A Note on the Spectrum of Magnetic Dirac Operators
In this article, we study the spectrum of the magnetic Dirac operator and the magnetic Dirac operator with potential over complete Riemannian manifolds. We find sufficient conditions on the potentials as well as the manifold so that the spectrum is either maximal or discrete. We also show that magne...
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| description | In this article, we study the spectrum of the magnetic Dirac operator and the magnetic Dirac operator with potential over complete Riemannian manifolds. We find sufficient conditions on the potentials as well as the manifold so that the spectrum is either maximal or discrete. We also show that magnetic Dirac operators can have a dense set of eigenvalues.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 19 (2023), 102, 12 pages
A Note on the Spectrum of Magnetic Dirac Operators
Nelia CHARALAMBOUS a and Nadine GROSSE b
a) Department of Mathematics and Statistics, University of Cyprus, Nicosia, 1678, Cyprus
E-mail: charalambous.nelia@ucy.ac.cy
b) Mathematisches Institut, Universität Freiburg, 79100 Freiburg, Germany
E-mail: nadine.grosse@math.uni-freiburg.de
Received June 02, 2023, in final form December 14, 2023; Published online December 22, 2023
https://doi.org/10.3842/SIGMA.2023.102
Abstract. In this article, we study the spectrum of the magnetic Dirac operator, and
the magnetic Dirac operator with potential over complete Riemannian manifolds. We find
sufficient conditions on the potentials as well as the manifold so that the spectrum is either
maximal, or discrete. We also show that magnetic Dirac operators can have a dense set of
eigenvalues.
Key words: Dirac operator; potentials; spectrum
2020 Mathematics Subject Classification: 58J50; 35P05; 53C27
1 Introduction
In this article, we consider the magnetic Dirac operator, and the magnetic Dirac operator with
potential over a complete Riemannian manifold with an associated Clifford bundle. Our main
goal is to study the spectral properties of these operators depending on the behavior of both
the magnetic potential and the additional potentials. In particular, we are interested in finding
sufficient conditions on the potentials so that the essential spectrum of the operator is either
maximal, in other words R, or discrete.
The spectral properties of magnetic Schrödinger operators, over R2 and R3 have been exten-
sively studied, see, for example, [7, 10, 20] and references therein. In this case, the operators
are classical Schrödinger operators (the Laplacian with a scalar-valued potential) plus a mag-
netic field which is a vector field acting via some Clifford-type action on C2-valued L2-integrable
functions over the Euclidean space. In [7], Cycon, Froese, Kirsch and Simon show that when-
ever the potential is relatively compact with respect to the Laplace operator and the magnetic
field vanishes at infinity, the spectrum of the magnetic Schrödinger operator coincides with the
spectrum of the Laplacian and is [0,∞), it is in other words maximal [7, Theorem 6.1]. On
the other hand, Miller and Simon show that depending on the decay rate of the magnetic field,
the Hamiltonian operator of a spinless particle can have purely absolutely continuous spectrum,
dense point spectrum in [0,∞), or dense point spectrum in a closed interval and absolutely
continuous spectrum in its complement [13] (see also [7, Theorem 6.2]). The case of ‘pure’
Schrödinger operators over Euclidean spaces was extensively studied (see, for example, [1, 22]).
The case of the Dirac operator with varying types of potentials over Euclidean spaces was
considered in [11, 24]. General notes on the occurrence and meaning of different type of potentials
for the Dirac operator in R1,3 and discussions on self-adjointness can be found in [23, Sections 4.2
and 4.3], see also Remark 2.3. Yamada showed for example in [24] that if the mass-type potential
This paper is a contribution to the Special Issue on Global Analysis on Manifolds in honor of Christian Bär
for his 60th birthday. The full collection is available at https://www.emis.de/journals/SIGMA/Baer.html
mailto:charalambous.nelia@ucy.ac.cy
mailto:nadine.grosse@math.uni-freiburg.de
https://doi.org/10.3842/SIGMA.2023.102
https://www.emis.de/journals/SIGMA/Baer.html
2 N. Charalambous and N. Große
becomes unbounded at infinity and its derivative as well as the one of the electric potential do
not grow faster than the potential itself, then the spectrum will be purely discrete.
While several results for Schrödinger operator both with scalar potential and with magnetic
fields were transferred to manifolds, see, e.g., [19, 21], to the best of our knowledge little has
been done for magnetic Dirac operators over complete Riemannian manifolds and their spectrum.
Over compact odd-dimensional spin manifolds, with certain restrictions on the eigenvalues of
the contact endomorphism, Savale proves Weyl-type of estimates for the number of eigenvalues
of the coupled Dirac operator when the magnetic form is a contact 1-form, and he also provides
a limit formula for the eta invariant of the magnetic Dirac operator when the magnetic potential
is a non-resonant contact form in [17, 18]. In the context of compact manifolds with boundary,
spectral estimates for Callias-type Dirac operators that can be interpreted as Dirac operators
with mass-type potentials were studied for example by Cecchini and Zeidler in [3].
In this short note, we start by generalizing the small selection of the theorems in [7] and [24]
which we mentioned above, to a more general class of complete Riemannian manifolds with
Dirac-type operators.
