
2(1 − A
Q
η)γu, AQη := sin(η)σ · t(s) + cos(η)σ3.
The operator DQ acts as the free Dirac operator H on the domain:
(1.3) D(DQ) = {u ∈ H1(Ω, C2)|PηQγu = 0}.
For the Lorentz-scalar interaction, we will consider functions u = (u+, u−), with u+, u− be
defined respectively on Ω+and Ω−. In this case, u will have an interior and an exterior boundary
trace, linked by the boundary condition PL
µγ(u+, u−) = 0, where
(1.4) PµLγ(u+, u−) = (−iσ · n + µσ3) γu+− (−iσ · n − µσ3) γu−.
The operator DL act as the free Dirac operator H on
(1.5) D(DL) =(u+, u−) ∈ H1(Ω+, C2) × H1(Ω−, C2)|PηQγ(u+, u−) = 0
, for some µ ∈ (−1, 1). For the physical interpretation, 2µ is the mass of the δ shell.
The divergence theorem gives, for general u, v in H1(Ω, C2),
hu, HviL2(Ω)− hHu, viL2(Ω)= −i Z
∂Ω(u, σ · nv)C
2. Since AQ
η defined in (1.2) anti-commutes with σ · n, DQ is symmetric. For other computations,
it is convenient to expand the boundary conditions (1.2) and (1.7). Indeed PQ η γu1 γu2 = 0 if and only if
(1.6) γu2= Btγu1, where t := t1+ it2, and B :=
sin η 1 − cos η,
and PL
η γ(u+, u−) = 0 if and only if
(1.7) γu−= cosh(α)γu++ sinh(α)σ ·t γu+, where tanh(α) =
2µ 1 + µ2.
With this notation, it is straightforward to check that also DL is symmetric.
Let A be a piecewise C2 domain with finitely many corners. We define
K(A) := {u ∈ L2(Ω, C2)|Hu ∈ L2(Ω, C2)}. It is easy to see that H1(A, C2) ⊂ K(A).
For a C2 domain A, the starting point in [7] was to show that functions in K(A) admit
boundary traces in H−1/2(A, C2). The same result still holds true when A is a piecewise C2
domain with finitely many corners. Its proof is identical to the proof of [7] but we have to take into account the fact that multiplication by σ·n is no longer continuous in any Hs(∂Ω, C2).
Lemma 1.1. Let A be a piecewise C1 domain with finitely many corners. The map σ·n γ : H1(A, C2) → L2(∂A, C2) extends to a bounded map T : K(A) → H−1/2(∂A, C2).
For the Quantum-dot operator DQ defined as in (1.3),
D((DQ)∗) = {u ∈ K(Ω)|PηQσ·n T u = 0},
where the boundary conditions hold in the sense that, for all f ∈ H1/2(∂A) such that PQ η f ∈
H1/2(∂A), we have
T v[(1 − PηQ)f ] = 0.
For the Lorentz scalar operator DL defined in (1.5),
D((DL)∗) = {u ∈ K(Ω+) × K(Ω−)|PµLσ·n T u = 0}. where the boundary conditions are interpreted in a similar fashion.
Remark 1.2. If v ∈ D((DQ)∗) is supported away from the corners, multiplication of T v by σ·n
and PηQ makes sense and the boundary conditions hold in the usual sense.
Remark 1.3. If C is the set of corner points of ∂A, if f and PηQf are in H1/2(∂A), f vanishes
on C in the sense that it can be written as a H1/2-limit of functions with compact support in
∂A \ C.
Lemma 1.1 will be proven in Appendix A. To end this preliminary section, we group a few useful identities.
Lemma 1.4. Let A be a piecewise C1-domain, nA the outward normal. Let Ω be piecewise C2
and κ the piecewise continuous curvature of the boundary. (i) For all u, v ∈ H1(A) , we have
(1.8) hu, HviL2(A)− hHu, viL2(A) = −i Z
∂A(u, σ · n Av)C2; (ii) For all u ∈ H1(A)
(1.9) hHu, HuiL2(A) = h∇u, ∇uiL2(A)+ Z
∂A
(iii) For all u ∈ D(DQ) (1.10) DQu, B2 0 0 1 DQu L2(Ω)= ∇u, 1 00 B2 ∇u L2(Ω)+ Z ∂Ωκ|u 1|2;
(iv) For all u ∈ D(DL)
(1.11) DLu, DLu L2(Ω+)×L2(Ω−
) = h∇u, ∇uiL2(Ω+)×L2(Ω−)+ Z
∂Ω
κ (u+, σ3u+)C2.
Proof. Direct from the divergence theorem and the (anti)-commutation relations of the Pauli matrices. The last two identities also use ∂tt= −κn, on each C2-piece of the boundary
sepa-rately.
1.2. Statement of results. With all definitions in place, we can state our results.
Theorem 1.5. Let Ω be a piecewise C2-domain with finitely many, non-degenerate corners and let DQ and DL be defined respectively as in (1.3) and (1.5). Then
• Quantum-Dot Dirac Operator: The operator DQ0 ⊃ DQ defined by
D(DQ0) = D((DQ)∗) ∩ H1/2(Ω, C2),
is self adjoint. Moreover, If all corners are convex, then DQ= D0Q. • Lorentz-Scalar δ-shell Dirac Operator: The operator DL0 ⊃ DL defined by
D(D0L) = D((DL)∗) ∩
H1/2(Ω+, C2) × H1/2(Ω−, C2), is self-adjoint.
Theorem1.5will follow, after localizing and straightening the boundary, from a detailed study of the problem on the wedge of opening ω ∈ (0, 2π) \ {π}, given in polar coordinates (r, θ) as
Wω = {(r, θ)|r > 0, 0 < θ < ω}.
