Infiniteness of zero modes for the Pauli operator with singular magnetic field

0.33 MB·English

by  Grigori Rozenblum

INFINITENESS OF ZERO MODES FOR THE PAULI OPERATOR
WITH SINGULAR MAGNETIC FIELD
GRIGORIROZENBLUMANDNIKOLAISHIROKOV
5
0
0 Abstract. We establish that the Pauli operator describing a spin-1/2 two-
2 dimensionalquantumsystemwithasingularmagneticfieldhas,undercertain
n conditions,aninfinite-dimensionalspaceofzeromodes,possibly,bothspin-up
a andspin-down,moreoverthereisaspectralgapseparatingthezeroeigenvalue
J from the rest of the spectrum. In particular, infiniteness takes place if the
fieldhasinfiniteflux,whichsettlesthispreviouslyunknowncaseofAharonov-
4
Cashertheorem.
2

1
v
9
Contents
5
0 1. Introduction 1
1
2. Definition of the operator 3
0
3. Zeromodesandthespectralgap. Methodsofthetheoryofsubharmonic
5
0 functions 8
/ 4. Perturbations of the field and zero modes 19
h
References 28
p
-
h
t
a
m 1. Introduction
:
v Thepresenceofzeromodes,eigenfunctionswithzeroeigenvalues,isatypicalfea-
i turefortwo-dimensionalspin1/2quantumsystemsinvolvingmagneticfields. Such
X
eigenvalues were first found by Landau (see [15]) for the Pauli operator with con-
r
a stantmagneticfield,andthemultiplicityturnedouttobeinfinite. LaterAharonov
and Casher [3] calculated the number of zero modes for a bounded compactly sup-
ported magnetic field, and this number turned out to be finite and determined by
the total flux of the field. The conditions on the magnetic field were gradually re-
laxed,see [16,5], untilin [8] the caseof measure-valuedmagnetic fields wassettled
and an Aharonov-Casher type formula was established for a magnetic field being
a regular measure with finite total variation, thus producing a finite number of
zero modes. On the other hand, a weak perturbation of the constant magnetic
field leaves the space of zero modes infinite-dimensional ([14]). In the paper by
Shigekawa [19] it was established that if the field is sufficiently locally regular and
separatedfromzeroatinfinity(ortendsatinfinitytozerosufficientlyslowly)then,
again, the space of zero modes is infinite-dimensional. On the other hand, an ex-
ample in [8] had shown that if the total variation of the field is not finite, there
may be no zero modes at all, even if the total flux of the field, defined as a condi-
tionally convergent integral, is nonzero. Under some rather restrictive conditions,
infiniteness of zero modes was established for a periodic magnetic field, see [6], [7].
A new type of magnetic fields was recently considered in relation to the study
of zero modes by Geyler and Grishanov in [9]. They have studied a system of
equalAharonov-Bohmmagneticsolenoidsplacedatthepointsofaninfinitedouble-
periodicallatticeintheplane. Neitherofthepreviousresultsapplyforthiscase,the
1
2 ROZENBLUMANDSHIROKOV
fieldbeing verysingularandthe totalflux being infinite. Nevertheless,the authors
of[9]provedthat suchfield produces aninfinite-dimensionalzeroenergysubspace.
Moreover,bothspin-upandspin-downnullsubspacesareinfinite-dimensional. This
propertyis provedto be stable whenone adds a constant(positive)magnetic field,
of arbitrary size for the spin-down component, and not too large for the spin-up
component. Furtheron,in[10]this resultwasextendedto certainperturbationsof
this periodic structure.
Notso much is knownaboutthe rest ofthe spectrum ofthe Paulioperator. For
theLandauoperator,withconstantmagneticfield,thespectrumconsistsofLandau
levels, eigenvalues with infinite multiplicity placed at the points of an arithmetical
progression. Under a weak perturbation of the field, these eigenvalues, except the
lowest one, may split, producing a cluster of the discrete spectrum around the
Landaulevels (see [19]). Thus zeroremainto be anisolatedpoint ofthe spectrum.
On the other hand, a weak magnetic field without a background constant field
leaves the whole positive semi-axis belonging to the spectrum, so no spectral gap
arises. If the magnetic field grows unboundedly at infinity, the whole spectrum,
except zero, is discrete (see, again [19]). Under rather restricting conditions the
presence of the spectralgap was establishedin [6], [7] for a periodic field. However
in a more or less general case this question is still open.