First, we give some preliminaries on the different types of potentials for Dirac operators. We
restrict ourselves to potentials that are functions or one-forms acting via Clifford multiplication
as in Remark 2.1. Then, in Theorem 3.1 we prove that the spectrum of the magnetic Dirac
operator is discrete if the mass-type potential goes to infinity at infinity and dominates the
magnetic and electric potential as well as the derivatives of all these potentials. On the other
hand, in Theorem 3.3 we show that a pure magnetic Schrödinger operator over asymptotically
flat manifolds with magnetic field vanishing at infinity has essential spectrum [0,∞). Moreover,
we will see in Theorem 3.4 that similarly to magnetic Schrödinger operators, magnetic Dirac
operators can also have dense eigenvalues. Our results illustrate that the vast variability in
the behavior of the spectrum of Schrödinger operators, can be generalized to magnetic Dirac
operators with potential over complete manifolds. Certainly, there are many intermediate cases
depending on the behaviour of the various potentials at infinity which are not discussed here,
and which could merit further investigation.
2 Preliminaries
We consider a complete Riemannian manifold (Mm, g). Let S →M be a Clifford bundle. That
means S is a bundle of Clifford modules over M which has a fiberwise Hermitian metric ⟨·, ·⟩
and a metric connection ∇S on S such that the Clifford action of a tangent vector is skew-
adjoint with respect to ⟨·, ·⟩, and ∇S is compatible with the Levi-Civita connection on M , cf.
[16, Definition 3.4]. The sections of S are called spinors and the space of smooth section will be
denoted by Γ(S). The Riemannian metric induces an L2-inner product on spinors over Mn by
(φ, η) :=
∫
M
⟨φ, η⟩ dv,
where dv is the Riemannian measure.
Moreover, there is an associated Dirac operator acting on the sections of S which we denote
by D. When M is a spin manifold and S is the associated spinor bundle, the associated Dirac
operator is known as the classical Dirac operator. It is well known that the square Dirac operator
satisfies the Weitzenböck formula
D2 = ∇∗∇+R,
where R ∈ End(S) is the Clifford contraction which acts as a tensor on spinors. When M is
spin and D is the classical Dirac operator, the Clifford contraction is simply a constant multiple
A Note on the Spectrum of Magnetic Dirac Operators 3
of the scalar curvature of the manifold, R = scal/4. The Weitzenböck formula allows us to treat
the square Dirac operator as a Schrödinger-type of operator, with the Clifford contraction as its
potential. Controlling this potential allows us to then obtain analytical and spectral properties
for the operator. When one considers the classical Dirac operator, this process reduces to
controlling the scalar curvature of the manifold.
Next we present the different types of potentials. We introduce the notation step by step,
since for magnetic and electric potentials less structure is needed than for mass-type potentials.
2.1 Magnetic Dirac operators
Let (Mm, g) be a Riemannian manifold with Clifford bundle S and associated Dirac operator D.
Let A ∈ Ω1(M) be a smooth real-valued one-form on M .
The magnetic Dirac operator with magnetic potential A is defined as
DA := D + iA· : Γ(S) → Γ(S)
and the spinorial magnetic connection Laplacian as
HA :=
(
∇S + iA
)∗(∇S + iA
)
: Γ(S) → Γ(S),
where A : ϕ ∈ Γ(S) 7→ A ⊗ ϕ ∈ Γ(T ∗M ⊗ S) and A· : ϕ ∈ Γ(S) 7→ A · ϕ ∈ Γ(S) is given by the
Clifford multiplication.
Remark 2.1. We note that adding the iA-term is a special type of change of connection
for the bundle: In general, if ∇1 and ∇2 are two connections on the same bundle S, then
∇1 − ∇2 = ω ∈ Ω1(M,EndS). Adding iA corresponds to the special choice ωs := iA ⊗ s for
all s ∈ Γ(S). Since ⟨iA(X)ϕ, ψ⟩ = −⟨ϕ, iA(X)ψ⟩ for all p ∈ M , X ∈ TpM and ϕ, ψ ∈ Sp,
the connection ∇ + iA is also metric. Moreover, it is still compatible with the Clifford action.
The spinc construction is related to the above setting. In that case A is the connection on an
auxiliary line bundle which is tensored to the (locally) spinor bundle. In principle, we could
also generalize the potentials used here and in the following by using bundles constructed from
twisting the spinor bundle. But here we restrict to the case of no additional bundle.
The operators DA and HA are essentially self-adjoint on complete manifolds [5, pp. 411–412].
Next we calculate the Lichnerowicz formula (see, for example, [12, 16]) for the explicit choice
of the Dirac operator which we will be using in this article.
Lemma 2.2. It is
D∗
ADA = HA +R+ i dA · and HA =
(
∇S
)∗∇S − 2i∇S
A − i divA+ |A|2,
where R ∈ End(S) is the Clifford contraction and where for a local orthonormal frame ei of TM
the action of dA· is given by
dA · ϕ :=
∑
i<j
dA(ei, ej)ei · ej · ϕ.
We note that HA is not of standard Schrödinger type, since a first derivative ∇S
A appears.
