ap-5-10.dvi


Acta Polytechnica Vol. 50 No. 5/2010

Self-adjoint Extensions of Schrödinger Operators with δ-magnetic
Fields on Riemannian Manifolds

T. Mine

Abstract

We consider the magnetic Schrödinger operator on a Riemannian manifold M . We assume the magnetic field is given
by the sum of a regular field and the Dirac δ measures supported on a discrete set Γ in M . We give a complete
characterization of the self-adjoint extensions of the minimal operator, in terms of the boundary conditions. The result
is an extension of the former results by Dabrowski-Šťovíček and Exner-Šťovíček-Vytřas.

Keywords: Spectral theory, functional analysis, self-adjointness, Aharonov-Bohm effect, quantum mechanics, differen-
tial geometry, Schrödinger operator.

1 Introduction
Let (M, g) be a two-dimensional, oriented, connected
complete C∞-Riemannian manifold, where g is the
Riemannian metric onM . Let dμ be the measure in-
duced from the Riemannian metric. If we take a local
chart (U, ϕ), ϕ = (x1, x2), the measure dμ is writ-
ten as dμ =

√
Gdx1dx2 in U , where G = det(gmn),

gmn = g(∂m, ∂n), and ∂m = ∂/∂xm. We denote
L2(M) = L2(M ; dμ). The set of all 1-forms on M
is denoted by Λ1(M). In the coordinate neighbor-
hood U , A ∈ Λ1(M) is written as

A = A1dx1 +A2dx2.

In general, the coefficients A1, A2 are complex-
valued. We say A is real-valued if the coefficients are
real-valued. We say A is of the class CkΛ1(M) if the
coefficients are of the class Ck(U) for any local chart
(U, ϕ). We define the class Lq

locΛ
1(M) (1 ≤ q ≤ ∞)1,

etc. similarly. The 2-form dA is called the magnetic
field. If A ∈ L1locΛ

1(M), dA can be defined at least
in the distribution sense. In U , the magnetic field is
given by

dA = (∂1A2 − ∂2A1)dx1 ∧ dx2.

Let Γ = {γk}K
k=1 be a sequence of mutually dis-

tinct points in M . The number K may be infinity,
and in this case we assume additionally Γ has no ac-
cumulation points in M . Let A be a 1-form on M
given by the sum of two 1-forms
(A) A = A(0) +A(1).

The part A(0) corresponds to the δ magnetic fields,
that is, we assume the following.

(A0) A(0) ∈ C∞Λ1(M \Γ)∩L1locΛ
1(M), real-valued,

and

dA(0) =
K∑

k=1

2παkδγk
, (1)

where αk ∈ R, and δγ is the Dirac measure
concentrated on the point γ.

More precisely, (1) means

−
∫

M

dϕ ∧ A(0) =
K∑

k=1

2παkϕ(γk),

for any ϕ ∈ C∞
0 (M) (since A(0) ∈ L1locΛ

1(M), the
left hand side is well-defined). Notice that this equa-
tion is independent of the Riemannian metric g. For
the regular part A(1) and the scalar potential V , we
assume the following:

(A1) A(1) ∈ C1Λ1(M), real-valued.

(V) V is real-valued, V ∈ L2loc(M), and is bounded
in some open neighborhood of γk for every
k = 1, . . . , K.

Using the local coordinate (x1, x2), we define the
Schrödinger operator L in each coordinate neighbor-
hood by

Lu = − 1√
G

∑
m,n=1,2

(∂m + iAm) ·(√
Ggmn(∂n + iAn)u

)
+ V u,

1The measure dμ is omitted, since the class Lq
locΛ

1(M ; dμ) is independent of the choice of dμ. The coefficient Am is a function

on U ⊂ M , however, we denote the pull-back (ϕ−1)∗Am = Am ◦ ϕ−1 on ϕ(U) ⊂ R2 by the same symbol Am, for simplicity of
notations. This convention is frequently used in this paper.

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Acta Polytechnica Vol. 50 No. 5/2010

where (gmn) is the inverse matrix of (gmn). This
definition is independent of the choice of local coor-
dinates (see section 2). Define the minimal operator
Hmin by

Hminu = Lu, D(Hmin) = C∞
0 (M \ Γ),

where the overline denotes the closure with respect to
the graph norm. Define the maximal operator Hmax
by Hmax = H∗

min. Then we can show that

Hmaxu = Lu,

D(Hmax) = {u ∈ L2(M) | Lu ∈ L2(M)},

where L is a differential operator on D′(M \ Γ). We
assume

(SB) The operator Hmin is bounded from below.

In the case M is the flat Euclidean plane, it is
well-known that the operator Hmin is not essentially
self-adjoint and the structure of the self-adjoint ex-
tensions ofHmin can be determined via the celebrated
Krein-Von Neumann theory of self-adjoint extensions
(see e.g. Reed-Simon [13]). In the textbook by Al-
beverio et al. [3], the case A(0) = A(1) = 0 and
V = 0 (but Γ 
= ∅) is exhaustively studied. Adami-
Teta [1] and Dabrowski-Šťovíček [7] study the case
K = 1, α1 
∈ Z, A(1) = 0, and V = 0. Exner-
Šťovíček-Vytřas [8] study the case K = 1, α1 
∈ Z,
dA(1) = Bdx1 ∧ dx2 for some non-zero constant B
(the constant magnetic field), and V = 0. Moreover,
Lisovyy [11] studies the caseM is the Poincaré disk, g
is the Poincaré metric, V = 0 and dA = Bωg+2παδ0,
where B is a non-zero constant and ωg is the surface
form induced from the Poincaré metric g.
In all the results above, they first determine the

deficiency subspaces Ker(Hmax ∓ i) and apply the
Krein-Von Neumann theory. This method cannot be
applied in the case K ≥ 2 and αk 
∈ Z, however, this
case (and A(1) is the constant field, V = 0) on the flat
Euclidean plane is studied by the author [12], and the
structure of the self-adjoint extensions is determined.
Our main purpose in this paper is to generalize the
result in [12] on general complete Riemannian mani-
folds and for more general A and V .
Our first result is about the deficiency indices

n±(Hmin) = dimKer(Hmax ∓ i).

