aa r X i v : . [ m a t h - ph ] O c t Asymptotics for Erd˝os-Solovej Zero Modes inStrong Fields
Daniel M. EltonAugust 26, 2018
Abstract
We consider the strong field asymptotics for the occurrence of zero modesof certain Weyl-Dirac operators on R . In particular we are interested in thoseoperators D B for which the associated magnetic field B is given by pullingback a 2-form β from the sphere S to R using a combination of the Hopffibration and inverse stereographic projection. If R S β = 0 we show that X ≤ t ≤ T dim Ker D tB = T π (cid:12)(cid:12)(cid:12)(cid:12)Z S β (cid:12)(cid:12)(cid:12)(cid:12) Z S | β | + o ( T )as T → + ∞ . The result relies on Erd˝os and Solovej’s characterisation of thespectrum of D tB in terms of a family of Dirac operators on S , together withinformation about the strong field localisation of the Aharonov-Casher zeromodes of the latter. Keywords:
Weyl-Dirac operator, zero modes.
Suppose B is a (smooth) magnetic field on R , viewed either as a divergence freevector field B = ( B , B , B ) or as a closed 2-form B = B dx ∧ dx + B dx ∧ dx + B dx ∧ dx . Choose a corresponding magnetic potential (or 1-form) A = A dx + A dx + A dx which generates B in the sense that B = dA (such potentials exist by Poincar´e’sLemma). A Weyl-Dirac operator operator can then be defined by D R ,B = X j =1 σ j ( − i ∇ j − A j ) , (1)1here σ , σ and σ are the Pauli matrices and ∇ = ( ∇ , ∇ , ∇ ) denotes the usualgradient operator on R . The operator D R ,B acts on 2 component spinor-fieldswhich, on R , can be viewed simply as C valued functions. Standard arguments(see [T, Theorem 4.3] for example) show that D R ,B is essentially self-adjoint on C ∞ . We also use D R ,B to denote the corresponding closure which is an unboundedself-adjoint operator on L ( R , C ).We are interested in the question of when 0 is an eigenvalue of D R ,B or, equiv-alently, of determining when D R ,B has a non-trivial kernel. Definition.
Any eigenfunction of D R ,B corresponding to 0 is called a zero mode . Remark.
The potential A (and hence the operator D R ,B ) is not uniquely determinedby B . However if dA = B = dA ′ then A − A ′ = dφ for some φ ∈ C ∞ ( R ) (usingPoincar´e’s Lemma). Multiplication by e iφ then establishes a unitary equivalencebetween the operators D R ,B defined using the potentials A and A ′ . It follows thatspectral properties of D R ,B , and in particular the existence of zero modes, dependonly on B .Zero modes have been studied in a number of contexts in mathematical physicsincluding the stability of matter ([FLL], [LY]) and chiral gauge theories ([AMN1],[AMN2]). Most early work concentrated on the construction of explicit examples, in-cluding the original example ([LY]), examples with arbitrary multiplicity ([AMN2]),compact support ([E1]) and a certain rotational type of symmetry ([ES]; furtherdetails below). Some subsequent work moved toward studying the set of all zeromode producing fields (or potentials) within a given class; in particular, this setis nowhere dense ([BE1], [BE2]) and is generically a co-dimension 1 sub-manifold([E2]; slightly different classes of potentials were considered in these works).To further our understanding of which fields produce zero modes it is reasonableto consider the problem in various asymptotic regimes. We focus on the strong fieldregime (which, via a simple rescaling of the zero mode equation, is equivalent to thesemi-classical regime). For a fixed field B define a counting function N B by N B ( T ) = X ≤ t ≤ T dim Ker D R ,tB for any T ∈ R + . The behaviour of N B ( T ) as T → + ∞ is more regular than thatof dim Ker D R ,tB and clearly gives information about the occurrence of zero modesfor strong fields.In [ET] an upper bound of the form N B ( T ) ≤ C k A k L T was obtained, valid forany T ≥ A ∈ L (with B = dA ). The purpose of the present work isto determine the precise leading order asymptotic behaviour of N B ( T ) as T → + ∞ for a large class of symmetric magnetic fields first considered in [ES]. Before definingthis class we need to introduce some supporting ideas and notation.Let Ω ( S ) denote the set of 2-forms on S and let v S ∈ Ω ( S ) denote thestandard volume 2-form. Any β ∈ Ω ( S ) can then be written as β = f v S for a2nique f ∈ C ∞ ( S ). The flux of β is defined to beΦ( β ) = 12 π Z S β = 12 π Z S f v S . We also define | β | to be the (not necessarily smooth) 2-form given by | β | = | f | v S . Definition.
Let h : S → S and π : S \ { (0 , , , − } → R denote the Hopffibration and stereographic projection respectively. Set B ′ ES = (cid:8) ( π − ) ∗ h ∗ β : β ∈ Ω ( S ) , Φ( β ) = 0 (cid:9) (where ∗ denotes pullback). Define B ES similarly except without the conditionΦ( β ) = 0.Elements of B ES are closed 2-forms on R and can thus be viewed as magneticfields (note that, all 2-forms on S are closed). Furthermore fields B ∈ B ES aresmooth and satisfy bounds of the form | B ( x ) | = O ( | x | − ) as | x | → ∞ , while it isalways possible to find a smooth potential A with B = dA which satisfies boundsof the form | A ( x ) | = O ( | x | − ) as | x | → ∞ . It follows that fields in B ES (and theirassociated potentials) fall into the classes considered in [BE1], [BE2] and [E2].Our main result is the following. Theorem 1.1.