4 N. Charalambous and N. Große
Proof. We perform a local calculation at a given point p ∈ M . In a neighbourhood of p, we
choose a local orthonormal frame {ei}i such that ∇eiej(p) = 0 for all i, j. Let A =
∑
iAie
♭
i,
where
{
e♭i
}
i
denotes the dual cotangent frame. Using the fact that D∗
A = DA, we compute,
pointwise,
D∗
ADAϕ = (D + iA·)(D + iA·)ϕ
= D∗Dϕ+ iei · ∇S
ei(Ajej · ϕ) + iA ·Dϕ−
∑
i,j
AiAjei · ej · ϕ
=
(
∇S
)∗∇Sϕ+Rϕ+ i dA · ϕ− i divAϕ− 2i
∑
i
Ai∇S
eiϕ+ |A|2ϕ.
Note, that A∗(ei ⊗ ϕ) = Aiϕ and
(
∇S
)∗
(ei ⊗ ϕ) = −∇S
eiϕ. Hence,
HAϕ =
(
∇S + iA
)∗(∇S + iA
)
ϕ =
(
∇S
)∗∇Sϕ− 2i
∑
i
Ai∇S
eiϕ− i divAϕ+ |A|2ϕ. ■
2.2 With electric potential
Instead of only working with magnetic potentials we can also add an electric potential A0, in
other words a function on M and consider the more general operator
DA,A0 = DA +A0.
Remark 2.3. Above A is interpreted as the magnetic potential and A0 as the electric potential
which can be seen as follows: If we work on R1,3, with coordinates x0, . . . , x3 the (physical)
Dirac operator with potential A = Aµdx
µ looks like iγµ(∂µ + iAµ) (by borrowing the notation
from physics using gamma-matrices, cf. [23]). A spinor in its kernel satisfies iγ0∂tϕ =
(
−iγ⃗∇⃗+
γ⃗A⃗+ γ0A0
)
ϕ, where the right hand-side is DA,A0ϕ as defined above. If A itself does not depend
on time, A0 gives the part of the electric field in F = dA and (A1, A2, A3) the magnetic field
(also compare to [23, Section 4.2]).
Remark 2.4. Let V be a zero order symmetric operator on S acting as (V ϕ)(x) = V (x)ϕ(x)
with V (x) ∈ Hom(Sx, Sx). Assume that for each point p ∈M there is a Ṽp ∈ L2
loc that coincides
with V in a neighbourhood of p and such that DA + V is essentially self-adjoint on C∞
c (S).
Then by [6, Theorem 2.1], DA,V is essentially self-adjoint.
In particular, this is true for V smooth [5, Theorem 2.2]. In case the potential is singular,
the essential self-adjointness depends both on how fast the singularity blows up as well as the
coupling constant, in other words the scalar multiple in front of it (see [23, Section 4.3.3]). Here
we will only consider C1-potentials for simplicity.
2.3 With (mass-type) potential
Let again (Mm, g) be a Riemannian manifold with Clifford bundle S and associated Dirac
operator D.
Assume that S is Z2-graded, in other words there is a parallel and orthogonal Z2-grading
S = S+ ⊕S− and the Clifford multiplication by a tangent vector is an odd map with respect to
this grading. Then the associated Dirac operator has the form
D =
(
0 D−
D+ 0
)
A Note on the Spectrum of Magnetic Dirac Operators 5
with respect to this splitting. Moreover, we define ν · (ϕ, ψ) = i(ϕ,−ψ). This corresponds to
a choice of representation of the Clifford action as in [9, 23]. Then, ν· commutes with Clifford
multiplication for all X ∈ Γ(TM), since it is an odd map. Then we define for V ∈ L2
loc,
DA,V,A0 := D + iA ·+iV ν ·+A0 : Γ
(
S̃
)
→ Γ
(
S̃
)
.
If A0 = 0, we write DA,V := DA,V,0. If A = 0 and A0 = 0, we write DV := D0,V,0.
Remark 2.5. Let M be spin (with a chosen spin structure) and S the spinor bundle. In case
dimM is even, the spinor bundle is automatically Z2-graded. Then ι
∗S, ι : M ↪→M ×R, is the
spinor bundle of M × R and the ν· from above is the Clifford multiplication by the unit vector
in the R direction. In case dimM is odd, then S̃ := S ⊕ S and X · (ϕ, ψ) = (X · ϕ,−X · ψ) for
X ∈ Γ(TM) defines a Z2-graded Clifford bundle as above, cf. [9, Section 1.4].
For R1,3, in other words for the manifold M = R3 with C4-valued spinors, the V -term
above corresponds to the scalar potential in [23, Section 4.2.1]. Over Riemannian manifolds the
operator D + iV ν· is often referred to as a Callias-type operator, see [3].
For the essential self-adjointness Remark 2.4 is applicable.
Lemma 2.6. For V ∈ C1(M), we have D∗
A,VDA,V = D∗
ADA + i dV · ν ·+V 2.