Theorem 1.1 Assume (A), (A0), (A1), (V), and
(SB). Then, both deficiency indices n±(Hmin) are
equal to 2K1 +K2, where

K1 = #{αk | αk 
∈ Z}, K2 = #{αk | αk ∈ Z}.

Note that Bulla-Gesztesy [4] obtain a similar result
in the case A = 0 and V has singularities, and Iwai-
Yabu [9] also obtain a similar result on the two-
dimensional torus.

Next, we shall give a complete characterization of
the self-adjoint extensions of Hmin. To this purpose,
we introduce some nice coordinates around singular-
ities and some auxiliary functions. For simplicity, we
assume K = #Γ is finite for a while.
For k = 1, . . . , K, let (Uk, φk), φk = (x1, x2), be

a local chart around γk such that Uk is simply con-
nected, φk(γk) = 0, V is bounded in Uk, and {Uk}K

k=1
are disjoint. Let (r, θ) be the radial coordinate in Uk

defined by x1 + ix2 = reiθ, r ≥ 0, 0 ≤ θ < 2π. We
assume

gmn(0, 0) = δmn, (2)

∂jgmn(0, 0) = 0 (m, n, j = 1, 2),

where δmn is the Kronecker delta. Condition (2)
is satisfied, for example, if we take the normal
coordinate2 as (x1, x2).
Let βk be the fractional part of αk, that is,

αk = [αk] + βk, [αk] ∈ Z and 0 ≤ βk < 1. Put3

Ã(0) = βkr−2(−x2dx1 + x1dx2),

Ã(1) = A(1) − A(1)(0).

It is well-known that dÃ(0) = 2πβkδ0 (see e.g.
Aharonov-Bohm [2, 1] or [7]). Define a phase func-
tion ψk ∈ C∞(Uk \ {0}) by

ψk(x) = exp
1
i

(
A
(1)
1 (0)x

1 +A
(1)
2 (0)x

2 + (3)∫ x

x0

(
A(0) − Ã(0)

))
,

where A(1) = A
(1)
1 dx1 + A

(1)
2 dx2, x0 is some point

in Uk \ {0}, and the path of the line integral
∫ x

x0
lies in Uk \ {0}. Notice that the value of the line
integral is independent of the choice of paths mod-
ulo 2πZ, by the Stokes theorem and the assumption
d(A(0) − Ã(0)) = 2π[αk]δ0 in Uk. Then we have

A = Ã+ iψ−1
k dψk, Ã = Ã(0) + Ã(1) (4)

and

L = ψkL̃ψ−1
k (5)

in Uk \ {0}, where L̃ is the operator L corresponding
to the vector potential à and the scalar potential V .
Let K1, K2 be the numbers in Theorem 1.1.

In the sequel, we rearrange the index k so that
0 < βk < 1 for 1 ≤ k ≤ K1. As we prove later, the

2The coordinate defined by the local inverse map of the exponential map from the tangent space at γk to M .
3More precisely, the 1-form A(1) − A(1)(0) is defined as (A(1)1 (x

1, x2) − A
(1)
1 (0, 0))dx1 + (A(1)2 (x

1, x2) − A
(1)
2 (0, 0))dx2, in the

coordinate neighborhood Uk.

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Acta Polytechnica Vol. 50 No. 5/2010

asymptotics of u ∈ D(Hmax) in Uk as r → 0 is given
by

u =

⎧⎪⎪⎨⎪⎪⎩
ψk(ck

1r
βk−1e−iθ + ck

2r
−βk+

ck
4r
1−βke−iθ + ck

5r
βk) + ξ (1 ≤ k ≤ K1),

ψk(ck
3 log r + ck

6) + ξ (K1 + 1 ≤ k ≤ K),

where ck
1 , . . . , c

k
6 are constants and ξ is a regular func-

tion in the sense ξ ∈ D(Hmin). Define

Φj(u) =

{
t(c1j , . . . , c

K1
j ) ∈ C

K1 (j = 1, 2, 4, 5),
t(cK1+1

j , . . . , cK
j ) ∈ CK2 (j = 3, 6),

Φ(u) = t(tΦ1(u) · · · tΦ6(u)) ∈ C4K1+2K2 .

Define a (2K1 + K2) × (2K1 +K2)-diagonal matrix
D by

D = diag (1− β1, . . . , 1− βK1 , β1, . . . , βK1 , (6)

−1/2, . . . , −1/2).

Now our theorem is stated as follows.

Theorem 1.2 Assume (A), (A0), (A1), (V), (SB)
and K < ∞. Let Φ(u), D given above.

(i) Let X=

(
X1

X2

)
, where X1, X2 are (2K1 +K2)×

(2K1 +K2) matrices satisfying

rankX = 2K1 +K2, X∗
1DX2 = X∗

2DX1. (7)

Then, the operator HX defined by

HXu = Lu,

D(HX) = {u ∈ D(Hmax) | Φ(u) ∈ RanX}

is a self-adjoint extension of Hmin.

(ii) For any self-adjoint extension H of Hmin, there
exists some matrix X satisfying (7) and H =
HX .

