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Impenetrability of Aharonov-Bohm solenoids: proof
of norm resolvent convergence
C´esar R. de Oliveira∗
Departamento de Matem´atica – UFSCar, S˜ao Carlos, SP, 13560-970 Brazil
Marciano Pereira
Departamento de Matem´atica e Estat´ıstica – UEPG,
Ponta Grossa, PR, 84030-000 Brazil
October 25, 2010
Abstract
Consider bounded solenoids in the space or planar approximate
models of infinitely long solenoids. It is proved that the solenoid impen-
etrability, by means of a sequence of Hamiltonians with diverging po-
tentials, converges in the norm resolvent sense to the usual Aharonov-
Bohm model with Dirichlet boundary condition. The framework is
that of nonrelativistic quantum mechanics.
Keywords: Aharonov-Bohm Hamiltonian, norm resolvent convergence, bounded
solenoids.
MSC: 82D99; 47B25; 81Q10.
1 Introduction
The Aharonov-Bohm (AB) effect is an authentic quantum phenomenon and
was first considered in [19, 18, 1]. There are many discussions about the
justification of the famous Aharonov-Bohm Hamiltonian and interpreta-
tions (see, for instance, [2, 7, 8, 9, 11, 15, 20, 22, 26, 28, 30] and references
therein); the questions are particularly interesting for the more realistic case
of solenoids Sof radii greater than zero. Sometimes it involves the quan-
tization in multiply connected regions, and the main points to be clarified
∗Corresponding author. Email: oliveira@pq.cnpq.br
1
are the presence of the vector potential Ain the operator action (occasion-
ally in regions with no magnetic field), and the (natural) choice of Dirichlet
boundary conditions at the solenoid border.
Apparently, Kretzschmar [24] was the first one to propose a model of
the solenoid impenetrability via increasing positive potentials Vnso that
Vn→ ∞ precisely on the interior S◦of the solenoid (see also comments
in [7]), but a mathematical rigorous approach was only published many
years later by Magni and Valz-Gris [26], whose main result was a proof a
strong resolvent convergence.
In [13], the present authors have considered an increasing sequence of
cylindrical solenoids of finite length together with the same shielding process
of [26], and it was proved that both limits commute and converge, in the
strong resolvent sense, to the usual AB Hamiltonian HAB given by
dom HAB =H2(S0)∩ H1
0(S0),
HAB =p−q
cA2
,p=−i∇,
where S0denotes the exterior region of the solenoid. Note that the Dirich-
let boundary conditions are selected (due to the space H1
0(S0) in the limit
operator domain). cand qare the light speed and particle electric charge,
respectively. Recently [15], other boundary conditions at the cylindrical
solenoid that are compatible with quantum mechanics, in the sense that
they give rise to self-adjoint Hamiltonians, were investigated, and the scat-
tering of the AB Hamiltonian with Robin boundary conditions has been
considered and compared with the traditional Dirichlet case.
The goal of this note is to prove that, in some cases of interest, one has the
norm resolvent convergence instead of just the strong one. This is important
since spectral and eigenfunctions convergences are usually guaranteed only if
norm resolvent convergence holds. The important assumption, necessary to
get such norm convergence, is that the region border S=∂S◦is a bounded
set, since some arguments involving compactness will be evoked. Hence, the
results obtained below apply, in particular, to any finite cylindrical solenoid
as well as to toroidal solenoids in R3[27], and, due to the symmetry along the
vertical coordinate, to approximate models in R2of infinitely long cylindrical
solenoids.
The toroidal case is particularly important, since for the physics com-
munity this setting is considered to present the only trustful experimental
evidence of the AB effect [27, 31, 32, 10]; in case of straight solenoids there
is a leakage of magnetic flux and the experiments are not generally accepted.
Note that the recent experiments reported in [10] show that the experimen-
tal results of [31, 32], with toroidal solenoids, can not be explained by the
action of a force. It is also worth mention the rigorous proofs [4, 5, 6] that
the original Ansatz of Aharonov and Bohm [1], adapted to the toroidal case,
can be approximated by real solutions of the Schr¨odinger equation. Hence,
2
it is the three-dimensional case with some bounded solenoids the physically
important one, and this setting is included in Theorem 1 below.
In the next section the results of this work are precisely stated, whereas
their proofs appear in Section 3. Concluding remarks are discussed in the
last section of this note.