Proof. It is
D∗
A,VDA,V = D∗
ADA − V A · ν · −V ν ·A ·+iV ν ·D + iD(V ν·) + V 2
= D∗
ADA + i dV · ν ·+V 2,
where we use the anticommutativity between D and Clifford multiplication by ν in the last
line. ■
Remark 2.7. If V = const, then iV ν· can be interpreted as a mass term and DA,V as a massive
(magnetic) Dirac operator. In this case, the term dV in D∗
A,VDA,V vanishes and the operator
becomes a truly Schrödinger-type operator. Thus, in general, such a potential can be interpreted
as a position-dependent mass term.
2.4 Diamagnetic inequality
For a metric connection ∇ on a vector bundle, we have |d|ϕ|| ≤ |∇ϕ| for all ϕ ∈ C∞(S) and
at all points where ϕ ̸= 0. Since ∇ + iA is just another metric connection on the bundle, the
following diamagnetic inequality holds:
|d|ϕ|| ≤ |(∇+ iA)ϕ|.
3 Spectral results
In this section, we will explore whether and how the results of [7, Section 6] and [24] generalize
to complete manifolds under appropriate conditions.
3.1 Potentials and discrete spectrum
It is known that the self-adjoint Schrödinger operator HV :=
(
∇S
)∗∇S + V 2 : domHV ⊂
L2(S) → L2(S) has discrete spectrum whenever the potential term blows up at infinity (condi-
tion (i) in Theorem 3.1) since the quadratic form associated to the operator can be made as large
as possible on the complement of a large enough ball (see, for example, [15, Theorem XIII.16]).
6 N. Charalambous and N. Große
In this section we show that this result also holds for the magnetic Dirac operator with potential
under the additional assumption that the magnetic potential is bounded and that the derivative
of the magnetic potential and the potential A0 is dominated by V .
We note that having a magnetic and/or electric potential that goes to infinity at infinity is
not enough to ensure that the spectrum is discrete.
In the following let |x| := d(x, p) for a fixed p ∈M .
Theorem 3.1. Assume that M is complete, connected and noncompact. Let S be a Z2-graded
Clifford bundle as in Section 2.3. Let V,A0 ∈ C1(M) and A ∈ Ω1(M). Assume
(i) lim|x|→∞ |V (x)| = ∞;
(ii) R and A are both bounded;
(iii) |dV |+ |dA|+ |dA0|+ |divA| = O(V ) as |x| → ∞;
(iv) |A0| ≤ ϵ|V | on |x| ≥ R0 for an ϵ < 1 and some R0 > 0.
Then the spectrum of DA,V,A0 = D + iA ·+iV ν ·+A0 is discrete.
This generalizes a result forM = R3 with D = /D by [24], and our proof follows the same idea
as in this paper. In [24], only an electric potential and a mass-type potential are considered.
Proof. In order to prove the theorem, we will show that (DA,V,A0 − i)−1 is compact. For that
let ϕn ∈ L2(S) be a bounded sequence and set ψn := (DA,V,A0 − i)−1ϕn ∈ H1
loc(S). Since DA,V,A0
is essentially self-adjoint, its spectrum is real and hence (DA,V,A0 − i)−1 is bounded. Thus, ψn is
a bounded sequence in L2. We need to show that this sequence has a convergent subsequence.
As mentioned above HV :=
(
∇S
)∗∇S + V 2 − A2
0 has discrete spectrum. By [15, Theo-
rem XIII.64], the set
Sb =
{
ϕ ∈ domHV ⊂ L2(S) | ∥ϕ∥L2 ≤ 1,
(
ϕ,
((
∇S
)∗∇S + V 2 −A2
0
)
ϕ
)
≤ b
}
is compact for all b. We want to show that the sequence ψn is in a set Sb for some b, then our
claim follows by the compactness of this set.
We apply DA,V − A0 + i = DA,V,−A0 + i on both sides of DA,V,A0ψn − iψn = ϕn, and use
Lemmas 2.6 and 2.2, as well as the self-adjointness of DA,V to obtain((
∇S
)∗∇S − 2i∇S
A + |A|2 +R− i divA+ i dA·
+ 1 + i dV · ν ·+V 2 −A2
0 + 2A0i + dA0·
)
ψn = (DA,V,−A0 + i)ϕn.
Define H :=
{
ϕ ∈ L2(S) | ∥ϕ∥2V := ∥ϕ∥2H1 +
∥∥√V 2 −A2
0ϕ
∥∥2
L2 < ∞
}
. Then H ⊂ H1(S) is
a Hilbert space with inner product (ϕ, ψ)V := (ϕ, ψ)H1 +
(
(V 2 − A2
0
)
ϕ, ψ)L2 . By (iv), V 2 ≤
a
(
V 2 − A2
0
)
≤ aV 2 for some a > 1. Thus, the norms ∥ϕ∥V and (∥ϕ∥2H1 + |V |∥ϕ∥2L2)
1
2 are
equivalent.