We can consider the case K =∞, but some technical
assumptions are necessary. We shall argue this case
in section 5.
Thus we can characterize the self-adjoint exten-

sions in terms of the boundary conditions. We can
easily prove that the Friedrichs extension corresponds
to the case X1 = O, X2 = Id. In the case M = R

2

and K = 1, similar results are obtained in [7] and [8],
and our theorem is a generalization of their results.
As stated in their paper, the choice of matrices X is
of course not unique: there are infinitely many ma-
trices X giving same RanX .
The difficulty in the proof is that we cannot de-

termine the deficiency subspaces explicitly. To over-
come this difficulty, we describe the condition of
the self-adjointness only using the quotient subspace

D(Hmax)/D(Hmin). This quotient subspace is essen-
tially the same object as the sum of deficiency sub-
spaces, but much easily tractable than the deficiency
subspaces themselves. This idea is also used in [4]
or [12].
We note that recently self-adjoint extensions of

the Schrödinger operators on R2 with δ magnetic
fields are studied from the viewpoint of the hidden
supersymmetric structure; see Correa et al. [5, 6].
The rest of the paper is organized as follows.

In section 2, we review basic notations and facts
from the differential geometry and the theory of self-
adjoint extensions. In section 3, we shall prove the
structure of the self-adjoint extensions depends only
on the singular part of the vector potentials. In sec-
tion 4, we shall prove the main theorems. In sec-
tion 5, we shall consider the case K = ∞ and give
a complete characterization of the self-adjoint exten-
sions, under some homogeneity conditions.

2 Basic facts

2.1 Formulas in differential geometry

We quote some formulas used in Shubin [14] for the
convenience of the readers. Take a local chart (U, ϕ),
ϕ = (x1, x2), around p ∈ M . Put gmn = g(∂m, ∂n),
and let (gmn) be the inverse matrix of (gmn). For
α, β ∈ Λ1p(M) (the cotangent space at p), we define
the scalar product

〈α, β〉 =
∑

m,n=1,2

gmnαmβn,

where α = α1dx1 + α2dx2 and β = β1dx1 + β2dx2.
Put |α|2 = 〈α, α〉, where α = α1dx1 + α2dx2. For a
1-form ω = ω1dx2 + ω2dx2, we define a function d∗ω
by

d∗ω = − 1√
G

∑
m,n=1,2

∂m

(√
Ggmnωn

)
.

This definition is independent of the choice of local
coordinates. Actually, operator d∗ is characterized
by the following relation:∫

M

〈du, ω〉dμ =
∫

M

ud∗ω dμ

for any u ∈ C∞
0 (M) and ω ∈ C∞

0 Λ
1(M).

Let A be a 1-form satisfying our assumptions. For
a function f , we define a 1-form dAf by

dAf = df + ifA,

where d is the exterior derivative, and i =
√

−1. For
a 1-form ω, we define

d∗Aω = d∗ω − iA∗ω, A∗ω = 〈A, ω〉.

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Acta Polytechnica Vol. 50 No. 5/2010

Then we obtain a representation of our Schrödinger
operator L independent of local coordinates:

L = d∗AdA + V.

For operator d∗A, the following Leibniz formulas
hold: for an appropriate function f and 1-form ω, we
have

d∗A(fω) = fd∗ω − 〈df, ω〉 − if〈A, ω〉 =
fd∗Aω − 〈df, ω〉 = fd∗ω − 〈dAf, ω〉, (8)

d∗AdA(fg) = fd∗AdAg − 2〈df, dAg〉+ gd∗df. (9)

Proposition 2.1 Let U , U ′ be open subsets of M \Γ
such that U is a compact subset of U ′, and V is
bounded in U ′. Then, there exists a constant C > 0
such that∫

U

|dAf |2 dμ ≤ C

∫
U ′
(|f |2 + |Lf |2) dμ (10)

for f ∈ D(Hmax).

Proof. According to [14, (5.3)],4 we have

(L(φf), φf) = �(φLf, φf) +
∫

M

|dφ|2|f |2 dμ

for f ∈ D(Hmax) and φ ∈ C∞
0 (M \ Γ). Take

φ ∈ C∞
0 (U

′) such that φ = 1 on U . Then the conclu-
sion follows from the above equality,∫

suppφ

|dA(φf)|2 = (L(φf), φf) − (V φf, φf)

and assumption V is bounded. �

2.2 Theory of self-adjoint extensions

We quote some notation from the textbook [13]. Let
H be a separable Hilbert space and denote its inner
product by (·, ·), and norm by ‖ · ‖. All the linear
operators in this subsection are on the Hilbert space
H. For a linear operator X , D(X) denotes the do-
main of definition of X , X the closure of X , X∗ the
adjoint operator of X . For a linear operator X , the
graph inner product of X is defined by

(x, y)X = (Xx, Xy) + (x, y)

for x, y ∈ D(X), and the graph norm by ‖x‖X =

(x, x)1/2X .
We introduce some equivalent for the sum of the

deficiency subspaces, which is also introduced in [4]
or [12]. Let X be a closed, densely defined symmet-
ric operator. Let D = D(X∗)/D(X), where the right
hand side denotes the quotient space. The space D
is a Hilbert space equipped with the norm

‖[x]‖2D = min
y∈[x]

‖y‖2X∗ = ‖Qx‖2X∗ ,

where x ∈ D(X∗), [x] = x+D(X) denotes the equiv-
alence class of x in the quotient space D(X∗)/D(X),
and Q denotes the orthogonal projection onto the
orthogonal complement of D(X) in D(X∗). For
u, v ∈ D, define

[u, v]D = (X∗x, y)− (x, X∗y),

u = [x], v = [y], x, y ∈ D(X∗).

The value [u, v]D is independent of the choice of the
representatives x, y. Let P be the canonical projec-
tion from D(X∗) to D. For a closed subspace V of
D, we define a closed linear operator XV by

D(XV ) = {x ∈ D(X∗) | Px ∈ V }, XV x = X∗x.

We also define

V [⊥] = {u ∈ D | [u, v]D = 0 for any v ∈ V }.

Then the following proposition immediately follows
from the definition of the self-adjointness.
Proposition 2.2 1. For a closed subspace V of D,

the operator XV is a self-adjoint extension of X
if and only if

V [⊥] = V. (11)

2. For any self-adjoint extension X̃ of X, there ex-
ists a closed subspace V of D such that XV = X̃.

In terms of the above notations, the Krein-Von
Neumann theory can be rephrased as follows.