In the present paper we study the zero modes of the Pauli operator with a non-
regular magnetic field with an infinite total flux. The typical example of the fields
in question is a, probably infinite, discrete configuration of AB solenoids on the
background of a more regular magnetic field. The Pauli operator for a strongly
singular field is not essentially self-adjoint, there is anongoing discussionon which
self-adjoint extension of the Pauli operator in the presence of AB solenoids more
adequately describes the real physical situation – see [1], [4], [22] and references
therein. It turns out that depending on which approximation to AB field by more
regular fields is chosen, with simultaneous adjustment of some other physical pa-
rameters, different self-adjoint extensions can arise. Our main analysis deals with
the so called maximal extension. Its advantage is its invariance with respect to
singulargaugetransformationsreducingtheABfluxes. Wehandlealsoanotherex-
tensionconsideredinthepaper[8],alsogaugeinvariant,butwithdifferentspectral
properties. We discuss the relations of these two extensions in Sect. 2, as well as
describe the connections ofthe study of zeromodes with problems in the theory of
analytical functions.
In Sect. 3 we find rather general conditions for the infiniteness of zero modes
and for zero being an isolatedpoint in the spectrum, with a possibility to estimate
the size of the spectral gap. We start by settling the long-standing hypothesis
(see the discussion in [8]) that a field of constant direction with infinite total flux
produces infinitely many zero modes. Further on, we show that this infiniteness
is preserved under addition of a field with different direction, having a finite flux.
ThisestablishesAharonov-Cashertheoremforthecaseofaninfinitetotalfluxofthe
field. We pass then to the case when this ’wrong’ component may have an infinite
flux. Here, we suppose that the flux of the field through any disk of a fixed size is
non-negative,atleastfarenoughfromthe origin,moreoverthe flux oftheaveraged
field is infinite. This requirement, together with some additional local conditions,
grants that the spin-down zero subspace is infinite-dimensional. If, additionally,
the above local fluxes are separated from zero, then zero is an isolated point of
the spectrum of the Pauli operator. For regular fields, this condition prevents the
infiniteness ofthe spin-upzeromodes,sincespininversioncorrespondstochanging
thesignofthefield. However,ifthediscretecomponentofthefieldislargeenough,
in other words, if sufficiently many Aharonov-Bohm solenoids are present, then it
ZERO MODES OF THE PAULI OPERATOR 3
turnsoutthatthemainconditioncanbesatisfiedforthespin-upcomponentofthe
’maximal’operatoraswell,sothereareinfinitelymanyspin-upzeromodestoo. We
alsoexplainhow the resultschangewhenwe passto the self-adjointPaulioperator
considered in [8]. Here the situation with both spin-up and spin-down zero modes
does not appear. We conclude Sect.3, by some examples, in particular the case of
a periodic andquasi-periodic magnetic field fits into the generalapproach,andthe
results of [6] and [7] are substantially extended.
Further on, we pass to the situation when the general results are not sufficient,
since the main condition of positivity of local flux may be violated. Supposing
that the magnetic field in question is a perturbation of some initial field where a
quadratic lower estimate for the potential is known, we establish such quadratic
estimate for the potential of the perturbed field, thus ensuring the infiniteness of
zero modes, but, probably, without the spectral gap. Among others, the constant
one,AB-lattice,probably,onthebackgroundofaconstantfield,aperiodicorquasi-
periodic field with some mild local regularity may serve as the unperturbed field.
Admissible perturbations are rather general, in particular they allow existence of
arbitrarylarge regions on the plane with field having ’wrong’direction. In the end
we discuss some examples where the perturbation theorems can be applied. The
estimates for the potential obtained on this way of reasoning,may be useful in the
further study of the perturbation of the Pauli operator by an electric field.
The starting point of our study was an attempt to understand the possibility of
perturbing the results of [9], by means of changing the intensities and positions of
the AB solenoids. We thank V. Geyler who attracted our attention to this kind
of problems. Further on, when it turned out that much more general situations
can be taken care of, the proof of the crucial theorem 3.2 appeared in the process
of discussions with F. Nazarov. We highly appreciate also the discussions with
B. Berndtsson on the spectral gaps and with L. Erdo¨s about the definition of the
Pauli operator. The second author (N.Sh.) was supported by the stipend from
the Swedish Royal Academy of Sciences. Both authors thank the Mittag-Leffler
Institute for hospitality when the work on the paper was in its most active phase.
2. Definition of the operator
We identify the real two-dimensional space R2 with co-ordinates x = (x ,x )
1 2
with the complex plane C, setting z = x +ix ; as usual, ∂¯= ∂ = (∂ +i∂ )/2,
1 2 z¯ 1 2
∂ =∂ =(∂ −i∂ )/2,∂ =∂ , and the Lebesgue measure will be denoted by dx.
z 1 2 j xj
Formally,thePaulioperatorinL (R2),withgyro-magneticratiog =2,isdefined
2
as the square of the Dirac operator 1
D=σ·(−i∇+A)=(σ (−i∂ +A )+σ (−i∂ +A )).