By assumptions (iii) and (iv), there exist R > R0 and C > 0 such that (|dV | + |dA| +
| divA| + |dA0| + |A0|)(x) ≤ CV (x) for all |x| ≥ R. Let η be a smooth cut-off function with
η(x) = 1 for |x| ≥ R+ 1 and η(x) = 0 for |x| ≤ R. Let u ∈ C∞
c (S). Then
(ηψn, u)V =
((
∇S
)∗∇S(ηψn), u
)
L2 +
((
1 + V 2 −A2
0
)
ηψn, u
)
L2
=
(
η
(
∇S
)∗∇Sψn, u
)
L2 +
((
1 + V 2
)
ηψn, ψ
)
L2 +
(
ψn∆η − 2∇S
∇ηψn, u
)
L2
=
(
ϕn, (DA,V,−A0 − i)(ηu)
)
L2 +
(
ψn∆η − 2∇S
∇ηψn, u
)
L2
+
((
2i∇S
A − |A|2 −R+ i divA− i dA · −2A0i− dA0 · −i dV · ν·
)
ψn, ηu
)
L2
=
(
ϕn, (DA,V,A0 − i)(ηu)
)
L2 +
(
ψn, u∆η + 2∇S
∇ηu
)
L2
A Note on the Spectrum of Magnetic Dirac Operators 7
+
(
ψn, (2i∇S
A − i divA− i dA ·+2A0i + dA0 · −i dV · ν · −R − |A|2)(ηu)
)
L2 .
Using our assumptions together with
∣∣∇S
Xu
∣∣ ≤ |X|
∣∣∇Su
∣∣, we obtain
|(ηψn, u)V | ≲ ∥ϕn∥L2∥(DA,V,−A0 − i)(ηu)∥L2 + ∥ψn∥L2∥ηu∥2V
and, as a result
(ηψn, u)V ≤ C(∥ϕn∥+ ∥ψn∥)∥u∥V ≤ C ′∥u∥V .
Thus, ∥ηψn∥V is bounded for all n. Hence, this sequence lies in the compact subset Sb for some b
and therefore has a convergent subsequence in L2(S). ■
3.2 Magnetic Schrödinger operator with vanishing magnetic field at infinity
In this section, we prove that the essential spectrum of a magnetic Schrödinger operator with
vanishing magnetic field at infinity will be maximal over asymptotically flat manifolds. Our
result generalizes [7, Theorem 6.1] who considered the case of Euclidean space. The main idea is
to compare the spectrum of the magnetic Schrödinger operator to that of certain gauge perturbed
operators, using the fact that the magnetic field vanishes at infinity. Since the spectrum of the
Laplacian on functions is maximal over asymptotically flat manifolds, and we have ‘nice’ spinors
over this space, we can then find an appropriate family of approximate eigenspinors to prove
that the essential spectrum of our operator is also maximal.
Definition 3.2. A manifold (Mm, g) with m ≥ 2 is asymptotically flat of order τ if there is
a compact set K ⊂M and a diffeomorphism φ : M \K → Rm \Bro(0) such that
(φ∗g)ij(x) = δij +O
(
|x|−τ
)
and
∂k(φ∗g)ij(x) = O
(
|x|−(τ+1)
)
, ∂kl(φ∗g)ij(x) = O
(
|x|−(τ+2)
)
for some τ > 0, where |x| is the Euclidean distance of x to 0.
Note that under the above assumption on the metric, the curvature tensor of the manifold
and hence its scalar curvature tend to zero as |x| → ∞.
To prove that the spectrum of the magnetic Schrödinger operator is maximal requires, as
we will see, the construction of a large class of L2 integrable approximate eigenspinors. Even
though the spectrum of the Laplacian on functions would be [0,∞) over these manifolds [4], we
cannot use the bounded test functions with L1 estimates constructed in that article. Instead, we
use the idea of [7] choosing gauges such that the magnetic potential becomes small on growing
balls, as well as the volume doubling property which holds for asymptotically flat manifolds.
This volume doubling property is defined as follows: There is an Ro > ro such that for all R > 0
and p ∈M \ φ∗(Rn \B3R+Ro(0)
)
,
volB2R(p) ≤ C volBR(p), (3.1)
where C is a uniform constant, independent of p and R. For asymptotically flat manifolds
this follows completely analogously to the estimates in [14, Proof of Proposition 1]. The only
difference between our volume doubling property and the statement in [14] is that the balls in
the latter are all centered at zero.
Theorem 3.3. Let (Mm, g) be an asymptotically flat spin Riemannian manifold of order τ > 0,
with Dirac operator D. Let A ∈ Ω1(M) such that |B = dA| → 0 as |x| → ∞, in other words the
magnetic field vanishes at infinity. Then, [0,∞) = σess(D
∗
ADA). In particular, R = σess(DA) in
dimension 0, 1, 2 mod 4.
8 N. Charalambous and N. Große
Proof. First, we will argue that W = scal
4 + iB· is relatively compact to HA, in other words
W (HA+1)−1 is a compact operator on L2. The argument is mainly analogous to [15, Example 6,
p. 117 and Problem 41, p. 369] but will be given here for completeness. We approximate
W = scal
4 + iB· : L2 → L2, by Wn = χBn(p)W where χBn(p) is the characteristic function of
the geodesic ball Bn(p) ⊂ M . Since the scalar curvature and the magnetic field B tend to
zero at infinity, limn→∞ ∥W − Wn∥ → 0 in the operator norm. Let now ϕk be a bounded
sequence in domHA = {ϕ ∈ L2 | HAϕ ∈ L2} equipped with the graph norm. Then, ϕk is locally
also bounded in H1 and hence locally strongly converges in L2 to some ϕ (after possibly taking
a subsequence). In particular,Wnϕk converges toWnϕ = χBn(p)Wϕ in L2. This implies thatWn
is compact as an operator from domHA (equipped with the graph norm) to L2 and hence, that
Wn(HA + 1)−1 is compact. Since limn→∞ ∥W −Wn∥ → 0, then also W (HA + 1)−1 is compact.