In this case, we explicitely find all self-adjoint extensions. In the following theorem and trhough-out the paper, we identify R2 with the complex plane C.
Theorem 1.6. Let ω ∈ (0, 2π), Ω = Wω, and let DQ and DL be defined respectively as in (1.3)
and (1.5). Let φ : R+ → R be a smooth cut-off function, supported in [0, 2] and equal to 1 in
[0, 1]. Then:
• Quantum-Dot:
(i) For ω < π, DQ is self-adjoint.
(ii) For π < ω < 2π, DQhas a one-parameter family of self-adjoint extensions {DQα}α∈[0,π)
with domains D(DQα) = D(DQ) + span(cos(α)uQ0 + i sin(α)u Q −1), where (1.12) uQk(z) := φ(|z|) z λQk−1/2 BzλQk−1/2 ! , λQk ≡ (1 + 2k)π/(2ω).
• Lorentz-Scalar: DL has a one-parameter family of self-adjoint extensions {DL α}α∈[0,π) with domains D(DαL) = D(DQ) + span(cos(α)uL0 + i sin(α)uL−1). where (1.13) uLk = (uLk,+, uLk,−) and uLk,±(z) := φ(|z|) zλk−1/2a k,± zλL k−1/2bk,± , with λLk the solutions of
(1.14) 2|µ| 1 + µ2 = |cos(πλLk)| |sin((π − ω)λL k)| ,
numbered such that λL0 ∈ (0, 1/2), and the coefficients ak,±, bk,± are such that γuLk
satisfies the boundary condition (1.4).
Remark 1.7. The functions uQk and uLk, given in (1.12) and (1.13) respectively, are supported near the origin, C∞ away from the origin, and
|uQk(r, θ)| ∼ r(2k+1)π/(2ω)−1/2,
|uLk(r, θ)| ∼ rλ
L
k−1/2, with − λ−1 = λ0∈ (0, 1/2). For both models, uQ
0 and uL0 are in H1/2, while uQ−1and uL−1 are not. Thus, DQ0 and DL0 coincides
with the self-adjoint realizations described in Theorem1.5.
The remainder of the paper is organized as follows. In the short Section 2, we show how functions satisfying boundary conditions can be composed with conformal maps. This will be our main tool in Section 3, where Theorem 1.6 is established. In Section 4, we show how Theorem 1.5follow from Theorem1.6by localizing and straightening the boundary.
The proof of Lemma 1.1 will be deferred to Appendix A, since it is essentially contained in [7] for smooth domains. We give a shorter proof that requires less regularity of the boundary. AppendixBcontains the construction of an explicit coordinate transform from a curved boundary with a corner to a straight wedge.
2. Transplantation
A conformal map F : Ω17→ Ω2 is a holomorphic function with holomorphic inverse. Locally, it
is a dilation by |F′| and a rotation by arg(F′). Since the Dirac operator transforms nicely under
dilations and rotations, it is no surprise that it also intertwines nicely with conformal maps. We define τF : L1loc(Ω2) 7→ L1loc(Ω1) by
(2.1) τFu(z) = F′(z)1/2 0 0 F′(z)1/2 u(F (z)).
If Ω1 is simply connected, it is always possible to define F′(z)1/2 as a continuous function. The
following proposition shows how convenient this definition is.
Proposition 2.1 (Properties of transplantation maps). Let Ω1 and Ω2 be simply connected
domains and F : Ω1 7→ Ω2 a conformal map between them, such that |F′| has a continuous
(i) Intertwining with H: For u in L1
loc(Ω2), in distribution sense
HτFu = |F′|τF(Hu).
(ii) Quadratic form: For u ∈ L2(Ω2), and v ∈ K(Ω2)
hu, HviL2(Ω
2)= hτFu, HτFviL2(Ω1).
(iii) Quantum-dot Boundary Condition: For u ∈ K(Ω2), if PηQγu = 0, then PηQγ(τFu) = 0
(iv) Lorentz-scalar Boundary Conditions: For s on ∂Ω1, if F extends to a conformal map
on B(s, ρ) for some ρ > 0 and u has support in F (B(s, ρ)), then PµLγu = 0 implies PµLγ(τFu) = 0.
Proof. Using the complex notation ∂z = 12(∂x− i∂y) and ∂z¯= 12(∂x+ i∂y), from (1.1) we deduce
that H = −2i 0 ∂z ∂z¯ 0 .
F and F are respectively holomophic and anti-holomorphic, then ∂z¯F = ∂zF = 0. Thank to this
and applying the chain rule we directly have (i).
Let us now prove (ii): thanks to (i)we have that hτFu, HτFviL2(Ω 1)= hτFu, |F ′|τ F(Hv)iL2(Ω 1) = h|F ′|2u(F (·)), (Hv)(F (·))i L2(Ω 1) = hu, HviL2(Ω 2),
where in the last line we use the change of variable z2= F (z1).
To prove (iii)and (iv)it is sufficient to plug the identity
tΩ2(F (z)) = F′(z)
|F′(z)|tΩ1(z), for z ∈ ∂Ω1
into (1.6) and (1.7) respectively.
3. The problem on the wedge
The key observation to prove Theorem1.6, is the following descripition of the maximal domains D((DQ)∗) and D((DL)∗).
Proposition 3.1. Let ω ∈ (0, 2π), Ω = Wω, η, µ ∈ R, and let DQ and DLbe defined respectively
as in (1.3) and (1.5). Then • Quantum-Dot: (i) for 0 < ω < π: D((DQ)∗) = D(DQ); (ii) for π < ω < 2π: D((DQ)∗) = D(DQ) + span(uQ0, uQ−1), with uQk defined in (1.12).
• Lorentz-Scalar: For ω 6= π:
D((DL)∗) = D(DL) + span(uL0, uL−1). with uLk defined in (1.13).