Proposition 2.3 Let N± = Ker(X∗ ∓ i) the defi-
ciency subspaces of X, n± = dimN± the deficiency
indices of X. Then, the following holds.
(i) The projection operator P gives a Hilbert space
isomorphism from the direct sum N+ ⊕ N− to
D. In particular, dimD = n+ + n−.

(ii) There exists a one-to-one correspondence be-
tween the closed subspaces V of D satisfying (11)
and the unitary operators U from H+ to H−,
given by

V = P (1 + U)H+.

This proposition says the space D can play the same
role as the sum of deficiency subspaces in the the-
ory of self-adjoint extensions. Particularly when N±
is difficult to determine explicitly (as in our case),
the space D is more tractable, since the element of
this space has ambiguity by D(X). Actually, in the
next section we shall see that the structure of D for
our Schrödinger operator Hmin and the form [·, ·]D is
determined only from the singular part A(0) of the
vector potential.

4Since the function φ avoids the singularities, the proof of [14, (5.3)] is also available in our case.

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Acta Polytechnica Vol. 50 No. 5/2010

3 Reduction

3.1 Division to the local potential

Let (Uk, φk), φk = (x1, x2), the local coordinate in-
troduced in section 1. Let à be the 1-form given
by (4). Take a positive number εk so small that
the closed disc {r ≤ 2εk} is contained in Uk. Let
ηk ∈ C∞

0 (U) such that 0 ≤ ηk ≤ 1, ηk = 1 for r ≤ εk,
ηk = 0 for r ≥ 2εk. Define functions ĝmn, Âm and V̂
on R

2 by

ĝmn = ηkgmn + (1− ηk)δmn,

Âm = Ã(0)m + ηkÃ(1)m ,

V̂ = ηkV.

Define a differential operator Lk on R
2 by

Lk = − 1√
Ĝ

∑
m,n=1,2

(
∂

∂xm
+ iÂm

)
·

√
Ĝĝmn

(
∂

∂xn
+ iÂn

)
+ V̂ ,

where Ĝ = det(ĝmn), and (ĝmn) is the inverse ma-
trix of (ĝmn). Define a linear operator Lk,min on

L2(R2; dμk), dμk =
√

Ĝdx1dx2, by

Lk,minu = Lku, D(Lk,min) = C∞
0 (R

2 \ {0}).

Let Lk,max = L∗
k,min. Then

Lk,maxu = Lku,

D(Lk,max) = {u ∈ L2(R2; dμk) | Lku ∈ L2(R2; dμk)},

where Lk is regarded as a differential operator on
D′(R2 \ 0). Let D = D(Hmax)/D(Hmin), Dk =
D(Lk,max)/D(Lk,min). Let χk ∈ C∞

0 (M) such that
0 ≤ χk ≤ 1, χk = 0 for r ≥ εk and χk = 1 for
r ≤ εk/2. Define a map Tk from D to Dk by

Tk[f ] = [ψ
−1
k χkf ],

where the function ψk is given by (3). Define a map

T from D to the direct sum
K⊕

k=1

Dk by

T [f ] =
k⊕

k=1

Tk[f ].

We also define a map S from
K⊕

k=1

Dk to D by5

S

K⊕
k=1

[fk] =

[
K∑

k=1

ψkχkfk

]
.

In the sequel, we sometimes write [f, g]D = [[f ], [g]]D
etc. for simplicity of notations.
Lemma 3.1 1. Assume K < ∞. Then, the maps

S, T defined above are well-defined and mutually
inverse. Moreover, we have

[f, g]D =
k∑

k=1

[Tk[f ], Tk[g]]Dk
(12)

for any [f ], [g] ∈ D.
2. Assume K =∞. Then the map S is well-defined
and injective.

Proof. (i) We divide the proof into three steps.
Step 1. The map

D(Hmax) + f �→ ψ−1
k χkf ∈ D(Lk,max)

is well-defined and continuous.
Proof. Clearly ψ−1

k χkf ∈ L2(R2; dμk), so it suffices
to show that Lk(ψ

−1
k χkf) ∈ L2(R2; dμk). By (5) and

the Leibniz rule (9), we have

Lk(ψ
−1
k χkf) = ψ−1

k L(χkf) =

ψ−1
k (χkLf − 2〈dχk, dAf〉+ (d∗dχk)f) .

The first term and the third in the parenthesis of the
right hand side are in L2(R2; dμk) and continuous
with respect to ‖ ·‖Hmax. Moreover, we can prove the
second term is also in L2 and continuous with respect
to ‖ · ‖Hmax by using (10). �

Step 2. Let f ∈ D(Hmin). Then, we have ψ−1
k χkf ∈

D(Lk,min).
Proof. By definition, there exists a sequence
{fn}∞n=1 ⊂ C∞

0 (M \Γ) such that fn → f in D(Hmin).
Then, ψ−1

k χkfn ∈ C∞
0 (R

2 \ {0}) and ψ−1
k χkfn →

ψ−1
k χkf in D(Lk,max), by Step 1. Since D(Lk,min) is
a closed subspace of D(Lk,max), we have the conclu-
sion. �

Step 1 and 2 imply the map T is well-defined.
We can similarly prove that the map S is also well-
defined.
Step 3. The operator ST is the identity map on D.
Proof. By definition, we have

(I − ST )[f ] = [ψf ], ψ = 1−
K∑

k=1

χ2k.

So it suffices to prove that g = ψf ∈ D(Hmin).

5When K = ∞, we define the map S for the elements of

∞⊕
k=1

Dk having only finite nonzero components [fk]. So there is no

difficulty in the definition of S.