1 1 1 2 2 2
0 1 0 −i
Here σ ,σ arethe Paulimatrices,σ = , σ = , andA are
1 2 1 1 0 2 i 0 j
(cid:18) (cid:19) (cid:18) (cid:19)
real functions, components of the magnetic potential A=(A ,A ). So, P =D2 =
1 2
−(σ·(∇+iA))2.
Introducing the notations Π = −i∂ +A , Q = Π ±iΠ , we can represent
j j j ± 1 2
the Pauli operator P as
P 0 Q Q 0
+ = − + . (2.1)
0 P 0 Q Q
− + −
(cid:18) (cid:19) (cid:18) (cid:19)
Formally, the operators Q are adjoint to each other.
±
1Different sign conventions are used in the literature. We follow the sign choice made in [8],
whichistheoppositetotheonemadein,say,[5].
4 ROZENBLUMANDSHIROKOV
ThemagneticfieldisdefinedasB=curlA=∂ A −∂ A ,anditisconsideredin
1 2 2 1
the classical physics as the only actual physical reality, the potential being merely
a mathematical fiction. This is the fact also in the quantum physics, provided
the magnetic field (and therefore the potential) are not too singular. The latter
statement means that if for two magnetic potentials A ,A the equality B =
1 2
curlA = curlA holds in the proper distributional sense then the corresponding
1 2
Pauli operators are gauge equivalent: there exists a real function φ such that the
multiplication by exp(iφ) transformsone ofthe correspondingPaulioperatorsinto
another. To assign a rigorous meaning to the above statement, one has to define
the Pauli operator as a self-adjoint operator in the Hilbert space L (R2), with a
2
certain domain. The gauge transformation, the multiplication by exp(iφ), should
transformdifferentialexpressionintheproperwayaswellastransformthedomain
of one operator to the domain of the other one.
The standard definition of the Pauli operator by means of quadratic forms re-
quires A ∈ L and is described, for example in [8] or [21]. Having in mind the
2,loc
representation (2.1) of the Pauli operator, the quadratic form
ψ
p [ψ]= |σ·(−i∇+A)ψ|2dx=kQ ψ k2+kQ ψ k2,ψ = + (2.2)
A + + − − ψ
−
Z (cid:18) (cid:19)
is introduced. If the magnetic potential A is sufficiently regular, say, A ∈ L ,
4,loc
curlA ∈ L ,divA ∈ L , the quadratic form (2.2) can, actually, be obtained
2,loc 2,loc
from the expression (Pψ,ψ), ψ ∈ C∞ by means of the integration by parts, thus
0
justifying the use of (2.2) for more singular potentials. So, for A ∈ L one can
2,loc
chosesomedomaindfortheformp ,wherethisformisclosed,andaccepttheself-
A
adjoint operator corresponding to this form as the Pauli operator. Unlike the case
ofthemagneticSchr¨odingeroperator,whereallreasonablechoicesofthedomainof
theformturnouttobeequivalent(see[20]),forthePaulioperatorsuchequivalence
is not established. In [8] the authors argue that the choice of the maximal domain
consisting of all functions ψ ∈ L for which (2.2) is finite, is physically reasonable
2
since this corresponds to the states with finite energy.
For magnetic fields possessing local singularities, the ones which we are going
to study further on, this description is not satisfactory, as it was explained in
[8], and another approach, based upon the scalar potential, was proposed. We
will use the same way of defining the operator, with certain modifications. In
what follows, the magnetic field will be represented by a Borel signed measure
µ having locally finite variation. We suppose, moreover, that the support of the
discrete part of the measure does not have finite accumulation points. To the
field µ we associate a scalar potential Ψ(x), a solution of the equation ∆Ψ = µ
in the sense of distributions. The corresponding vector potential is defined as
A=(A ,A )=sgradΨ=(∂ Ψ,−∂ Ψ), again in the sense of distributions.
1 2 2 1
Thequadraticform(2.2),undercertainregularityconditions,canbetransformed
to
p[ψ]=p [ψ ]+p [Ψ ]=4 |∂¯(e−Ψψ )|2e2Ψdx+4 |∂(eΨψ )|2e−2Ψdx. (2.3)
+ + − − + −
Z Z
For a field µ with singularities, is is the form (2.3) that is used for defining the
Pauli operator. The decomposition of the measure µ, µ=µ +µ leads to a
cont disc
similardecompositionofthepotential,Ψ=Ψ +Ψ .Wewillusethepotential
disc cont
Ψ constructed in [8]. This is a function satisfying the equation ∆Ψ =
cont cont
µ in the sense of distributions. It is established in [8] that such a function
cont
exists and possesses certain regularity properties, in particular, exp(±2Ψ ) ∈
cont
L (R2), ∇Ψ ∈L ,p<2.