Remark 3.2. Assume that u satisfies the QD boundary conditions (1.2) for B ∈ R. Then we find
that
1 0
0 B
u
satisfies the QD boundary conditions (1.2) with B = 1. Therefore, we only have to consider the DQ with parameter B = 1, and DL with general parameters. In this way, the result for the
Quantum Dot boundary condition is a direct consequence of the result for the MIT-bag model in [12]. However, we prefer to treat it in details with our approach, since this gives a notationally and computationally simpler test case for the method.
In order to prove Theorem1.6, we use the logarithm to map Wω conformally to the strip
Cω := {z ∈ C : 0 < Im(z) < ω},
with boundary ∂Cω = {z ∈ C : Im(z) = 0} ∪ {z ∈ C : Im(z) = ω}. To this end, we define
G : Cω → Wω by G(ζ) = eζ and its inverse F : Wω→ Cω defined as F (z) = log(z). Here and in
the rest of the paper, the logarithm is defined with a branch cut on the positive real axis.
By (iii) of Proposition 2.1, τF and τG preserve the Quantum-Dot boundary conditions. By
(iv), these maps also preserve the Lorentz scalar boundary conditions on the upper boundary of Wω. On the lower boundary of Wω, F does not have a continuous extension. However, a short
computation shows that, if PL
µu = 0 on the lower boundary, then PµL(τGu+, −τGu−) = 0 on the
lower boundary. The functions uQ
k and uLk defined in the statement of Theorem1.6are obtained by separating
variables in the cylinder Cω and transplanting the resulting spinors back to the wedge.
Indeed, define the operator NQ = −iσ
3∂θ acting on L2([0, ω]) with QD boundary conditions
(for the B = 1 case). In this case, (1.6) reads as
(3.1) f2(0) = f1(0), f2(ω) = −f1(ω).
Analogously, define the operator NL = −iσ
3∂θ acting on L2([0, ω]) × L2([ω, 2π]) with
Lorentz-scalar boundary condition. In this case, taking into account a change of sign at 0 ∼ 2π, (1.7) reads as (3.2) f−(ω) = cosh(α) − sinh(α) − sinh(α) cosh(α) f+(ω), f−(2π) = − cosh(α) sinh(α) sinh(α) cosh(α) f+(0).
By integration by parts, NLand NQare self-adjoint on H1with the boundary conditions (3.1)
and (3.2) respectively. Moreover one can see that NQ has eigenfunctions fQ
k and eigenvalues λ Q k given by fkQ(θ) = √1 2ω eiλkθ e−iλQkθ ! , λQk = (2k + 1) π 2ω, k ∈ Z. Note that (3.3) fkQ(θ) = σ1f−k−1Q (θ).
For NL, the eigenfunctions are of the form fL k = (fk,+L , fk,−L ) with fk,±= a±eiλkθ b±e−iλkθ . The boundary conditions imply that
cosh(α) sinh(α) sinh(α) cosh(α) a+ a− =
− cosh(α)ei2πλk sinh(α)ei(2π−2ω)λk sinh(α)e−i(2π−2ω)λk − cosh(α)e−i2πλk
a+
a−
. This is only possible if
cosh2(α)|1 + ei2πλk|2− sinh2(α)|1 − ei(2π−2ω)λk|2 = 0,
which reduces to (1.14) by using the definition of α in (1.7). For definiteness, we assume that the fL
k are normalized, that is
ω(|a+|2+ |b+|2) + (2π − ω)(|a+|2+ |b+|2) = 1.
This determines the fL
k up to a phase. Since {σ1, NL} = 0 (note that multiplication by σ1
preserves the boundary conditions (3.2)), we can define the eigenvectors for k < 0 by
(3.4) fkL(θ) = σ1f−k−1L (θ).
Thanks to this, we have that
(3.5) (τGuQk)(t + iθ) = φ(et)etλ
Q kfQ
k (θ) and (τGuLk)(t + iθ) = φ(et)etλ
L kfL
k(θ).
Now, we are ready to prove Proposition 3.1.
Proof of Proposition 3.1. Fix u ∈ D((DQ)∗). Without loss of generality, we may assume that u has support within B(0, 1). Thanks to Remark3.2, we only have to consider DQ with parameter
B = 1, and DLwith general parameters. The first part of the proof is the same for both models. For this reason we will use the superscript X to identify Q or L indistinctly.
The idea of the proof is to use an explicit calculation to write u as the sum of a regular part that is H1 in a neighbourhood of the origin, and a linear combination of uX
0 and uX−1.
Using the transplantation maps defined in (2.1), we set
w(ζ) := HτGu(ζ) = |eζ|τG(Hu)(ζ).
By a change of variables,
(3.6) |e−ζ/2
|w ∈ L2(Cω).
In order to solve H(τGu) = w, since σ2= iσ1σ3, we write
H = −iσ1(∂t+ iσ3∂θ).
By Proposition 2.1, for each fixed t ∈ R, τGu(t + i·) satisfies the boundary conditions (3.1) or
(3.2). We solve
iσ1w(t + iθ) = (∂t− NX)(τGu)(t + iθ),
by projecting onto the eigenfunctions of NX. Explicitly, we write
τGu(t + iθ) = X k∈Z ak(t)fkX(θ), iσ1w(t + iθ) = X k∈Z bk(t)fkX(θ),
where, the coefficients of each mode are related by the ordinary differential equation (∂t− λXk)aXk(t) = bXk(t).