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Acta Polytechnica Vol. 50 No. 5/2010

Let (r, θ) be the radial coordinate in Uk and
put Bk,ε = {x ∈ Uk | r < ε}. Then we have

suppψ ⊂ M \
K⋃

k=1

Bk,εk/2. For c > 0, let ξc ∈ C∞(M)

such that 0 ≤ ξc ≤ 1, ξc = 1 in M \
K⋃

k=1

Bεk/c, ξc = 0

in
K⋃

k=1

Bk,εk/(2c). Let L0, H0,min and H0,max be the

operators corresponding to the potentials ξ4A and
ξ4V . These potentials have no singularities, so we
have H0,min = H0,max by [14]. Since Lg = L0g ∈ L2,
we have g ∈ D(H0,max) = D(H0,min). Thus we can
take a sequence {gn} such that gn → g in ‖ · ‖H0,min .
Then ξ2gn ∈ C∞

0 (M \ Γ) and ξ2gn → ξ2g = g in
‖ · ‖Hmin . Thus we have g ∈ D(Hmin). �

We can prove TS = I similarly. Then (12) follows

from (5) and the equality [f, g]D =
K∑

k=1

[χkf, χkg]D

(notice that f −
∑

k

χkf ∈ D(Hmin) can be proved as

in Step 3).
(2) LetK =∞. For any positive integer n, we can

define T (n) from D to
n⊕

k=1

Dk, and S(n) from
n⊕

k=1

Dk

to D similarly, and prove T (n)S(n) = Id. This implies
the map S is well-defined and injective. �

3.2 Analysis of operators on R
2

We shall analyze the operator Lk (or Lk,min, Lk,max)
defined in the previous subsection. For simplicity of
notation, we omitˆand˜in the definition of Lk in the
sequel. Then our assumptions are the following:
1. Lk = d∗AdA + V on R

2 \ {0}, A = A(0) +A(1),
2. Lk,min and Lk,max are operators on L2(R2; dμk),

dμk =
√

Gdx1dx2,
3. A(0) = βkr−2(−x2dx1 + x1dx2), 0 ≤ βk < 1,
4. A(1) ∈ C10Λ

1({r < 2εk}), real-valued, A(1)(0) =
0,

5. V is bounded, real-valued,
6. gmn(0) = δmn, ∂jgmn(0) = 0, and gmn = δmn

for r ≥ 2εk.
We shall show that gmn, A(1) and V have nothing

to do with the structure of the self-adjoint extensions.
To this purpose, define a differential operatorMk on
R
2 by

Mk = −
∑

n=1,2

(
∂

∂xn
+ iAn

)2
.

Define a linear operatorMk,min on L2(R2; dx1dx2) by

D(Mk,min) = C∞
0 (R

2 \ {0}),
Mk,minu = Mku for u ∈ D(Mk,min).

Put Mk,max = M∗
k,min, and Ek = D(Mk,max)/

D(Mk,min). We also define M(0)
k , M

(0)
k,min, M

(0)
k,max,

and E(0)k , by replacing An by A(0)n in the above defi-
nition.
The operator M(0)

k is already studied in [1] and
[7]. Here we quote their results and calculate the
form [·, ·]E(0)

k

.

Proposition 3.2 Let χ ∈ C∞
0 (R

2) such that χ = 1
in some neighborhood of 0.
1. Assume 0 < βk < 1. Put

f1k = χe−iθrβk−1, f2k = χr−βk ,

f4k = χe−iθr1−βk , f5k = χrβk .

Then, the deficiency indices n±(M
(0)
k,min) = 2,

dim E(0)k = 4 and the vectors {[fn
k ]}n=1,2,4,5 form

a basis of E(0)k . Moreover, for m, n ∈ {1, 2, 4, 5}
with m ≤ n,6 we have

[fm
k , fn

k ]E(0)
k

=

⎧⎪⎪⎨⎪⎪⎩
4π(βk − 1) for (m, n) = (1, 4),

−4πβk for (m, n) = (2, 5),

0 otherwise.

2. Assume βk = 0. Put

f3k = χ log r, f6k = χ.

Then, the deficiency indices n±(M
(0)
k,min) = 1,

dim E(0)k = 2, {[f j
k ]}j=3,6 form a basis of E(0)k ,

and

[f3k , f6k ]E(0)
k

= 2π, [f3k , f3k ]E(0)
k

= [f6k , f6k ]E(0)
k

= 0.

Proof. (i) The first statement follows from the re-
sult in [7] or [1]. For the calculation of [u, v]E(0)

k

, we

use some notation in vector analysis. We use the
gradient vector ∇ = t(∂1, ∂2), and identify a 1-form
A with the component vector t(A1, A2). The dot ·
denotes the Euclidean inner product. Then we have

[u, v]E(0)
k

= lim
ε→0

∫
r≥ε

(
−v(∇+ iA(0)) · (∇+ iA(0))u +

u(∇+ iA(0)) · (∇+ iA(0))v
)
dx1dx2 =

6Notice that [fn
k , fm

k ]E(0)
k

= −[fm
k

, fn
k
]E(0)

k

by definition.

67



Acta Polytechnica Vol. 50 No. 5/2010

lim
ε→0

∫
r=ε

(
vn · (∇+ iA(0))u−

un · (∇+ iA(0))v
)

r dθ =

lim
ε→0

∫
r=ε

(
v∂ru − u∂rv

)
r dθ, (13)

where n = (cos θ, sin θ), and the line integral is taken
counterclockwise. We used the Green formula and
the fact n · A(0) = 0. Then we can easily prove the
second statement by using (13).
(ii) The first part of the statement follows from

the results in [3]. The second statement can be jus-
tified by using (13). �

Next, we prove that the regular part A(1) does
not affect the structure of Ek and the corresponding
form.

Proposition 3.3 All the statements of Proposition
3.2 hold even if we replace M

(0)
k,min by Mk,min, and

E(0)k by Ek.

Before the proof, we prepare a perturbative lemma,
which is an immediate corollary of [10, Theo-
rem IV.5.22].