1,loc cont p,loc
ZERO MODES OF THE PAULI OPERATOR 5
Ifonly thecontinuouspartofthe measureispresent,the naturaldomainforthe
form (2.3) consists of all functions ψ for which (2.3) is finite. Although this defi-
±
nitionisratherimplicit,thisdomainpossessesaneasilydescribablecore: the space
of functions ψ for which e−Ψψ and eΨψ are smooth functions with compact
± + −
support (see [8]).
We assume next that only the discrete part of the measure µ=µ is present,
disc
µ =2π α δ(x−λ), x∈R2. (2.4)
disc λ
λ∈Λ
X
The support Λ of µ will be supposed to be a discrete set, without finite accu-
disc
mulation points, moreover,uniformly discrete:
dist(λ,Λ\λ)≥r0,r0 >0, for any λ∈Λ. (2.5)
Each component of the discrete measure is an Aharonov-Bohm(AB) solenoid (see
[2])withflux2πα andintensityα . Weconsiderthecaseofonesolenoidfirst. The
λ λ
ABmagneticpotentialcorrespondingtoonetermin(2.4),B=2παδ(x−x0); x0 =
(x0,x0),withintensityαcanbechosenasA(x)= −αx2−x02,αx1−x01 ,r =|x−x0|.
1 2 r2 r2
The corresponding scalar potential is Ψ(x) = Ψ(cid:16)(z) = αln|z − z0(cid:17)| (here z0 =
x0+x0,fromnowon,thecomplexpictureismoreconvenient.) Thustheexpressions
1 2
exp(±Ψ) have singularities of the form |z−z0|±α at z0.
Formally, the Pauli operator with AB field admits gauge transformations. For
an integer m, we set φ(z) = exp(−imarg(z −z0)). Then the multiplication by
φ transforms the AB Pauli operator with intensity α to the one with intensity
α+m. ThepotentialcorrespondinglytransformsasΨ7→Ψ|z−z0|m. Whetherthis
transformationisaunitaryequivalenceofoperatorsdependsonhowtheself-adjoint
operator corresponding to the form (2.3) is defined.
Tomakeourdescriptionofsuchself-adjointoperatorsmoreprecise,weintroduce
the following notations. In what follows, the notations ∂¯,∂ have the meaning of
derivatives in the sense of the space of distributions D′(R2). For a closed set E,
we denote by ∂¯ ,∂ the derivatives in the sense of D′(R2\E).
E E
We explain now the way of defining the Pauli operator, proposed in [8]. For
α ∈ [−1/2,1/2), one accepts as a domain of the form (2.3) the space of such
functionsψ ∈L (R2)forwhichthederivatives∂¯(e−Ψψ )and∂(eΨψ )(thustaken
2 + −
in the sense of distributions in D′(R2)) are functions, and the form is finite:
p[ψ]<∞. (2.6)
With such domain, which we denote here by d (α), the form (2.3) is closed and
EV
definestheself-adjointoperatorwhichwedenotebyP =P (α). Forαoutside
EV EV
the above interval, the operator is defined by means of the gauge transformation.
For a given α, let α∗ be the unique number in the interval [−1/2,1/2) such that
α∗−α=m is aninteger. Thenthe PaulioperatorP (α) isdefined asP (α)=
EV EV
exp(imarg(z−z0))P (α∗)exp(−imarg(z−z0)).Withthisdefinition,theoperator
EV
is automatically gauge invariant. However, for α ∈/ [−1/2,1/2), the description of
the domain does not agree with (2.6). In fact, if the distributional ∂¯(e−Ψψ ) is a
+
function,forΨ=|z−z0|α∗,thegaugetransformationleadstotheexpression∂¯((z−
z0)me−Ψψ ), which is not necessary a function, it may contain the δ-distributions
+
and its derivatives. This might be considered as a minor inconvenience, however
it leads to the unnatural absence of invariance of the number of zero modes under
the change of sign of the magnetic field, as can be seen from the version of the
Aharonov-Casher theorem in [8] (or, more easily, from the non-symmetry of the
main interval [−1/2,1/2),chosen arbitrarily - see [18] for more details).
6 ROZENBLUMANDSHIROKOV
We consider, along with the above operator, an alternative one. For a given α
we define P =P (α) as the operator corresponding to the quadratic form
max max
p [ψ]=p [ψ ]+p [ψ ]
max +max + −max −
=4 |∂¯ (e−Ψψ )|2e2Ψdx+4 |∂ (eΨψ )|2e−2Ψdx. (2.7)
{z0} + {z0} −
Z Z
defined on such functions ψ that the derivatives in (2.7) (thus understood in the
sense of D′(R2\{z0})) are functions and p [ψ] is finite. The operator P is
max max
again gauge invariant (see, again, [18] for corresponding calculations).