By [15, Corollary 2, p. 113], we get σess(D
∗
ADA) = σess(HA).
Next, we want to show that [0,∞) ⊂ σess(HA). As in [7, Step 3, p. 117], we want to use
the following fact. Let T ≥ 0 be a self-adjoint operator on a Hilbert space H. Whenever
there exists an orthonormal sequence ψn in H which converges weakly to zero and satisfies∥∥(T + 1)−1(T − λ)ψn
∥∥ → 0 then λ ∈ σess(T ).
Before we find the test functions ψn, we first exploit the condition that the magnetic field
vanishes at infinity.
For any p ∈ M we define |p| = |φ(p)|. Since |B| → 0, there is a sequence of points qn ∈ M
with |qn| → ∞ and supx∈B2n(qn) |B| ≤ 1
n2 . By taking a subsequence, we may choose qn such
that |qn| > 6n+Ro.
Since the spinor bundle is trivial on M \K, we can choose a spinor Φ that is constant with
norm 1 on M \K. We note that by our assumptions on the decay of the metric, and hence for
the Christoffel symbols corresponding to our coordinates, we get∣∣∇SΦ
∣∣(x) = O
(
|x|−(τ+1)
)
and
∣∣(∇S
)∗∇SΦ
∣∣(x) = O
(
|x|−(τ+2)
)
for |x| → ∞ (see [12, Section 4] for the spinorial connection ∇S in local coordinates).
Now let gn : M → [0, 1] be a radial (with respect to qn) smooth cut-off function which is
constant on Bn(qn), 0 on M \B2n(qn) and satisfies |∇gn| ≤ 2
n and |∆gn| ≤ 4
n .
So far we know that B is small on the support of gn, but the term A also appears in HA
on its own. Next we choose another vector potential An corresponding to the same magnetic
field B which is small on Bn(qn).
In local coordinates, we have B = Bµνdx
µ ∧ dxν with Bµν = −Bνµ. Then 0 = d2A = dB =
∂κBµνdx
κ ∧ dxµ ∧ dxν . Thus,
∂κBµν − ∂µBκν − ∂νBµκ = 0.
Using the asymptotically flat coordinates on B2n(qn), shifted such that qn has coordinates
zero, we define on B2n(qn),
An
µ(x) := −
∑
ν>µ
∫ xν
0
Bµν
(
x1, . . . , xν−1, t, 0, . . . , 0
)
dt.
Then
∂κA
n
µ(x) = −
∑
ν>min{µ,κ}
∫ xν
0
∂κBµν
(
x1, . . . , xν−1, t, 0, . . . , 0
)
dt
+ δµ>κBµκ
(
x1, . . . , xκ, 0, . . . , 0
)
.
A Note on the Spectrum of Magnetic Dirac Operators 9
Thus for κ < µ,
∂κA
n
µ(x)− ∂µA
n
κ(x) = −
∑
ν>κ
∫ xν
0
(∂κBµν − ∂µBκν)︸ ︷︷ ︸
=∂νBµκ
(
x1, . . . , xν−1, t, 0, . . . , 0
)
dt
+Bµκ
(
x1, . . . , xκ, 0, . . . , 0
)
= −
∑
ν>κ
(
Bµκ
(
x1, . . . , xν , 0, . . . , 0
)
−Bµκ
(
x1, . . . , xν−1, 0, . . . , 0
))
+Bµκ
(
x1, . . . , xκ, 0, . . . , 0
)
= −Bµκ(x) = Bκµ(x).
Thus An = An
µdx
µ is an equivalent potential to A on B2n(qn) and by construction it satisfies
|A| ≤ c 2n for some uniform constant c > 0.
Hence, by the Poincaré lemma, An − A = dfn for a smooth function fn on B2n(qn). Let f̃n
be a smooth function on all of M that coincides with fn on Bn(qn) and set Ãn = A+df̃n. Then
HA = eif̃nHÃn
e−if̃n .
For k ∈ Rm, we define ϕn = eikµx
µ
gnΦ. By our assumptions on gn and Φ, we have
∥ϕn∥2 ≥ Co volBn(qn)
for a uniform constant Co. Setting λ = |k|2, we obtain
(H0 − λ)ϕn(x) =
(
(∆g − λ)
(
g̃ne
ikµxµ))
Φ− 2∇S
∇(g̃ne
ikµxµ )
Φ+ g̃n
(
∇S
)∗∇SΦ
=
(
eikµx
µ
∆g g̃n + 2
〈
∇g̃n,∇eikµx
µ〉
+ g̃n(∆g − |k|2)eikµxµ)
Φ
− 2∇S
∇(g̃ne
ikµxµ )
Φ+ g̃n
(
∇S
)∗∇SΦ.