Lemma 3.4 Let H be a separable Hilbert space and
‖ ·‖ its norm. Let X, Y be densely defined symmetric
operators on H. Assume D(X) ⊂ D(Y ) and there
exist positive constants C, δ with 0 < δ < 1 and

‖Y u‖ ≤ δ‖Xu‖+ C‖u‖

for every u ∈ D(X). Then, we have D(X + Y ) =
D(X) and n±(X + Y ) = n±(X), where the overline
denotes the operator closure.

Proof of Proposition 3.3 We prove only state-
ment (i). Statement (ii) can be proved similarly.
By the Leibniz formula (8), we have for u ∈

C∞
0 (R

2 \ {0})

(Mk − M(0)
k )u = i(d∗A(1))u − (14)

2i〈A(1), dA(0)u〉+ |A(1)|2u.

We denote ‖u‖2 =
∫
R
2

|u|2dx1dx2 for a function u,

and ‖ω‖2 =
∫
R
2

|ω|2dx1dx2 for a 1-form ω (notice

that |ω|2 = 〈ω, ω〉). We denote the essential supre-
mum norm of |u| and |ω| by ‖u‖∞ and ‖ω‖∞, respec-
tively. Then we have by the Schwarz inequality

‖(Mk − M(0)
k )u‖ ≤

‖d∗A(1)‖∞‖u‖+2‖A(1)‖∞‖dA(0)u‖+ ‖A(1)‖2∞‖u‖ ≤
(‖d∗A(1)‖∞ + ‖A(1)‖2∞)‖u‖+
‖A(1)‖∞(ε‖M(0)

k u‖2 + ε−1‖u‖2)

for any ε > 0, where we used the inequality

‖dA(0)u‖ = (M(0)
k u, u)1/2 ≤

(ε‖M(0)
k u‖)1/2(ε−1‖u‖)1/2 ≤

1
2
(ε‖M(0)

k u‖+ ε−1‖u‖).

Take ε > 0 sufficiently small and apply Lemma 3.4.
Then we have n±(Mk,min) = n±(M

(0)
k,min) = 2, thus

dim Ek = 4 by (i) of Proposition 2.3. Moreover we

have D(Mk,min) = D(M (0)
k,min), so the functions {f j

k}
(j = 1, 2, 4, 5) do not belong to D(Mk,min). And we
can proveMkf j

k ∈ L2(R2) by using (14) and the fact
|A(1)| = O(r) near the origin. Thus {[f j

k ]} form a
basis of Ek. For the form [·, ·]Ek

, we can prove the
formula

[u, v]Ek
=

lim
ε→0

∫
r=ε

(
v(∂ru)− u(∂rv)− 2i(n · A(1))uv

)
rdθ

in a similar way as in (13). Thus the value [fm
k , fn

k ]Ek

is not affected by A(1), since |A(1)| = O(r) and
|fm

k fn
k | is at most O(r−max(2βk,2(1−βk))). �

Next we shall consider the non-flat case. We shall
show that metric g also does not affect the structure
of Dk and the corresponding form.

Proposition 3.5 All the statements of Proposition
3.2 hold even if we replace M

(0)
k,min by Lk,min and E(0)k

by Dk.

Since V is bounded, we can assume V = 0. In the
sequel, we use the following notation:

L = G−1/2(D +A) · G1/2g−1(D +A),

where D is the column vector t(D1, D2), Dj = −i∂j,
A is identified with the component vector t(A1, A2),
and g−1 is the inverse matrix of g = (gmn).
We shall prepare some elliptic a priori estimate.

Lemma 3.6 Let m, n ∈ {1, 2}. Then, there exist
Cm > 0 and Cmn > 0 such that

‖(Dm +Am)u‖ ≤ Cm(ε‖Mku‖+ ε−1‖u‖),
‖(Dm +Am)(Dn +An)u‖ ≤ Cmn(‖Mku‖+‖u‖) (15)

for every u ∈ C∞
0 (R

2 \ {0}) and every ε > 0, where
‖ · ‖ = ‖ · ‖

L2(R2
;dx1dx2)

.

The difficulty is the singularity of our vector poten-
tial A at the origin. We can overcome this difficulty
by using some commutator technique.
Proof of Lemma 3.6 Put Πj = Dj+Aj (j = 1, 2).

Then, since

‖Πju‖2 = (ε1/2Π2ju, ε−1/2u) ≤
1
2

(
ε‖Π2ju‖2 + ε−1‖u‖2

)
for u ∈ C∞

0 (R
2), it suffices to prove (15).

68



Acta Polytechnica Vol. 50 No. 5/2010

Define auxiliary operators

A = iΠ1 +Π2, A† = −iΠ1 +Π2.

Let [X, Y ] = XY − Y X be the commutator of oper-
ators X and Y . Then we have

[Π1,Π2] = [D1, A2]− [D2, A1] = −i(b+ 2πβkδ0),

where b = ∂1A
(1)
2 − ∂2A

(1)
1 is the magnetic field cor-

responding to A(1). Thus we have

[A, A†] = 2i[Π1,Π2] = 2(b+ 2πβkδ0).

Particularly for u ∈ C∞
0 (R

2 \ {0}), we have

(AA† − A†A)u = 2bu.

Moreover, we have by definition

(AA† +A†A)u = 2Mku.

These equalities imply

AA† =Mk + b, A†A =Mk − b (16)

on C∞
0 (R

2 \ {0}).
Since ΠmΠn can be written as a finite linear com-

bination of the operators of the formXY , whereX, Y

are A or A†, it suffices to show that there exists some
constant C > 0 such that

‖XY u‖ ≤ C(‖Mku‖+ ‖u‖) (17)

for u ∈ C∞
0 (R

2\{0}). For (X, Y ) = (A, A†), (A†, A),
(17) follows from (16), since b is bounded. To esti-
mate ‖A2u‖2, we assume A(1) ∈ C∞ for a while.
Then, we have by (16)

‖A2u‖2 = (A2u, A2u) = ((A†)2A2u, u) =

(A†(AA† − 2b)Au, u) =

‖A†Au‖2 − 2(bAu, Au) ≤
‖A†Au‖2 + 2‖b‖∞‖Au‖2 ≤
‖A†Au‖2 + 2‖b‖∞‖A†Au‖‖u‖.