Both constructions can be carried over to the case of a finite or infinite system
of AB solenoids placed at the points of a discrete set Λ of the plane, as in (2.4).
We saythatthe vector-functionA(x)=(A (x),A (x)) isavectorpotentialforthe
1 2
magnetic field (2.4) if µ = curlA in the sense of D′(R2). The function Ψ
disc disc
satisfyingthePoissonequation∆Ψ =µ isthescalarpotential. Asabove,we
disc disc
define the quadratic form p by the expression in (2.3). The gauge transformations
enablechangingallintensitiesα byarbitraryintegers. Ifm ,λ∈Λ,isacollection
λ λ
of integers then the gauge transformation changing the intensity at the point λ by
2πm consists in the multiplication by a function W(z). The function W(z), as
λ
proposedin[8],equals L(z) ,L(z)=F(z)G¯(z)whereF(z)isananalyticalfunction
|L(z)|
havingzerosoforderm atthepointsλwherem >0,andG(z)haszerosoforder
λ λ
−m atthepointsλwherem <0. Foranycollectionofα ,byaddingproperm ,
λ λ λ λ
oneobtainsthereducedintensitiesα∗ ∈[−1/2,1/2),usedfordefiningtheoperator.
λ
As in the case of a single AB solenoid, the operator P ({α ,λ ∈ Λ}) is defined
EV λ
as the one gauge equivalent to P ({α∗,λ ∈ Λ}), the latter determined by the
EV λ
quadratic form (2.3) on all functions ψ for which this form is finite. Alternatively,
the maximal operator P ({α ,λ∈Λ}) is defined for any set of (non-integer) α
max λ λ
by the quadratic form p in (2.7) on the functions for which this latter form is
Λ
finite.
Finally, when both discrete and continuous components of the measure µ are
present, the discrete one located at the points λ of a discrete set Λ, the Pauli
operatorsP andP are definedin asimilar way,with only difference thatthe
EV max
scalar potential Ψ is now the sum of the potentials Ψ and Ψ corresponding
disc cont
to the discrete and continuous parts of the measure µ. We do not touch upon the
question on which of these operators (if any) describes the actual physical picture.
we keep however in mind that for a continuous measure as a field these operators
coincide.
In the generalsituation it is hard to describe the domain of these two operators
explicitly. However,morecanbesaidaboutthenullsubspaceoftheseoperators,in
otherwords,aboutthezeromodes. Note,firstofall,thatthequadraticformofp
max
is an extension of the form p . Since both forms are non-negative, this implies
EV
that Ker(P )⊂Ker(P ). This can also be seen from the direct description of
EV max
thezeromodes. Ifafunctionψ =(ψ ,ψ )liesinthe nullsubspaceofthe operator
+ −
P or P than ψ must annule the corresponding quadratic forms p , p ,
EV max EV max
which means
∂¯(e−Ψψ )=0,∂(eΨψ )=0 (2.8)
+ −
for P and
EV
∂¯ (e−Ψψ )=0,∂ (eΨψ )=0 (2.9)
Λ + Λ −
for P . Both (2.8) and (2.9) mean that the function f = e−Ψψ must be
max + +
analytical, f = eΨψ must be anti-analytical, but on different sets. For the
− −
operator P , by (2.8), these functions must be entire functions of variables z,z¯
EV
respectively. On the other hand, for the operator P , by (2.8) these functions
max
ZERO MODES OF THE PAULI OPERATOR 7
may have poles at the points λ∈Λ, but not too strong ones, so that still after the
multiplication by exp(±Ψ), they get into L .
2
Tomakethingsmoreconcretewesupposethatfromtheverybeginningthegauge
transformationismade,sothatallintensities α areinthe interval(0,1)forP .