2 Statement of the Results
If λ∈Cand Tis a linear operator on a Hilbert space, Rλ(T) = (T−λ1)−1
denotes the resolvent operator of Tat λ. Let S◦denote a (nonempty)
bounded open subset of Rd,d= 2,3, Sits boundary, S0its exterior, that is,
S0=Rd\(S ∪ S◦), and its closure will be indicated by ˜
S=S◦∪ S, whereas
the closure of its exterior by ˆ
S=S0∪ S.
Let Bdenote a (bounded) magnetic field on Rdand Aa (bounded)
magnetic potential so that B=∇ × A; furthermore, assume that divA∈
L2
loc, and so Hnbelow is essentially self-adjoint when defined on C∞
0(Rd) [25].
In R2it is supposed that the only possible nonzero component of Bis the
one perpendicular to the plane, so that, in fact, the magnetic field is reduced
to a scalar function in this case.
As already indicated, think of Sas a finite solenoid of general shape,
and the most interesting situation here is when Bvanishes on S0(but not
necessarily on S◦), but the results below are not restricted to this setting.
Particularly relevant situations are: (1) Sis a torus in R3, and (2) Sis a
circle in R2.
In order to prevent a quantum particle to penetrate the interior re-
gion S◦, as in the Aharonov-Bohm effect, consider an increasing sequence
of potential barriers Vn(x) = nχ(x), where χ=χS◦is the characteristic
function of this set, that is, χ(x) = 1 if x∈ S ◦and χ(x) = 0 otherwise. The
question is: What does happen with the particle Hamiltonian as n→ ∞?
This is a way to model the impenetrability of the region S◦[24, 26], and so to
find the quantum Hamiltonian describing the particle motion in a magnetic
field and in Rdwith a hole S◦.
The Hamiltonian of a quantum particle in this situation is given by the
self-adjoint operator, for n≥0 and d= 2,3,
dom Hn=H2(Rd), Hn=p−q
cA2
+Vn=H0+Vn,
where H2denotes a usual Sobolev space in L2(Rd), i.e., the domain of the
free Hamiltonian −∆ (the negative laplacian). Note that the operator H0is
implicitly defined by this relation. Recall that, under the above hypotheses
on A, C∞
0(Rd) is a core of Hn[25].
The impenetrable limit n→ ∞ was rigorously considered in [26] for a
particular region, however the same arguments hold in a much more general
3
situation. By using the Kato-Robinson theorem [12, 14], it was shown that
Hnconverges to HAB in the strong resolvent sense as n→ ∞, and since
elements of H1
0(S0) vanish at the solenoid border (in the sense of Sobolev
traces), Dirichlet boundary conditions have showed up in this limit. Note,
however, that Hnand HAB act in different Hilbert spaces; since Vn→ ∞
in S◦, one restricts the resolvent convergence to the subspace of functions
that vanish a.e. in S◦, and if P0denotes the projection onto the subspace
L2(S0) of L2(Rd), the resolvent convergence will be understood in the sense
R−λ(Hn)P0→R−λ(HAB)P0, for λ > 0 (see [12, 26, 29] for details); but in
order to simplify the notation, here it will be simply written R−λ(Hn)→
R−λ(HAB). This construction works since Hnis a monotonically increasing
sequence of self-adjoint operators, and so the strong resolvent convergent is
directly related to the convergence of a monotonic family of quadratic forms
[12, 29]. A discussion on the nontrivial relation between resolvents and
quadratic forms for general bounded from below sequences of self-adjoint
operators, with some applications to singular quantum limits, can be found
in [16].
The main goal of this note is to show that, in case of bounded regions S,
the above strong convergence can be substantially improved, that is, it will
be proven that the norm resolvent convergence is in effect.
Theorem 1. Let Sbe a (nonempty) bounded open subset of Rd,d= 2,3,
and Hnand HAB as above. Then, Hnconverges to HAB in the norm resol-
vent sense as n→ ∞.
3 Proofs
3.1 Preliminary Lemmas
In this subsection two technical results that will be used to prove Theorem 1
will be discussed. Such results are based on references [3, 25]. Without loss
set c= 1 = q.
Let Band Cbe bounded operators on L2(Rd) and denote by B∞
∞(Rd)
the set of bounded Borel functions that vanish at infinity (with the sup
norm). Write B˙
≤Cto indicate that |Bψ| ≤ C|ψ|for all ψ∈L2(Rd),
that is, |Bψ|(x)≤(C|ψ|) (x). The following results will be used ahead, and
references to their proofs are indicated:
(i) If B˙
≤Cand Cis a compact operator, then Bis also compact (The-
orem 2.2 in [3]).
(ii) If λ > 0, a∈L2
loc(Rd)dand the function V∈L1
loc(Rd) is nonnegative,
then
(−i∇ − a)2+V+λ−1˙
≤(−∆ + λ)−1
(Lemma 6 in [25]).