By the decay estimates of ∆g,∇g̃n and the uniform upper and lower bound on Φ and gn, we get∥∥(eikµxµ
∆g g̃n + 2
〈
∇g̃n,∇eikµx
µ〉)
Φ
∥∥2 ≤ C
n
volB2n(qn) ≤
C
n
volBn(qn) ≤
C
n
∥ϕn∥2,
where C is a constant that changes from line to line but is always uniform in n. Similarly, the
upper bounds on the derivatives of Φ give∥∥∇S
∇(g̃ne
ikµxµ )
Φ+ g̃n
(
∇S
)∗∇SΦ
∥∥2 ≤ C
n(τ+1)
volB2n(qn) ≤
C
n(τ+1)
volBn(qn) ≤
C
n
∥ϕn∥2.
Moreover,(
∆g − |k|2
)
eikµx
µ
= eikµx
µ(
kµkν
(
gµν − δµν
)
− ikµg
µν
,ν − ikνg
λν∂λ(ln det g)
)
.
By the asymptotic flatness of M we get∣∣(∆g − |k|2
)
eikµx
µ∣∣ = O
(
|x|−τ
)
.
Together with the volume estimate above and the choice of qn, this implies∥∥(∆g − |k|2
)
eikµx
µ∥∥2 ≤ C(|qn| − 2n)−τ volB2n(qn) ≤ Cn−τ∥ϕn∥2
in a similar way as above. Combining all of the above we get that for any ϵ > 0 and we can
find n large enough such that
∥(H0 − λ)ϕn∥ ≤ ϵ∥ϕn∥.
Set ψn(x) := eif̃n(x)eikµx
µ
gn(x)Φ/∥ϕn∥. By construction, ψn is an orthonormal sequence.
10 N. Charalambous and N. Große
We have∥∥(HA + 1)−1(HA − λ)ψn
∥∥ =
∥∥(HÃn
+ 1)−1(HÃn
− λ)ϕn
∥∥.
Using HÃn
= H0 − 2i∇Ãn
− i div Ãn + |Ãn|2 = H0 + i(∇+ iÃn)
∗Ãn − iÃn∇, we obtain∥∥(HA + 1)−1(HA − λ)ψn
∥∥ ≤
∥∥(HÃn
+ 1)−1
∥∥ · ∥(H0 − λ)ϕn∥
+
∥∥(HÃn
+ 1)−1
(
∇+ iÃn
)∗∥∥ ·
∥∥∣∣Ãn
∣∣g̃n∥∥+∥∥(HÃn
+ 1)−1
∥∥ ·
∥∥∣∣Ãn
∣∣∇(g̃nΦ)
∥∥.
Since HÃn
is a nonnegative operator, by the spectral theorem (HÃn
+ 1)−1 is bounded on L2
uniformly in n. Similarly, we can show that (HÃn
+1)−1HÃn
is bounded on L2, uniformly in n.
Thus, also (HÃn
+ 1)−1(∇ + iÃn)
∗ is bounded on L2 uniformly in n. Since Ãn|B2n(qn) → 0 as
n→ ∞ and using again the properties of g̃n, Φ as before, we obtain
∥∥(HA+1)−1(HA−λ)ψn
∥∥ → 0
and, hence, that λ ∈ σess(HA).
The remaining claim on the spectrum of DA follows from to the symmetry of the spectrum in
these dimensions. The symmetry of the spectrum D itself, follows form even by ω ·Dϕ = −D(ω ·
ϕ) by ω = e1 · e2 · · · edimM , and for m ≡ 1 mod 4 by the existence of a Spin(m)-equivariant real
structure that anticommutes with Clifford multiplication, cf. [8, Proposition, p. 31]. Since in even
dimensions ω· also anticommutes with Clifford multiplication as does the real structure just men-
tioned, DA anticommutes with these maps as well and has, as a result, symmetric spectrum. ■
3.3 Dense eigenvalues
In [13], see also [7, Theorem 6.2], an example of a magnetic potential A, such that the magnetic
Schrödinger operator (∇+ iA)∗(∇+ iA) on functions has pure point spectrum which is dense in
[0,∞) was given. The magnetic potential used there was
A = − y
(1 + ρ)γ
dx+
x
(1 + ρ)γ
dy,
where ρ = ρ(x, y) =
(
x2 + y2
) 1
2 and γ ∈ (0, 1). We will prove here that this phenomenon
still occurs for the Dirac operator (with the same magnetic potential). We note that
(
on R2
)
D∗
ADA = (∇ + iA)∗(∇ + iA) + idA·, therefore one cannot deduce the spectrum directly from
the result on the Schrödinger operator. But we can run an argument using similar ideas. In [7,
Theorem 6.2], it is used that (∇+ iA)∗(∇+ iA) commutes with the orbital angular momentum
which is not true for DA. But this can be circumvented by adding the spin angular momentum
as we will see in the following result.
Theorem 3.4. On R2 we consider the magnetic potential A provided above. Then, the magnetic
Dirac operator DA = D + iA has pure point spectrum which is dense in R.
Proof. We consider the total angular momentum Jz = Lz +Sxy, where Lz = −i
(
x ∂
∂y − y ∂
∂x
)
is
the orbital angular momentum and Sxy = 1
2 diag(1,−1) is the spin angular momentum. Direct
calculation gives [DA, Jz] = 0 (but Lz itself does not commute with DA).