λ max
Inthiscase,thefunctioneΨbehavesas|z−λ|αλ nearthepointλ∈Λ. Thereforethe
condition ∂ f =0, together with f exp(−Ψ)∈L leads to anti-analyticity of f
z − − 2 −
at the points of Λ as well: near a point λ ∈ Λ the function f2 must be summable
−
with weight having a singularity of the form |z −λ|−2αλ, therefore the possible
singularity of f is removable. On the other hand, for the spin-up component ψ ,
− +
the function f has to be holomorphic outside Λ but near the points of Λ it must
+
belong to L2 with weight |z−λ|2αl, which, due to αλ ∈(0,1), allows f+ to have a
simple pole at λ. This asymmetry can be reversed by changing the normalization
of the discrete part of the measure: by mean of a gauge transformation we can
decrease all intensities by 1 thus arriving at the measure µ′ with negative discrete
part having the intensity α′ = (α −1) ∈ (−1,0) at the point λ ∈ Λ. Then it is
λ λ
for the spin-up component that the null subspace is generated by entire functions,
and for the spin-down one by meromorphic functions with simple poles. We can,
moreover, take the first normalization when studying the spin-down component
and the second one for the spin-up component, thus only entire functions will be
involved. With this last agreement accepted, the derivatives involved in the forms
do not depend in the space of distributions where they are considered, so we can
painlessly omit the corresponding subscripts in our notations.
Tocomparethenullsubspacesoftwooperatorsunderconsideration,wesuppose
first that all intensities α lie in (0,1). In this case for the spin-down component
λ 2
P null subspaces coincide, being in both cases generatedby entire anti-analytical
−
functions. The null subspace of Ψ may be larger than the null subspace of
+max
Ψ ,sincethelattersubspacemayinvolvemeromorphicfunctionsf inaddition
+EV +
to entire functions for P . If some of the intensities α lie in [−1,0), both
+EV λ 2
P and P may have zero modes generated by meromorphic functions, so
+max −max
both null subspaces may turn out to be larger than the ones for P . See, again,
EV
[18] for more detailed comparison of these two self-adjoint extensions.
Nowwemakesomemoreobservationsaboutthe partΨ ofthe potential,the
disc
one responsible for the discrete part of the measure. It follows from the uniform
discreteness condition (2.5) that the discrete set Λ has a density not higher than
that of a regular lattice, more exactly,
N(R)≡♯{λ∈Λ,λ<R}=O(R2),r →∞. (2.10)
Consider the sum
z z 1 z 2
Ψ (z)=α log|z|+ α log|1− |+ℜ + ; (2.11)
disc 0 λ
λ λ 2 λ
λ∈XΛ,λ6=0 (cid:18) (cid:18) (cid:16) (cid:17) (cid:19)(cid:19)
if λ = 0 does not belong to the set Λ, the first term in (2.11) is omitted. The
series converges uniformly on any compact set in C not containing the points in
Λ, moreoverthe Laplace operatorcanbe applied term-wise,so (2.11)producesthe
required potential.
The particular case of a special interest is the one of a purely discrete measure,
a regular lattice, with all intensities equal,
Λ={λ }=ω m +ω m , α =α∈(0,1), (2.12)
m1m2 1 1 2 2 λm1m2
8 ROZENBLUMANDSHIROKOV
whereω ,ω arecomplexnumberswithnon-realω /ω . Inforthisconfiguration,as
1 2 1 2
it wasnoticed in [9] the potential Ψ is closely relatedto the Weierstrassσ-function
z z 1 z 2
σ(z)=z 1− exp + , (2.13)
λ λ 2 λ
λ6=Y0,λ∈Λ(cid:16) (cid:17) (cid:18) (cid:16) (cid:17) (cid:19)
so that Ψ˜ (z)=αlog|σ(z)| can serve as a potential for the magnetic field (2.4). It
0
was established in [17] that, the potential Ψ˜ possesses a very special property:
0
Ψ˜ (z)=α(ℜ(νz2)+m|z|2+ρ(z)), (2.14)
0
where ν is a certain coefficient determined by the lattice, m = π , S being the
2S
area of the elementary cell, and ρ a Λ-periodic function, with proper logarithmic
singularities at the points of the lattice. We do not care about the value of ν,
explicitly given in [17, 9]. Anyway, the first summand in (2.14) is a harmonic
function; we subtract it and for a regular lattice we will consider
Ψ (z)=α(m|z|2+ρ(z)), (2.15)
0
with ρ determined by (2.14). With such potential Ψ , for any entire function f(z)
0
subjectto|f(z)|≤Cexp(γ|z|2)withγ <αm,thefunctionψ (z)=exp(−α(m|z|2+
−
ρ(z)))f¯(z) belongs to L . This observation made in [9] proves infiniteness of zero
2
modes for P . Passing to the operator P , we make the gauge transforma-
−max +
tion reducing all fluxes to 2π(α − 1). For the reduced operator, the function
Ψ (z)=(α−1)(m|z|2+ρ(z)) serves as a potential, and thus any entire function
0+
f ofthe variablez producesthe L zeromode ψ =exp((α−1)(m|z|2+ρ(z)))f
+ 2 + +
of the operator P . As it is explained above, if α lies in the interval (0,1),
+max 2
infiniteness of zero modes holds also for P , and the operator P , as follows
−EV +EV
easily from the properties of Ψ , has no zero modes. In the case α ∈ [−1,0) the
0 2
spin-upandspin-downcomponents changetheir roles. InSect.4,we willshow that
suchestimates for the potential arepreservedunder certaintypes kinds ofthe per-
turbationsoftheregularABlattice,thus,inparticular,providinguswithexamples
of magnetic fields possessing arbitrarily large regions with ’wrong’ direction of the
field, but, nevertheless, with infinitely many zero modes.