When A(1) ∈ C1, we approximate A(1) by C∞-
potentials w.r.t. C1-norm on some neighborhood of
suppu, then we get the above inequality again. Then,

we have (17) by using (16). The case X = Y = A†
can be treated similarly. �

Proof of Proposition 3.5 First, by assump-
tion (vi), we have

g−1 = I + ĝ, max |ĝmn| = O(r2), (18)

G = 1 +O(r2), |DG| = O(r),

as r → 0.

Define a unitary operator U from
L2(R2;

√
Gdx1dx2) to L2(R2; dx1dx2) by

Uu = G1/4u.

Put L̃k = ULkU−1, L̃k,min = ULk,minU
−1, etc.

Then we have for v ∈ C∞
0 (R

2 \ {0})

L̃k,minv = G−1/4(D +A) ·
√

Gg−1(D +A)G−1/4v.

Thus we have

L̃k,min=G−1/4(D +A) · G1/4g−1(D +A) +

G−1/4(D +A) ·
√

Gg−1(DG−1/4). (19)

The second term of (19) is written as

G−1/4
(
D · (

√
Gg−1(DG−1/4))

)
+ (20)

(DG−1/4) · G1/4g−1(D +A).

The first term of (20) is bounded, and the second
is infinitesimally small w.r.t. Mk,min, by Lemma 3.6.
The first term of (19) is written as

(D +A) · g−1(D +A) + (21)

G−1/4(DG1/4) · g−1(D +A).

The second term of (21) is also infinitesimally small
w.r.t. Mk,min, by Lemma 3.6. The first term of (21)
is written as

Mk,min + (D +A) · ĝ(D +A).

The second term of this expression is written as∑
m,n=1,2

(Dmĝmn)(Dn +An) + (22)

∑
m,n=1,2

ĝmn(Dm +Am)(Dn +An).

The first sum of (22) is infinitesimally small w.r.t.
Mk,min. If we take εk sufficiently small, the second
sum isMk,min-bounded with relative bound less than
1, by Lemma 3.6. Now we can apply Lemma 3.4, and
conclude that D(L̃k,min) = D(Mk,min) = D(M (0)

k,min),

and n±(Lk,min) = n±(L̃k,min) = n±(Mk,min) =

n±(M
(0)
k,min). Moreover, one can show that multi-

plication by G1/4 is a bijective continuous map on
D(M (0)

k,min). Thus we have

D(Lk,min) = U−1D(L̃k,min) =

G−1/4D(M (0)
k,min) = D(M (0)

k,min).

And then we can prove Lkfm
k ∈ L2(R2; dμk) by the

Leibniz formula and (18), and thus {[fm
k ]}m form a

basis of Dk.

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Acta Polytechnica Vol. 50 No. 5/2010

In a similar way as in (13), we have

[u, v]Dk
= lim

ε→0

∫
r=ε

(
vn ·

√
Gg−1(∇+ iA)u−

−un ·
√

Gg−1(∇+ iA)v
)

r dθ.

Since
√

Gg−1 = I + O(r2), we can replace
√

Gg−1

by I in the calculation of [fm
k , fn

k ]Dk
, and we have

[fm
k , fn

k ]Dk
= [fm

k , fn
k ]Ek
. Thus we have the conclu-

sion. �

4 Proof of main theorems
Proof of Theorem 1.1 Since Hmin is semi-

bounded, we have n+(Hmin) = n−(Hmin) =
dimD/2. By Lemma 3.1 and Proposition 3.5, we
have for K < ∞

dimD =
K∑

k=1

dimDk = 4K1 + 2K2,

and for K =∞

dimD ≥
∞∑

k=1

dimDk =∞.

Thus we have the conclusion. �

Proof of Theorem 1.2 By Lemma 3.1 and Propo-
sition 3.5, we have for u, v ∈ D(Hmax)

[u, v]D = 4πΦ(u)∗
(

O −D

D O

)
Φ(v),

where Φ(u)∗ is the row-vector tΦ(u) and D is the ma-
trix given by (6). Let X = t(X1, X2) be the matrix
satisfying (7). Then we have

X∗

(
O −D

D O

)
X = O,

which implies V ⊂ V [⊥] for V = RanX . Moreover,
if rankX = 2K1 +K2, we have

dim V [⊥] = 4K1+2K2−dimV = 2K1+K2 = dimV.

Thus we have (11), and therefore HX is self-adjoint.
Conversely, for a given self-adjoint extension H of
Hmin, we can construct a (4K1+2K2)× (2K1+K2)-
matrix X by arranging the coefficients of an arbi-
trary basis of V = PD(H) with respect to the basis
{[ψkf j

k ]}. �

5 Infinite singularities

Let us consider the case K = ∞, and extend Theo-
rem 1.2. Even in this case, for u ∈ D(Hmax) and for
each k, we can define the asymptotic coefficients ck

j

at γk. However, the sequence Φj(u) is an infinite se-
quence. We shall find appropriate assumptions which
make these infinite sequences square summable.
In the sequel, Uk, βk, gmn are those introduced

in section 1. However, we may replace ψk defined by
(3) more appropriate one satisfying (4), if such one
exists. For simplicity, we assume V = 0.
(U) (i) There exists ε0 > 0, independent of k, such

that Uk = {r < ε0} for every k.
(ii) There exist β−, β+ such that 0 < β− ≤

βk ≤ β+ < 1 or βk = 0, for every k.
(iii) There exists C1 > 0 independent of k such

that gmn satisfies (2) and

|∂i∂jgmn| ≤ C1

in Uk, for every i, j, m, n = 1, 2.
(iv) There exists C2 > 0 independent of k, and

phase functions ψk ∈ C∞(Uk \ {0}) satis-
fying |ψk| = 1, (4) and

|∂jA
(1)
m | ≤ C2

in Uk, for j, m = 1, 2.