Since divA = 0, we have
D∗
ADA = ∇∗∇+ |A|2 − 2i∇A + idA(e1, e2)e1 · e2 · .
Using ie1 · e2· = diag(1,−1), dA = Bdx∧dy with B = − γρ
(1+ρ)γ+1 +
2
(1+ρ)γ and −i∇A = 1
(1+ρ)γLz
this gives
D∗
ADA = ∆+
2
(1 + ρ)γ
Jz +
ρ2
(1 + ρ)2γ
+
(
B − 1
(1 + ρ)γ
)
diag(1,−1).
A Note on the Spectrum of Magnetic Dirac Operators 11
The operator Jz has discrete spectrum. We denote the eigenvalues by µm, m ∈ Z. The dis-
creteness of the spectrum for Jz allows us to write H = L2
(
R2,C2
)
as the direct sum of the
eigenspaces Em corresponding to the eigenvalue µm. At the same time, the fact that [DA, Jz] = 0
allows one to claim that DA restricts to Em. Then,
D∗
ADA|Em = diag
(
T+
m , T
−
m
)
with T±
m = ∆ + 2
(1+ρ)γ µm + ρ2
(1+ρ)2γ
±
(
B − 1
(1+ρ)γ
)
. Thus, T±
m is an operator on L2
(
R2,C
)
of Schrödinger-type, in other words the Laplacian on functions plus a scalar potential. For
γ ∈ (0, 1), the term ρ2
(1+ρ)2γ
is the dominating term of this potential at infinity. Thus, this poten-
tial goes to infinity at infinity and hence, T±
m and thus also D∗
ADA|Em have discrete spectrum.
If D∗
ADA has discrete spectrum, so does DA (see, e.g., [2, Lemma B.8]).
On the other hand, |B| → 0 as |x| → ∞. By Theorem 3.3, this implies [0,∞) = σess(D
∗
ADA).
Over R2, Clifford multiplication on spinors anticommutes with the map β : (ϕ1, ϕ2) ∈ C2 7→
(ϕ1, ϕ2), see [8, Section 1.7]. Thus, DA(β(ψ)) = −β(DAψ) which implies that the spectrum
of DA is symmetric and, thus, σess(DA) = R. Hence, the pure point spectrum above must be
dense. ■
Acknowledgements
The authors thank Gilles Carron for asking whether for magnetic Dirac operators the spectrum
can consist of a pure point spectrum that is dense in R, which led to this article. They would
also like to thank the anonymous referees whose comments helped to improve the quality of this
paper.
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1 Introduction
2 Preliminaries
2.1 Magnetic Dirac operators
2.2 With electric potential
2.3 With (mass-type) potential
2.4 Diamagnetic inequality
3 Spectral results
3.1 Potentials and discrete spectrum
3.2 Magnetic Schrödinger operator with vanishing magnetic field at infinity
3.3 Dense eigenvalues
References
|
| id | nasplib_isofts_kiev_ua-123456789-212029 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-13T23:28:53Z |
| publishDate | 2023 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Charalambous, Nelia Grosse, Nadine 2026-01-23T10:08:24Z 2023 A Note on the Spectrum of Magnetic Dirac Operators. Nelia Charalambous and Nadine Grosse. SIGMA 19 (2023), 102, 12 pages 1815-0659 2020 Mathematics Subject Classification: 58J50; 35P05; 53C27 arXiv:2306.00590 https://nasplib.isofts.kiev.ua/handle/123456789/212029 https://doi.org/10.3842/SIGMA.2023.102 In this article, we study the spectrum of the magnetic Dirac operator and the magnetic Dirac operator with potential over complete Riemannian manifolds. We find sufficient conditions on the potentials as well as the manifold so that the spectrum is either maximal or discrete. We also show that magnetic Dirac operators can have a dense set of eigenvalues. The authors thank Gilles Carron for asking whether, for magnetic Dirac operators, the spectrum can consist of a pure point spectrum that is dense in R, which led to this article. They would also like to thank the anonymous referees whose comments helped to improve the quality of this paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications A Note on the Spectrum of Magnetic Dirac Operators Article published earlier |
| spellingShingle | A Note on the Spectrum of Magnetic Dirac Operators Charalambous, Nelia Grosse, Nadine |
| title | A Note on the Spectrum of Magnetic Dirac Operators |
| title_full | A Note on the Spectrum of Magnetic Dirac Operators |
| title_fullStr | A Note on the Spectrum of Magnetic Dirac Operators |
| title_full_unstemmed | A Note on the Spectrum of Magnetic Dirac Operators |
| title_short | A Note on the Spectrum of Magnetic Dirac Operators |
| title_sort | note on the spectrum of magnetic dirac operators |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/212029 |
| work_keys_str_mv | AT charalambousnelia anoteonthespectrumofmagneticdiracoperators AT grossenadine anoteonthespectrumofmagneticdiracoperators AT charalambousnelia noteonthespectrumofmagneticdiracoperators AT grossenadine noteonthespectrumofmagneticdiracoperators |