So, under our normalization conditions, the study of zero modes is reduced to
the study ofexistence ofentire functions which, after being multiplied by a certain
weight get into L . This study can be done by means of explicit estimates for the
2
potential Ψ, like in [3, 5, 16, 8, 9] or by indirect methods, cf. [19]. We are going to
combine both approaches.
3. Zero modes and the spectral gap. Methods of the theory of
subharmonic functions
In this section we establish the infiniteness of zero modes under rather general
conditions. We start by proving this for the magnetic field with constant sign and
infinite flux, and then relax the positivity restriction in different ways. Further on
we find conditions for the existence of the spectral gap.
We suppose that the general conditions on the measure as formulated in the
previous section are fulfilled.
Theorem 3.1. Let µ be non-negative locally finite Borel measure on C, with the
support of the discrete part not having finite accumulation points, and the normal-
ization agreements of the previous Section be fulfilled. Then the operators P ,
−EV
P have infinitely many zero modes.
−max
Asitisexplainedintheprevioussection,itissufficienttoestablishthefollowing
fact about(anti-)analyticalfunctions, whichis valid forany non-negativemeasure.
Theorem 3.1 follows from it, with an obvious replacement of µ by 2µ.
ZERO MODES OF THE PAULI OPERATOR 9
Theorem3.2. LetΨbeasubharmonicfunctionsuchthatforthemeasureµ=∆Ψ,
µ(C)= dµ(z)=∞. (3.1)
ZC
Then the spaces of entire analytical and anti-analytical functions f such that
|f(z)|2e−Ψdx<∞ (3.2)
C
Z
are infinite-dimensional.
Proof. Of course, it suffices to establish just one part, say, about analytical func-
tions, which we are going to do, for convenience of references. The strategy of
provingthe theoremis the following. We chose a sequence of points z , n=1,...,
n
in a special way. For any given n a collection of functions f , k =1,...,n satisfy-
k
ing (3.2) will be constructed in such way that f (z ) = 0, k < l,f (z ) 6= 0. Such
k l k k
system of functions is, obviously, linearly independent.
WedenotebyD(z,R)theopendisccenteredatzwithradiusR. Allthemeasures
in the proof are supposed to be non-negative.
We take z = 0. Then we chose R so that µ(D(0,R )) ≥ 46. Then for each
1 1 1
k >1 we find R so that µ(Ω )≥20, Ω =D(0,R )\D(0,R ), Ω =D(0,R ).
k k k k k−1 1 1
This can be done due to infiniteness of (3.1). Then we take any z , strictly inside
k
Ω , not in the support of µ , k ≥2.
k disc
Now we fix n and find a sufficiently smallpositiveδ suchthat the disks D(z ,δ)
k
k =1,...,nliestrictlyinsiderespectiveΩ . Thenwefixmeasureµ ≤µsupported
k 0
in Ω such that µ (Ω ) = 26. Denote by Ψ the logarithmic potential of measure
1 0 1 0
µ ,Ψ (z)=(2π)−1 ln|z−w|dµ (w).ThefunctionΨ behavesasΨ (z)∼ 13ln|z|
0 0 0 0 0 π
as |z|→∞.