We can write the general solution to this equation in the form
aXk(t) = Z t 0 eλX k(t−t ′ )b k(t′) dt′+ ckeλ X kt, for λX k ≥ 1/2, Z t −∞ eλX k(t−t ′ )b k(t′) dt′+ cXkeλ X kt, for λX k < 1/2. We define eaX k(t) = aXk(t) − cXk eλ X
kt. Going back to the original function τGu, we have found a decomposition valid for t sufficiently negative (such that φ(et) = 1),
τGu(t + iθ) = X k eaXk(t)fkX(θ) + X k cXkvkX(t + iθ), where vX
k = τGuXk, where we have used (1.12) and (1.13) to write
τGuXk(t + iθ) = eλktfkX(θ).
We will start by showing that the transplantation to the wedge of the first term in the decom-position is in H1 in a neighbourhood of the origin. Define
v(t + iθ) =X
k
eaXk(t)fkX(θ).
The decay of w given by (3.6) also holds for the Fourier coefficients bX
k. Writing bXk(t) = et/2ebXk(t),
with ebX
k ∈ L2(R−), gives the following estimates for t ≤ 0
|eaXk(t)|2≤ Z t 0 e2λXk(t−t ′ )+t′ dt′ kebXkk 2 L2 = |e t−e2λXk t| |1−2λX k| keb X kk 2 L2, for λXk > 1/2, Z t −∞ e2λX k(t−t ′ )+t′ dt′ kebXkk 2 L2 = e t 1−2λX k keb X kk 2 L2, for λXk < 1/2. Therefore, we have that
kvkL2(C ω∩Re ζ≤0)≤ C X k kebXkkL2(R −)≤ Ck|e −ζ/2|wk L2(C ω).
By a change of variables, also the transplantation back to original variables satisfies τFv ∈ L2(B(0, 1)).
Let us start by considering the Lorentz-scalar case. By dominated convergence, the identities (1.9) and (1.11) with curvature κ = 0 on the boundary of Wω, we have
k∇τFvk2L2(B(0,1)) = lim r→0 Z B(0,1)\B(0,r)|∇τFv| 2 = lim r→0 Z B(0,1)\B(0,r)|HτFv| 2 − Z {|z|=1} (τFv, iσ3∂θτFv)C2 + lim r→0r −1 Z {|z|=1} (τFv, iσ3∂θτFv)C2 ≤ kHuk2− Z 2π 0 v, (NL − 1/2)vC2(0 + iθ) dθ + lim t→−∞e −t Z 2π 0 v, (NL− 1/2)vC2(t + iθ) dθ . The last inequality follows from observation that
iσ3∂θ e−iθ/2 0 0 eiθ/2 − e−iθ/2 0 0 eiθ/2 iσ3∂θ= 1/2
The first two terms are finite, so we can restrict our attention to the last one. We have Z 2π 0 v, NLvC2(t + iθ) dθ = X k (λk− 1/2)|eak(t)|2 ≤ etX k |λk− 1/2| |1 − 2λk| kebkk 2 ≤ etCk|e−ζ/2|wk2L2(C ω)= Ce tkHuk2 L2(W ω), so k∇τFvk2L2(W ω∩B(0,1)) ≤ CkHuk 2 L2.
For the Quantum-Dot case, one could use the same approach by using (1.10) instead of (1.11). The only difference is the domain of integration for the angular variable.
In summary, we have found that each u ∈ D((DX)∗) admits a decomposition
u = τGv +
X
k
ckuXk, in B(0, 1),
and τGv ∈ H1 and therefore in τGv ∈ D(DX).
We now turn our attention to the harmonic part uH :=
X
k
ckuXk.
By definition, close to the origin
|uXk(r, θ)| ∼ rλ
X k−1/2. Since u ∈ L2(B(0, 1)) by assumption, we find that c
k = 0 for λXk < −1/2 or k ≤ −2. By using
uH ∈ H1(B(0, 1) \ B(0, s)) and again (1.10) and (1.11),
0 =DXuH, DXuH B(0,1)\B(0,s) = h∇uH, ∇uHiB(0,1)\B(0,s)+ X k≥−1 |ck|2(1 − s2λ X k−1)λX k.
Fixing s < 1 such that s2λX 1 −1≤ 1/2, shows that 1 2 X λX k≥1/2 |ck|2λXk ≤ h∇uH, ∇uHiB(0,1)\B(0,s) < ∞,
and, by the same computation as before, ∇ X λX k≥1/2 ckuXk 2 L2(B(0,1)) ≤ lim rց0|ck| 2λX k(1 − r2λ X k−1) + C(|c0|2+ |c−1|2) < ∞.
To conclude the proof we have to distinguish the two cases:
Lorentz-scalar case. The solutions of the eigenvalue equation (1.14) are symmetric around zero. There is a single solution in (0, 1/2), labeled λ0, and one solution, λ−1= −λ0 in (−1/2, 0).
Thus, we found the decomposition
u = v +X
k≥1
ckuLk + c0uL0 + c−1uL−1.
Since the first two terms are in H1 ⊂ D(DL), we conclude the proof.
The Quantum-Dot case. Recall that the eigenvalues of NQ are λQk = π
2ω(2k + 1)
If ω ∈ (0, π), then λQk ∈ [−1/2, 1/2] for any k. This means that u ∈ H/ 1(Ω). If ω ∈ (π, π), λ Q 0
and λQ−1 are the only eigenvalues of NL in (−1/2, 1/2). We have the decomposition
u = v +X
k≥1
ckuQk + c0uQ0 + c−1uQ−1,
as for the Lorentz-scalar case.
Having established this, it is straightforward to finish the proof of Theorem1.6.
Proof of Theorem 1.6. The case of Quantum-Dot for 0 < ω < π follows directly from Propo-sition 3.1. For this reason, we will use the superscript X to denote the Quantum-dot case for π < ω < 2π or the Lorentz-scalar case. We will denote Z = Wω in the Quantum-dot case,
Z = Wω∪ Wωc in the Lorentz-scalar case.