4
(iii) If f, g ∈B∞
∞(Rd), the operator f(x)g(p) on L2(Rd) is compact (Sec-
tion 11.4.1 in [14]).
Lemma 1. If f∈B∞
∞(Rd),a∈L2
loc(Rd)dand V∈L1
loc(Rd)is nonnegative,
then f(x)R−λ(−i∇ − a)2+V(x)is compact, for all λ > 0.
Proof. Let λ > 0 and f∈B∞
∞(Rd). By (ii) above,
R−λ(−i∇ − a)2+V˙
≤R−λ(−∆),
that is,
R−λ(−i∇ − a)2+Vψ
≤R−λ(−∆)|ψ|,∀ψ∈L2(Rd).
Hence
f(x)R−λ(−i∇ − a)2+Vψ
≤ |f(x)|R−λ(−∆)|ψ|,∀ψ∈L2(Rd),
that is,
f(x)R−λ(−i∇ − a)2+V˙
≤ |f(x)|R−λ(−∆).
Since the operator on the right hand side is compact by (iii), the lemma
follows by (i).
Lemma 2. Let λ,f,aand Vbe as in Lemma 1. Let H∞(a)be the strong
resolvent limit of the sequence of operators (−i∇ − a)2+nV as n→ ∞.
Then fR−λ(H∞(a)) is a compact operator.
Proof. Note that such strong resolvent limit H∞(a) is supposed to exist.
By (ii), for all none has
R−λ(−i∇ − a)2+nV ˙
≤R−λ(−∆),
that is,
R−λ(−i∇ − a)2+nV ψ
≤R−λ(−∆)|ψ|,∀ψ∈L2(Rd).
Take n→ ∞, and since the strong resolvent limit exists in L2(Rd), for each
ψthere exists a subsequence that converges a.e., so that
|R−λ(H∞(a)) ψ| ≤ R−λ(−∆)|ψ|,∀ψ∈L2(Rd).
Hence,
|f(x)R−λ(H∞(a)) ψ|≤|f(x)|R−λ(−∆)|ψ|,∀ψ∈L2(Rd),
and so
f(x)R−λ(H∞(a)) ˙
≤ |f(x)|R−λ(−∆).
The lemma follows by applying first (iii) and then (i).
5
The next simple remarks will be used several times in the arguments
of Subsection 3.2. If a bounded operator Ton a Hilbert space is positive
T≥0, then Tis self-adjoint and
kTk= sup
kξk=1
hT ξ, ξi= sup
kξk=1
hT1/2ξ, T 1/2ξi
= sup
kξk=1
kT1/2ξk2=kT1/2k2.
And if S, P are bounded operators, then kS P k2=kP∗S∗SP k.
3.2 Proof of Theorem 1
Denote Rn:= R−1(Hn), n≥0, and RAB := R−1(HAB), and recall that Rn
converges strongly to RAB [26].
Pick an open subset O ⊂ S◦so that dist (O,S)>0, and another open
subset U ⊃ ˜
S\O ⊃ S whose closure Uis compact. Choose a nonnegative
(real-valued) function ϕ∈C∞
0(Rd) with ϕ= 1 on U.
One may write
Rn−RAB =χO(Rn−RAB) + χS◦\O ϕ(Rn−RAB )
+χˆ
Sϕ(Rn−RAB) + χˆ
S(1 −ϕ) (Rn−RAB),
and the proof that follows consists in showing that each term on the right
hand side vanish in the norm of the space of bounded operators. This is the
motivation for the calculations below.
Lemma 3. Let ϕalso denote the multiplication operator by the function ϕ.
Then
0≤ϕ(Rn−RAB)ϕ≤ϕ(R0−RAB )ϕ,
and the operators ϕ(Rn−RAB)ϕare compact, for all n≥0. Furthermore,
ϕ(Rn−RAB)ϕconverges strongly to zero as n→ ∞.
Proof. It is known that Rn−RAB ≥0; see the proof of Theorem 10.4.2
in [14]. Thus, by Proposition 9.3.1 in [14], one has
hψ, ϕ (Rn−RAB )ϕψi=hϕψ, (Rn−RAB)ϕψi
=D(Rn−RAB)1/2ϕψ, (Rn−RAB)1/2ϕψE
=
(Rn−RAB)1/2ϕψ
≥0,∀ψ∈L2(Rd).