Thus we assume some homogeneity for g, A(0), and
A(1). Since the open sets {Uk}∞k=1 are required to
be disjoint, assumption (i) says the points of Γ are
uniformly separated in some sense. Assumption (ii)
seems a little strange, but we need this assumption if
we want to make the boundary value Φ(u) square
summable.7 Assumption (iii) binds the curvature
of M , and (iv) the intensity of the magnetic field.
In [12], the author considers a similar assumption
when M is the flat Euclidean plane and dA(1) is a
constant magnetic field.
In the sequel, we use the notation

C
∞ = l2 = {(cj)∞j=1 |

∞∑
j=1

|cj |2 < ∞},

and define its inner product by usual l2-inner prod-
uct. Let

H = C
K1 ⊕ C

K1 ⊕ C
K2 .

Proposition 5.1 Assume (A), (A0), (A1), (SB),
(U), V = 0, and K =∞. Then, the following linear
map

7If we consider another type of characterization, assumption (ii) may be dropped.

70



Acta Polytechnica Vol. 50 No. 5/2010

D + [u] �→ Φ(u) ∈ H ⊕ H, (23)

is a well-defined homeomorphism. Moreover,

[u, v]D = 4π(Φ(u), D̃Φ(v)), (24)

D̃ =

(
O −D

D O

)
,

where D is a bounded operator on H defined by (6).
Once this proposition is established, our theo-

rem can be proved similarly as in the proof of Theo-
rem 1.2. So we omit the proof.

Theorem 5.2 Assume the same conditions as in
Proposition 5.1. Then, the statements of Theorem
1.2 hold with the following changes: X1, X2 are
bounded operators on H, and condition (7) is re-
placed by the condition

RanX = KerX∗D̃,

where D̃ is the bounded operator on H ⊕ H defined
in Proposition 5.1.

We conclude this paper by proving Proposi-
tion 5.1.
Proof of Proposition 5.1. We divide the proof

into two steps.
Step 1. The map

D + [f ] �→
∞⊕

k=1

Tk[f ] ∈
∞⊕

k=1

Dk

is continuous, bijective and its inverse is also contin-
uous.
Proof. By our assumption (U) and the calculation
in section 3, we can prove there exists C > 0 inde-
pendent of k such that

‖ψ−1
k χkf‖2Lk,max

≤ C

∫
Uk

(|Lf |2 + |f |2)dμk.

Summing up these equalities with respect to k, we
conclude the map

D(Hmax) + f �→
∞⊕

k=1

ψ−1
k χkf ∈

∞⊕
k=1

D(Lk,max)

is continuous. Then the well-definedness of the
map (23) can be proved similarly as in section
3. Since D is identified with the closed subspace
D(Hmin)⊥ of D(Hmax) and the projection from
D(Lk,max) to Dk is continuous, we conclude the
map (23) is continuous. Moreover, we can prove the
inverse map is also well-defined and continuous, so
we have the conclusion. �

Step 2. There exists C > 1 independent of k such
that

C−1|ck| ≤ ‖[u]‖Dk
≤ C|ck|

for every [u] ∈ Dk, where ck = (ck
1 , c

k
2 , c

k
4 , c

k
5) for

0 < βk < 1, ck = (ck
3 , c

k
6) for βk = 0, and ck

j are
asymptotic coefficients of u defined in section 1.
Proof. We only consider the case 0 < βk < 1. Con-
sider the following formula for ck

1

ck
1 =

1
4π(1− βk)

[f4k , u]Dk
,

which can be verified by substituting all the basis
functions into u. By choosing the representative
u ∈ D(Lk,min)⊥ (so ‖u‖Lk,max = ‖[u]‖Dk

) and using
the Schwarz inequality, we have

|ck
1 | ≤ 1

2π(1 − βk)
‖f4k‖Lk,max‖[u]‖Dk

.

The fraction is bounded uniformly w.r.t. k, by our
assumption (ii) of (U). Moreover, we can prove
‖f j

k‖Lk,max is also uniformly bounded, by (U) and the
calculations in section 3 (first decompose Lk as in
section 3, and estimate all terms). Thus we have

|ck
j | ≤ C‖[u]‖Dk

.

for j = 1. The case j = 2, 4, 5 can be treated simi-
larly.
Conversely,∑

j=1,2,4,5

‖ck
j [f

j
k ]‖Dk

≤ |ck|(
∑

j=1,2,4,5

‖f j
k‖2Lk,max

)1/2,

and the sum in the right hand side is uniformly
bounded. Thus the conclusion holds. �

By Step 1 and 2, we have proved the map (23) is
well-defined and homeomorphism. Equation (24) is
confirmed by substituting each f j

k as u or v. �

Acknowledgement

This work was partially supported by Doppler In-
stitute for mathematical physics and applied math-
ematics, KIT Faculty Research Abroad Fellowship
Program, and JSPS grant Wakate 20740093.

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[12] Mine, T.: The Aharonov-Bohm solenoids in a
constant magnetic field. Ann. Henri Poincaré 6
(2005), no. 1, 125–154.

[13] Reed, M., Simon, B.: Methods of modern
mathematical physics. II. Fourier analysis, self-
adjointness, Academic Press, New York-London,
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[14] Shubin, M.: Essential Self-adjointness for Semi-
bounded Magnetic Schrödinger Operators on
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(2001), 92–116.

Dr. Takuya Mine
E-mail: mine@kit.ac.jp
Kyoto Institute of Technology
Matsugasaki, Sakyo-ku
Kyoto 606-8585, Japan

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