R
We set further Ψ = Ψ −Ψ , µ = µ− µ ≥ 0, so ∆Ψ = µ and, by our
1 0 1 0 1 1
construction,
µ (Ω )≥20,k=1,...,n. (3.3)
1 k
Furtheronwechosemeasuresν ≤µ ,k=1,...,n,supportedinrespectiveΩ ,and
k 1 k
such that ν (Ω )=20, and denote by U the logarithmic potential of the measure
k k k
ν , with the same asymptotic behavior U ∼ 10ln|z| for large |z|. For a positive
k k π
h < δ/2, we denote by ν′,k = 1,...,n, the measures supported in the respective
k
disks D(z ,h) and coinciding there with 20 times the Lebesgue measures,so that
k πh2
thelogarithmicpotentialsU′ ofthesemeasureshavethesameasymptoticbehavior
k
forlarge|z|asU ,U′(z)∼ 10ln|z|. WedenotebyUh(z)thesubharmonicfunction
k k π
n
Uh(z)=Ψ (z)+ (U′(z)−U (z)). (3.4)
1 k k
k=1
X
By our choice of measures, U and U′ differ controllably for large z. In fact,
k k
1 w
U′(z)−U (z)= ln 1− (dν′ −dν ), (3.5)
k k 2π z k k
ΩZk (cid:12)(cid:12) (cid:12)(cid:12)
and since for |z| ≥ 2(R +δ), |w| < R , we(cid:12) have (cid:12)ln|1−w/z| < ln2, so |U′(z)−
n n k
U (z)|<20ln2(2π)−1 <15. Adding up such estimates for all k, we obtain
k
|Uh(z)−Ψ (z)|≤15n (3.6)
1
for large |z|, |z|≥2(R +δ). Therefore, for any non-negative function v, and any
n
R≥2(R +δ),
n
vexp(−Ψ )dx≤e15n vexp(−Uh)dx, (3.7)
1
ZR≤|z|≤2R ZR≤|z|≤2R
10 ROZENBLUMANDSHIROKOV
For z small, |z| ≤ 2(R +δ) but lying outside the disks D(z ,δ), we note that
n k
each function U′(z), being the logarithmic potential of a measure supported in
k
the disk D(z ,h), h ≤ δ/2, is bounded by some constant depending on δ and
k
R , |U′(z)| ≤ C(d,R ). The potential U , being the logarithmic potential of a
n k n k
measure supported in the disk D(0,R ), is not necessarily bounded from below
n
but it is bounded from above, again, by some constant depending on δ and R ,
n
U′(z)≤C(δ,R ) for |z|≤2(R +δ). This gives us
k n n
Uh(z)−Ψ (z)= (U′(z)−U (z))≥−2nC(δ,R ), |z|≤2(R +δ),z ∈/ ∪D(z ,δ).
1 k k n n k
(3.8)
X
Next we fix a function ϕ ∈ C∞(C) such that ϕ vanishes in the disks D(z ,δ),
0 k
k < n, but ϕ(z) = 1 in D(z ,δ). To the function ∂¯ϕ we apply the theorem by
n
Ho¨rmander, see [13], Theorem 4.4.2, on the solutions of ∂¯-equation in weighed
spaces: we find a function g =g solving the equation ∂¯g =∂¯ϕ such that
h h
e−Uh
|g |2 dx≤ |∂¯ϕ|2e−Uhdx. (3.9)
h (1+|z|2)2
ZC ZC
We recall now that ∂¯ϕ(z) = 0 in the disks D(z ,δ). Therefore the estimate (3.9)
k
gives
e−Uh
|g |2 dx≤ |∂¯ϕ|2e−Uhdx. (3.10)
h (1+|z|2)2
ZC ZC\∪D(zk,δ)
Nowweusetheestimates(3.6)and(3.8)whichenableustoreplaceintheright-
hand side of (3.10) the weight exp(−Uh) by exp(−Ψ ). We obtain therefore the
1
inequality
e−Uh
|g |2 dx≤C |∂¯ϕ|2e−Ψ1dx=K (3.11)
h (1+|z|2)2
ZC ZC\∪D(zk,δ)
with a constant C depending on n,δ,R but not depending on h. The left-hand
n
side in (3.11) can be estimated from below for large R, using (3.7), which gives
|g |2e−Ψ1dx≤(1+R2)2K, (3.12)
h
Z
R≤|z|≤2R
as well as
e−Ψ1
|g |2 dx≤2K, (3.13)
h (1+|z|2)2
Z
|z|≥2(rn+δ)
so that the weighted norms of g over any annulus R < |z| < 2R are bounded
h
uniformly in h (of course, the bound may depend on R). Recalling now that ϕ is
a smoothfunction with compactsupport, we deduce from(3.12)that the weighted
L norms of g −ϕ over the annuli are bounded uniformly in h as well.
2 h
Wesetf =g −ϕ,∂¯f =∂¯g −∂¯ϕ=0,thusf isanentirefunction,moreover,
h h h h h
1/2 1/2
|f |dx≤ |f |2e−Ψ1dx eΨ1dx ≤C(R)K
h  h   
Z Z Z
R≤|z|≤2R R≤|z|≤2R R≤|z|≤2R
   
    (3.14)
Itfollowsfrom(3.14)thatinanyannulusR<|z|<2Rthefamilyofentirefunctions
{f },0<h<δ,hasboundedL -norms,thereforeithasboundedCN-normsofany
h 1
order N in a smaller annulus (see Theorem 1.2.4 in [13]) and thus, by maximum
principle, bounded CN-norms in anydisk |z|<R. Therefore,by Montel’s theorem
(see, e.g., [12], Theorem 15.2.5,) this family is compact with respect to uniform
convergence on compacts: there exists a sequence h → 0 and an entire function
l