To conclude the second inequality in the statement of the lemma, con-
sider the quadratic forms bnassociated with the operators Hn,n≥0, and
note that b0≤bn, for all n≥0. Hence, by Lemma 10.4.4 in [14], one has
Rn≤R0. Thus,
ψ, [ϕ(R0−RAB )ϕ−ϕ(Rn−RAB)ϕ]ψ
=ψ, ϕ (R0−Rn)ϕψ=
(R0−Rn)1/2ϕψ
≥0,
6
for all ψ∈L2(Rd).
Now apply Lemma 1 with f=ϕand R−1(−i∇ − ~
a)2+V=Rn, to
conclude that ϕRnis compact and, so, that ϕRnϕis compact since it is the
composition of a compact operator with a bounded one. Similarly, apply
Lemma 2 with f=ϕand R−1(H∞(~
a)) = RAB to conclude that both ϕRAB
and ϕRABϕare also compact. Since Hnconverges to HAB in the strong
resolvent sense as n→ ∞, for ψ∈L2(Rd) one has
kϕ(Rn−RAB)ϕψk≤kϕk k(Rn−RAB) (ϕψ)k → 0,
as n→ ∞, since ϕis a bounded function. Therefore, the sequence of
operators ϕ(Rn−RAB )ϕconverges strongly to zero as n→ ∞.
The above lemma, combined with Theorems VIII-3.3 and VIII-3.5 in [23],
ensures the norm convergence
kϕ(Rn−RAB)ϕk → 0, n → ∞,
and so
(Rn−RAB)1/2ϕ
2
=kϕ(Rn−RAB)ϕk → 0,
as n→ ∞. Hence, one also gets the following norm convergence
k(Rn−RAB)ϕk=
(Rn−RAB)1/2(Rn−RAB )1/2ϕ
≤
(Rn−RAB)1/2
(Rn−RAB)1/2ϕ
≤
(R0−RAB)1/2
(Rn−RAB)1/2ϕ
→0,
(1)
as n→ ∞.
By following the proof of Lemma 2.3 in [21], one finds that there exists
a constant κ≥1 so that
kRnχOk ≤ κ n−1/2,∀n≥1.
Indeed, it is possible to find nonnegative functions η1, η2∈C∞(Rd) so that
η2
1+η2
2= 1, with η1= 1 on Oand supp η1⊂ S ◦. Furthermore, it is possible
to assume that |∇η1|and |∇η2|are bounded functions. Thus,
Hn=η1Hnη1+η2Hnη2− |∇η1|2− |∇η2|2.
Since
hψ, Hnψi=hψ, H0ψi+hψ, nχS◦ψi
=hH1/2
0ψ, H 1/2
0ψi+nZS◦
|ψ|2≥nkψk2,∀ψ∈C∞
0(S◦),
7
one has
hψ, η1Hnη1ψi=hη1ψ, Hnη1ψi ≥ nhη1ψ, η1ψi
=nhψ, η2
1ψi ≥ nhψ, χOψi,∀ψ∈C∞
0(Rd),
that is,
η1Hnη1≥nη2
1≥nχO,
and one may find a constant κso that Hn≥nχO−κ1, that is, Hn+κ1≥
nχO; without loss, one may assume that κ≥1.
Write Qn:= R−κ(Hn) = (Hn+κ1)−1and S=Q1/2
nχO. It then follows
that
hψ, Q1/2
nχOQ1/2
nψi=DQ1/2
nψ, χOQ1/2
nψE
≤n−1DQ1/2
nψ, (Hn+κ1)Q1/2
nψE
=n−1Dψ, Q1/2
n(Hn+κ1)Q1/2
nψE=n−1hψ, ψi,
for all ψ∈C∞
0(Rd), that is,
(Hn+κ1)−1/2χO(Hn+κ1)−1/2≤n−1.
Note that Sis bounded, S∗=χOQ1/2
nand
SS∗=Q1/2
nχOχOQ1/2
n=Q1/2
nχOQ1/2
n
is self-adjoint and bounded. Hence,
kQ1/2
nχOk2=kSk2=kS∗Sk=kQ1/2
nχOQ1/2
nk ≤ n−1,
and so kQ1/2
nχOk ≤ n−1/2. Therefore,
kQnχOk=kQ1/2
nQ1/2
nχOk≤kQ1/2
nkkQ1/2
nχOk
≤ kQ1/2
nχOk ≤ n−1/2,
with kQ1/2
nk ≤ 1, since κ≥1. Thus,
(Hn+κ1)−1χO
≤
(Hn+κ1)−1/2χO
≤n−1/2.