By Proposition 3.1,
DX ⊂ DαX ⊂ (DX)∗. For simplicity, we define
wα := cos(α)uX0 + i sin(α)uX−1 and w⊥α := sin(α)uX0 − i cos(α)uX−1.
Then, by construction
The proof of the self-ajdointess of DX
α follows directly from
(3.8) h(DX)∗uXk, uXl iL2 − huXk, (DX)∗uXl iL2 = (
0 when k = l;
i when k 6= l; for k, l = 0, −1. We will defer the proof of (3.8) to the end of this proof.
Thanks to (3.7), to check that DX
α is symmetric, we have to prove that
(3.9) hDαX(u + wα), u + wαiL2(Z) = hu + wα, DαX(u + wα)iL2(Z), for u ∈ D(DX). In fact, hDXu, ui
L2(Z)= hu, DXuiL2(Z)due to the symmetry of DX; DX αwα, wα =wα, DαXwα thanks to (3.8); and DXu, w α L2(Z)= u, DX αwα L2(Z) since wα ∈ D(DαX) ⊂ D(D(X∗)). Then (3.9) is proved and so DX α ⊂ (DXα)∗.
Let us prove the opposite inclusion. Since both wα, wα⊥ ∈ D((DX)∗) and they are linearly
independent, we have that
D((DX)∗) = D(DX) + span(wα, wα⊥) = D(DαX) + span(wα⊥).
Thanks to this, it is sufficient to prove that w⊥
α ∈ D((D/ αX)∗). Indeed, thanks to (3.8) we have
that D DXαwα, w⊥α E L2(Z)− D wα, (DX)∗w⊥α E L2(Z)= 1, And so DX α = (DαX)∗.
To conclude the proof, we have to show (3.8). By using dominated convergence, Proposition2.1
and (1.8) from Lemma1.4, we have that (DX)∗uXk, uXl −uXk, (DX)∗uXl = lim rց0 HuXk, uXl Z\B(0,r)−uXk, HuXl Z\B(0,r) = lim t→−∞ HτGuXk , τGuXl {Re(ζ)≥t}− τGuXk, HτGuXl {Re(ζ)≥t} = lim t→−∞i Z {Re(ζ)=t} τGuXk, σ1τGuXl C2 = lim t→−∞iφ(e t)e(λX k+λXl )t Z {Re(ζ)=t} fkX, f−l−1X C2, where, in the last line, we used (3.5), (3.3) and (3.4). Since λX
−1 = −λX0 and the functions f0X
and fX
−1 are orthonormal, we directly have (3.8).
4. Localizing and straightening
In this section, we deduce Theorem 1.5from Theorem 1.6. We first check that the operators with domains in H1/2 are symmetric. In order to simplify the notation further, we assume that
Ω has a single corner centred at the origin. The case of several corners is again just a matter of extra notation.
Lemma 4.1. The operators DL0 and DQ0, as defined in Theorem 1.5are symmetric.
Proof. The proof is identical for the Lorentz scalar and Quantum-dot case. We give it here for the latter case, since the notation is more concise. Fix u, v ∈ D(DQ0). By the result for smooth
domains, u and v are in H1(Ω\B(0, r)) for all r > 0. We apply (1.8) from Lemma1.4to conclude
that
hu, HviL2(Ω\B(0,r))−hHu, viL2(Ω\B(0,r)) = −i Z ∂Ω\B(0,r)(u, σ · nΩ v)C2−i Z Ω∩∂B(0,r)(u, σ · er v)C2. The first term vanishes because of the boundary conditions, that hold in the classical sense away from the corner. In order to estimate the second term, we average the identity over r ∈ [s, 2s]. This gives 1 s Z 2s s Z Ω∩∂B(0,r)(u, σ · e rv)C2 dr ≤ 1 s Z Ω∩B(0,2s)\B(0,s)|u||v| ≤ 1skukL4(Ω∩B(0,2s))kvkL4(Ω∩B(0,2s))|Ω ∩ B(0, 2s))|1/2 ≤ CkukL4(Ω∩B(0,2s))kvkL4(Ω∩B(0,2s)).
The final bound tends to zero as s → 0, since u, v ∈ L4(Ω) by the Sobolev embedding H1/2(Ω) ⊂
L4(Ω).
This reduces the problem of self-adjointness to the issue of showing that the domain of the adjoint operator stays in H1/2. We know as well that the domain of the adjoint is included
in the maximal domain, so away from the corners, elements in the domain of the adjoint are even H1. Close to the corners, we have to transform coordinates to straighten the boundary.
This transformation will, in general, set up a unitary equivalence between the Dirac operator on the curvilinear wedge and the Dirac operator plus a perturbation on the straight wedge. The unbounded part of this perturbation consists of derivatives of the first order, multiplied by a function that measures the difference between the Jacobian matrix of the coordinate transfor-mation and the identity matrix. In the case of smooth boundaries, this perturbation is irrelevant by the elliptic regularity of the Dirac operator on the half-space.
Here, elliptic regularity does no longer hold, so we need a to work a little bit more.
For the quantum dot case, we can avoid issues by using bounded conformal transformation to send the interior of the domain to a subset of the wedge. This conformal transformation maps the maximal domain to the maximal domain on the wedge, where the classification from Theorem1.6
remains valid. For simplicity, we have stated Theorem1.5for the extension with domain in H1/2,
but in principle, this strategy allows for a complete characterization of self-adjoint extensions for the quantum dot case.