By the first resolvent identity, one has Rn=Qn+ (κ−1)RnQn,so that
RnχO=QnχO+ (κ−1)RnQnχO,
and since kRnk ≤ 1,
kRnχOk ≤ kQnχOk+ (κ−1)kRnkkQnχOk
≤n−1/2+ (κ−1)n−1/2=κ n−1/2,
8
that is, kRnχOk ≤ κ n−1/2. This inequality implies that
(2) k(Rn−RAB)χOk=kRnχOk → 0,
as n→ ∞, since RABχO≡0.
Recall that ˆ
S=S ∪ S0and write ρ= (1 −ϕ)χˆ
S; then one has ρ∈C∞.
Compute
(H0+1) [ρ(R0−RAB)] = (−∆ρ) (R0−RAB )−2 (∇ρ)· ∇ (R0−RAB )
+ρ(H0+1)R0−ρ(H0+1)RAB
+ 2i(A· ∇ρ) (R0−RAB),
but since (H0+1)R0=1and, due to the presence of the projection opera-
tor P0(see just before Theorem 1), (H0+1)RAB ≡(HAB +1)RAB =1, it
then follows that
(H0+1) [ρ(R0−RAB)] = (−∆ρ) (R0−RAB )−2 (∇ρ)· ∇ (R0−RAB )
+ 2i(A· ∇ρ) (R0−RAB),
and, since ρ= (1 −ϕ)χˆ
S, one has
(H0+1) [ρ(R0−RAB)] = (∆ϕ)χˆ
S(R0−RAB)
+ 2 (∇ϕ)χˆ
S· ∇ (R0−RAB )
−2i(A· ∇ϕ)χˆ
S(R0−RAB),
so that, after applying R0on the left, and using Lemmas 1 and 2 again, it
is found that the operator
ρ(R0−RAB) = R0χˆ
S(∆ϕ) (R0−RAB)
+ 2R0χˆ
S(∇ϕ)· ∇ (R0−RAB )
−2iR0χˆ
S(A· ∇ϕ) (R0−RAB)
is compact.
The same arguments above imply that ρ(Rn−RAB) is also compact,
and the inequality ρ(Rn−RAB)ρ≤ρ(R0−RAB)ρimplies that
(3)
(Rn−RAB) (1 −ϕ)χˆ
S
→0, n → ∞.
Now, one can write
Rn−RAB =χO(Rn−RAB) + χS◦\O ϕ(Rn−RAB )
+χˆ
Sϕ(Rn−RAB) + χˆ
S(1 −ϕ) (Rn−RAB),
and by equations (1), (2) and (3), it follows that
kRn−RABk → 0,
as n→ ∞, that is, Hnconverges to HAB in the norm resolvent sense. This
finishes the proof of Theorem 1.
9
4 Final Remarks
An important point in the proof of the norm resolvent convergence here
was the compactness of the closure of the region S◦, since this allowed the
choice of the nonnegative function ϕ∈C∞
0(Rd) with ϕ= 1 on the compact
set U ⊃ ˜
S. In some sense, such function ϕ∈C∞
0(Rd) is responsible for the
compactness of the auxiliary operators in the proof of Theorem 1.
The magnetic field may vanish outside ˜
Sor not. Note that Theorem 1
also applies to regions like (assume that a, L, > 0, and <a)
S=(x, y, z)∈R3:−L < z < L, a −<x2+y2< a,
which is a “fat” cylinder of radius aand finite length L. This may be a way
to take into account the difference between the particle running into the
finite cylindrical solenoid at a point with z=±L, from entering through
the lateral border of that cylinder.
Think of an infinitely long cylindrical solenoid in R3(i.e., the standard
Aharonov-Bohm setting). A common situation is to take into account the
symmetry along the vertical coordinate by considering approximations by
two-dimensional models [1, 28]. Note that the norm resolvent convergence
holds for such approximate models in R2, but the proof presented here does
not work for the original model in R3, since in this case the solenoid border
is not included in a compact subset.
Similarly, for fixed intensity n, if one considers a sequence of solenoids
whose lengths go to infinity in R3(e.g., as discussed in [13]), the norm
resolvent convergence is not expected to hold (furthermore, the convergence
of the corresponding sequence of vector potentials is not uniform in general).
In this way, the commutation diagram at the end of reference [13] should
hardly be improved to accommodate norm resolvent for all convergences
therein presented.
Finally, note that the result of Theorem 1 is easily adapted to subsets of
Rd, for any d≥2; the restriction to d= 2,3, was only due to the physical
interpretations related to the magnetic Aharonov-Bohm effect.
Acknowledgments
The authors acknowledge partial support from CNPq (Brazil).
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