The case of the Lorentz scalar operator is more delicate, because it is not, in general, possible to find a conformal transformation that maps both the interior and the exterior of the curvilinear domain to the interior and exterior of the wedge. On the other hand, it is always possible to find a C2 coordinate transformation that achieves this, but in this case, we have to treat the
perturbation terms carefully. We choose a coordinate transformation with the perturbation of the Jacobian matrix of order r, with r the distance to the corner. Combined with the H1/2regularity
in the whole domain, this gives us precisely what is needed to conclude. The perturbation terms are finite and symmetric on the image of the original domain, which allows concluding that the image of the original domain is included in D(DL
0) on the wedge. By using the decomposition
of spinors in this domain in a H1-part and a multiple of uL
0, we conclude that the perturbation
terms are relatively bounded with respect to the full operator, with a relative bound that can be made smaller by taking a smaller neighbourhood of the corner. Note that this strategy does not
give a classification of self-adjoint extensions, it only proves the existence of a single extension with the domain in H1/2.
In the next sub-section, we give the proof for the Quantum-dot case and in the last sub-section, we will analyse the Lorentz-scalar interaction.
4.1. Quantum Dot case.
Proposition 4.2. Let Ω be a piecewise C2-domain with a single corner centred at the origin.
The Quantum-dot operator D0Q defined in Theorem 1.5 is self-adjoint.
Proof. To clarify the notation, we will denote DQ,Ω0 and DQ,Wω
0 the operators DQ0 acting on Ω
and Wω respectively.
By Lemmas 1.1and 4.1, we only have to show that elements of D((DQ,Ω
0 )∗) are in H1/2. Let
φ ∈ Cc∞ such that φ = 1 in a neighbourhood of the origin.
For u in the maximal domain D((DQ,Ω)∗), the functions φu and is in K(Ω ∩ supp φ) and
satisfies boundary conditions on ∂Ω ∩ supp φ. Analogously (1 − φ)u ∈ K(Ω ∩ supp(1 − φ)) and satisfies boundary conditions on ∂Ω ∩ supp(1 − φ) and so (1 − φ)u ∈ H1(Ω, C2) by the result for
smooth domains.
Let us focus on φu. We may construct a conformal map F , C2 up to the boundary, that
maps a subset of Wω to ∂Ω ∩ supp φ. To see this, just extend ∂Ω ∩ supp φ to a closed C2-curve
enclosing a simply connected domain eΩ . Then, using for instance [17, Theorem 3.9], one finds that the1 conformal map f from the unit disk to e
Ω, that maps ζ ∈ S1 to the corner of opening ω = πα satisfies that
f (z) − f (ζ) (z − ζ)α and
f′(z) (z − ζ)α−1,
are continuous functions for z in a neighbourhood of ζ in the closed unit disk. Without loss of generality, we assume ζ = −1. The explicit conformal transformation g(z) = z1/α−i
z1/α+i maps Wω to the unit disk, and the corner to −1. Thus F = f ◦ g is a conformal transformation from Wω
to eΩ, that maps the vertex of the wedge to the corner. A straightforward use of the chain rule shows that F and F′ have continuous extensions to the boundary of W
ω in a neighbourhood of
the origin. In particular, the conformal factor F′ is bounded in this neighbourhood.
By Proposition2.1, τFφu ∈ K(Wω) and satisfies boundary conditions. By Proposition3.1, we
conclude that, if ω ∈ (π, 2π)
τFφu = v + c1uQ0 + c2uQ−1,
with v ∈ H1. If ω ∈ (0, π), we conclude that τ
Fφu ∈ H1, immediately.
To finish the proof for non-convex corners, we just have to check that c2 = 0. This follows
from the observation that (τF)−1uQ0 ∈ D(D0Q,Ω). We can then use that φu ∈ D((D0Q,Ω)∗) in 1unique up to a dilation.
combination with (ii)in Proposition 2.1, 0 =D(τF)−1uQ0, (D Q 0)∗φu E L2(Ω)− D D0Q(τF)−1uQ0, φu E L2(Ω) =DuQ0, (DQ0)∗v + c1uQ0 + c2uQ−1 E L2(W ω)− D D0Qu0, (v + c1uQ0 + c2uQ−1 E L2(W ω) = −ic2
where the last line follows from (3.8).
4.2. Lorentz-scalar case. We will write DL,Ω
0 and D
L,Wω
0 to distinguish the operators D0L
acting on Ω and Wω respectively.
As before, we may restrict our attention to a neighbourhood of the corner. A first technical step is to construct a coordinate transformation that maps the curved boundary inside this boundary to a straight boundary. Having this transformation at hand, we also have to transform spinors so that the transplanted functions satisfy the boundary conditions on the new domain. This is achieved by means of point-wise multiplication by a matrix that, at the boundary points, equals eiσ3γ/2, where γ is the angle measuring the rotation to pass from the tangent vector to the curved boundary to the tangent vector at the boundary of the wedge. We will denote this transformation by U. The map U can be chosen to be unitary. Details of this transformation can be found in AppendixB.
The result of this rather technical construction is to set up a unitary equivalence between DL,Ω0 (after restriction to a neighbourhood of the corner), and an operator in the wedge, that decomposes as
(4.1) U DL,Ω0 U∗ = H + X
j=1,2
Lj(x)∂j+ M (x),
with Lj and M defined in (B.1). The matrices Lj depend on the difference between the Jacobian
matrix of the coordinate transformation and the identity. The matrix M is a multiplication operator containing first and second derivatives of the functions giving the transformation. By the C2-regularity of the boundary, M is bounded, and the the transformation can be chosen to
tend linearly to the identity when approaching the origin. A priori, the expression H+PjLj(x)∂j
has to be taken in distribution sense, where only the sum of both is well-defined on UD(DL,Ω 0 ).
What we will use in the following, is that
(4.2) kLj(x)kC2→C2 ≤ C|x|, C > 0.
We first check that UDL,Ω
0 U∗, given by the differential expression (4.1), is well-defined on
D(DL,Wω
0 )
Lemma 4.3. Let Ω be a corner domain and assume the origin is at a corner. Assume that u ∈ D(D0L,Ω) with support in B(0, R), where R > 0 is sufficiently small such that the origin is
the only corner in supp u. Then
k|x|∇u(x)kL2 ≤ ∞. Proof. First, we note that
where the second term is clearly finite. Since u is H1 away from the origin, we may use (1.9)
and (1.11)from Lemma 1.4to write, for any r > 0, Z R2\B(0,r)|∇|x|u(x)| 2dx = Z R2\B(0,r)|D L (|x|u(x))|2dx − Z ∂Ω\B(0,r)κ|s| 2(u +, σ3u+)C2(s) ds + Z ∂B(0,r)∩Ω r2(u+, iσ3∂θu+)C2(r, θ) dθ + Z ∂B(0,r)\Ω r2(u−, iσ3∂θu−)C2(r, θ) dθ. Since DL(|x|u(x)) = |x|DLu + |x|−1
σ · xu, the first term is bounded independently of r. The second term is bounded as well, by using the representation of the boundary traces given in the proof of Lemma 1.1. Indeed, from (A.1), T is bounded from Hs(Ω) to Hs−1/2(Ω) for all s ≤ 1,
and thus, u+∈ H1/2(Ω, C2) has boundary traces in L2(∂Ω, C2).
In order to estimate the contribution from the boundary of B(0, r), we average over r ∈ (0, t) and write 1 t Z t 0 Z ∂B(0,r)∩Ω r2|u+||∂θu+|(r, θ) dθ dr ≤ Z B(0,t)∩Ω|u+||∂θ u+|(r, θ)r dθ dr ≤ ku+k2H1/2.
The same argument works for u−. Putting everything together, we have shown that, for all t > 0,
1 t Z t 0 k∇|x|u(x)k 2 L2(R2\B(0,r)dr ≤ C(kD0L|x|u(x)kL2 + kukH1/2). Since k∇|x|u(x)k2
L2(R2\B(0,r)increases as r decreases, this shows that the limit at zero is finite.
The previous lemma shows that U maps D(DL,Ω0 ) unitarily into D(D L,Wω
0 ). We can now use
Theorem 1.6to conclude that u ∈ D(DL,Ω
0 ) decomposes as
u = v + c0U−1uL0
for some v ∈ H1 and c
0 ∈ C. This decomposition also allows to show that the second term in
(4.1) is relatively bounded with respect to the first one. Lemma 4.4. Let u ∈ D(DL,Wω 0 ). Then we have X j Lj(x)∂ju(x) L2 ≤ C sup x∈supp u|x| kDL,Wω 0 ukL2 + 2C kukL2, with C > 0.
Proof. By Proposition 3.1, any u ∈ D(DL,Wω
0 ) has a decomposition u = v + c1uL0 with v ∈ H1.
The key point is that the entries of |x|uL
0(x) behaves as rλ
L
0+1/2e±i(λL0−1/2)θand therefore, |x|u(x) is in H1. In addition, u satisfies boundary conditions, so |x|u ∈ D(DL). Now, we use (4.2) and
(1.11) with κ = 0 to bound X j=1,2 Lj∂ju L2
≤ Ck|x|∇ukL2 ≤ Ck∇(|x|u)kL2 + CkukL2 ≤ CkDL|x|ukL2 + CkukL2
≤ Ck|x|DL0ukL2 + 2CkukL2. Proof of Theorem 1.5. As explained above, we will set up an equivalence between the operator DL,Ω0 near a corner, and an operator eD acting in the straight wedge. In order to construct a unitary transformation, it is useful to define a curvilinear wedge near the corner.
Let φ ∈∞
c such that φ = 1 in a neighbourhood of the origin. Then
(4.3) H = φHφ +p1 − φ2Hp1 − φ2.
Replacing H by D0L,Ω in (4.3), we have that the second addend describes a self-adjoint operator, since the corner does not belong to the support of 1 − φ2.
Let us now focus on the first addend. Since we are considering only functions that are localized close to the corner, we can assume that Ω = Wω outside a sufficiently large neighbourhood of
the origin. Let U be the unitary transformation defined in Appendix B; by (4.1), (4.2) and Lemma 4.3we have UD(DL,Ω
0 ) = D(D L,Wω
0 ). In particular, from (4.2) we have that
(4.4) U φD0L,ΩφU∗ = eφDL,Wω
0 φ + ee φ
X
j=1,2
Lj∂lφ + ee φM eφ,
with eφ = φ ◦ S−1 and S the coordinate transformation defined in Appendix B. The last two terms of the right-hand-side of (4.4) are symmetric on D(DL,Wωj
0 ) since both DL,Ω0 and DL,W0 ω
are symmetric. In addition,
||eφLj∂jφu||e L2 ≤ ||Lj∂jφe2u|| + C||∇φ||L∞||u||
L2 ≤ C sup x∈supp φ|x| ! ||eφDL,Wω 0 φu||e L2 + 2C||∇φ||L∞||u|| L2,
where in the last line, we used Lemma 4.4. So we have that PjφLe j∂jφ + ee φM eφ is relatively
bounded with respect to eφDL,Wω
0 φ. Choosing φ with sufficiently small support, the relativee
bound can be made smaller than 1 and so, by the Kato-Rellich theorem, see [11, Theorem 4.3] for instance, we conclude that φDL,Ω
0 φ is unitarily equivalent to a self-adjoint operator.
Appendix A. Extension of boundary traces Proof of Lemma 1.1. Fix a bounded extension operator (see [1, Section 5])
E : H1/2(∂A, C2) 7→ H1(A, C2). For v ∈ K(A), we define TEv ∈ H−1/2(∂A, C2) by
(A.1) TEv[f ] = i hv, HEf iL2(A)− i hHv, Ef iL2(A).
By definition, TE is bounded from K(A) to H−1/2(∂A, C2). By (1.8), it coincides with σ·n γ
on H1(A, C2). Since H1(A, C2) is dense in K(A), η
be denoted by T to stress this. Alternatively, take a second extension operator E1. We have
(E − E1)f ∈ H01(Ω), so for all v ∈ K(A)
TEv[f ] − TE1v[f ] = i hv, H(E − E1)f iL2(A)− i hHv, (E − E1)f iL2(A)= 0.
We will prove the characterization of the domain of the adjoint for the Quantum-dot operator. The proof for the δ-shell interaction is analogous. First, we have that H1
0(Ω, C2) ⊂ D(DQ), so
for v ∈ D((DQ)∗)
(DQ)∗v = Hv, which implies that D((DQ)∗) ⊂ K(Ω).
Now take v ∈ K(Ω) satisfying boundary conditions. For all u ∈ D(DQ), γu = (1 − PQ η )γu ∈
H1/2(∂Ω). By (A.1),
DQu, v L2(Ω)− hu, HviL2(Ω)= 0, so v ∈ D((DQ)∗).
Finally, take v ∈ D((DQ)∗) and f ∈ H1/2(∂Ω, C2) such that PQ
η f ∈ H1/2(∂Ω, C2). Then
u ≡ E(1 − PηQ)f ∈ D(DQ) and therefore,
T v[(1 − PηQ)f ] = i
v, DQu L2(Ω)− i
(DQ)∗v, u L2(Ω)= 0.
Appendix B. Straightening of a curvilinear wedge
Throughout this section, we consider a domain that is bounded by a pair of semi-infinite curves of class C2, intersecting at an angle ω at the origin. We assume that the tangent and curvature
have left and right limits at the origin. Up to interchanging the interior and exterior, we can assume that ω ∈ (0, 2π).
Contrary to the previous convention, in this appendix, we will take y-axis oriented along the bisector of Wω. We will orient Ω in the same way, with the angle of opening ω at the origin.
Then, we assume that Ω is bounded by a pair of semi-infinite curves that admit a parametrization (x, c(x)) for x ≥ 0 and (x, c(x)), for x ≤ 0 respectively. The border of Wω is parametrized by
x,tan(ω/2)|x| . Consider the coordinate transformation S : (x, y) ∈ Ω 7→ S(x, y) ∈ Wω defined by
S(x, y) =
x, y − c(x) +tan(ω/2)|x|
.
Since the boundary of Ω is C2 except at the origin, where it is tangent to the wedge,
c′(x) − sign(x) tan(ω/2) ≤ |x| supR\{0}|c′′(x)|. The Jacobian matrix of S is
J(x, y) = 1 0
−c′(x) + tan(ω/2)sign(x) 1 !
= 1 + O(|x|). The relative angle of the rotation of the boundary tangent is
Now, for u = (u+, u−) ∈ L2(Wω) × L2(Wωc), we define USu ∈ L2(Wω) × L2(Wωc) by (USu)±(x, y) := eiδ(x)σ3/2u±(S(x, y)) = eiδ(x)/2 0 0 e−iδ(x)/2 u±(S(x, y)).
One checks that
e−iδσ3 0 e−iω/2 eiω/2 0 eiδσ3 = 0 e−i(ω/2+δ) ei(ω/2+δ) 0 .
If u± have boundary traces that satisfy Lorentz-scalar boundary conditions at the boundary of
∂Ω, we have that
(−iσ · nWω + µσ3) γ(USu)+− (−iσ · nWω− µσ3) γ(USu)− = eiδ(x)σ3/2
((−iσ · nΩ+ µσ3) γu+(S(x, y)) − (−iσ · nΩ− µσ3) γu−(S(x, y))) = 0.
We can also compute
HUS(u) =eiδ(x)σ3US(Hu) − i
−c′(x) + sign(x) tan(ω/2) σ1eiδ(x)σ3/2∂yu ◦ S + (δ′(x)/2)σ1eiδ(x)σ3/2.
Then, expanding the first exponential around x = 0, we have US∗HUS = H + (eiδ(x)σ3 − 1)H − i −c′(x) + sign(x) tan(ω/2) σ1eiδ(x)σ3∂y+ (δ′(x)/2)σ1eiδ(x)σ3. We recover (4.1), setting L1 := −i(eiδ(x)σ3 − 1)σ1, L2 := −i(eiδ(x)σ3 − 1)σ2− i −c′(x) + sign(x) tan(ω/2) σ1eiδ(x)σ3, M := (δ′(x)/2)σ1eiδ(x)σ3. (B.1) Acknowledgements
We would like to thank Luis Vega for the enlightening discussions. This work was partially developed while F. P. was employed at BCAM - Basque Center for Applied Mathematics, and he was supported by ERCEA Advanced Grant 2014 669689 - HADE, by the MINECO project MTM2014-53850-P, by Basque Government project IT-641-13 and also by the Basque Govern-ment through the BERC 2018-2021 program and by Spanish Ministry of Economy and Compet-itiveness MINECO: BCAM Severo Ochoa excellence accreditation SEV-2017-0718. He has also has received funding from the European Research Council (ERC) under the European UnionâĂŹs Horizon 2020 research and innovation programme (grant agreement MDFT No 725528 of Math-ieu Lewin). The work of H. VDB. has been partially supported by CONICYT (Chile) through PCI Project REDI170157 and Fondecyt Project # 318–0059.
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F. Pizzichillo, CNRS & CEREMADE, Université Paris-Dauphine, PSL Research University, F-75016 Paris, France
E-mail address: pizzichillo@ceremade.dauphine.fr
H. Van Den Bosch, Centro de Modelamiento Matemático, CMM, FCFM, Universidad de Chile, Beaucheff 851, Santiago, Chile
E-mail address: hvdbosch@cmm.uchile.cl
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