Let B ∈ B ′ ES with B = ( π − ) ∗ h ∗ β for β ∈ Ω ( S ) . Then N B ( T ) = | Φ( β ) | Φ( | β | ) T + o ( T ) as T → + ∞ . (2)The lower asymptotic bound in (2), together with the explicit form of N B ( T )for the special case of the “constant” field β = v S , were obtained in [Ta]. It isalso clear where the argument for the upper bound in [ET] may gain an order in T ,although it remains unclear whether the O ( T ) upper bound might yet be sharp forsome magnetic field B .Fields in B ES are invariant under the symmetry of R induced by the rotationof S along the S fibres of the Hopf fibration. The main work in [ES] is to showhow this symmetry can be used to express the spectrum of D R ,tB in terms of thespectra of a family of Dirac operators on S (see Section 3 for further details). Tocalculate N B ( T ) we need to consider eigenvalues of the latter with modulus up to1 /
4. Aharonov-Casher zero modes (see Theorem 2.1) correspond to an eigenvalueof 0 and contribute | Φ( β ) | to the leading order coefficient on the right hand sideof (2); when β has a variable sign the remaining part of this coefficient comes from“approximate zero modes” which arise from the localising effects of strong fields (seeSection 4 for further details).This paper is organised as follows. Some background on Dirac operators on S is outlined in Section 2 while the key results we require from [ES] are stated at the3tart of Section 3. The proof of Theorem 1.1 is then reduced to determining thelarge k asymptotics of a spectral quantity N ( k ) B relating to a family of Dirac operatorson S ; see (7) and Theorem 3.3.The relatively straightforward lower bound in Theorem 3.3 is covered in Section4. Necessary information about the asymptotic number of approximate zero modesfor Dirac operators on S is given in Theorem 4.1 and justified in Section 8 usingequivalent results for the plane (from [E3]). Section 4 concludes with further es-timates relating to approximate zero modes; some of the arguments rely on ideasfrom differential geometry and are deferred to Section 9.The remaining sections are dedicated to the justification of the upper bound inTheorem 3.3. In Section 5 the quantity N ( k ) B is expressed as the number of eigenvaluesof a (non-self-adjoint) operator L within a particular set; see Proposition 5.1. Inturn this is estimated from the singular values of L via Weyl’s inequality; Section 6 isdevoted to estimating the singular values while the argument is tied up in Section 7. Notation
We use spec( T ) to denote the set of eigenvalues of an operator T with entries re-peated according to geometric multiplicity. The subset of positive eigenvalues isdenoted by spec + ( T ). General positive constants are denoted by C , with numeri-cal subscripts used when we wish to keep track of specific constants in subsequentdiscussions. The open disc in R with radius r and centre 0 is denoted D r , while I denotes the 2 × S S we firstly recall some notions from Rie-mannian geometry as well as the idea of a spin c structure (spin c spinor bundles,Clifford multiplication and spin c connections). A fuller introduction can be foundin [F] (see also [ES] for a discussion in a similar spirit to that presented here).Let h· , ·i S denote the standard Riemannian metric on (the tangent bundle of) S , with corresponding norm |·| S . The same symbols will be used for the inducedmetric on the exterior bundle ∧ ∗ T ∗ S . For n = 0 , , n ( S ) denote the set of n -forms (that is, sections of the n -form bundle ∧ n T ∗ S ). Note that, R S v S = 4 π while | β | = | β | S v S for any β ∈ Ω ( S ).A spin c spinor bundle Ψ on S is a hermitian vector bundle over S with fibre C on which we can define Clifford multiplication. The latter is a unitary map σ : T ∗ S → Hom(Ψ) which satisfies σ ( ω ) σ ( ρ ) + σ ( ρ ) σ ( ω ) = 2 h ω, ρ i S I for all 1-forms ω and ρ ; here Hom(Ψ) denotes the set of endomorphisms on Ψwith inner product given by h A, B i Hom(Ψ) = tr( A ∗ B ), and I ∈ Hom(Ψ) is the4dentity. (Clifford multiplication gives a unitary representation of the Clifford alge-bra Cl( T ∗ x S ) on C which is isomorphic to the standard representation and variessmoothly with x ∈ S .) Clifford multiplication extends naturally as a linear isomor-phism σ : ∧ ∗ T ∗ S → Hom(Ψ); in particular σ ( ω ) σ ( v S ) + σ ( v S ) σ ( ω ) = 0 (3)for any 1-form ω , while σ ( v S ) = I . The latter expression allows us to writeΨ = L + ⊕ L − where the line bundles L ± are defined by ξ ∈ L ± iff σ ( v S ) ξ = ± ξ .We use h· , ·i Ψ and |·| Ψ to denote the (fibrewise) inner-product and norm on Ψ, whileΓ(Ψ) is the space of spinors (sections of Ψ).Associated to a spin c spinor bundle Ψ is a line bundle which (for S ) is given as L = Ψ ∧ Ψ (the determinant bundle of Ψ). This line bundle determines Ψ up toisomorphism (note that, H ( S ; Z ) ∼ = Z which has no 2-torsion). On S there areinfinitely many mutually non-isomorphic spin c spinor bundles which we denote asΨ ( k ) for k ∈ Z , labelled so that the first Chern number of the associated line bundlesatisfies c ( L ( k ) )[ v S ] = 2 k .Fix k ∈ Z . A spin c connection on Ψ ( k ) is a connection e ∇ which is compatiblewith hermitian structure on Ψ ( k ) and the Clifford multiplication. For ξ, η ∈ Γ(Ψ ( k ) )and X ∈ T S the former compatibility means X h ξ, η i Ψ ( k ) = h e ∇ X ξ, η i Ψ ( k ) + h ξ, e ∇ X η i Ψ ( k ) , while the latter means [ e ∇ X , σ ( ω )] = σ ( ∇ X ω ) for all forms ω ; here ∇ is the Levi-Civita connection on S (for the metric h· , ·i S ). As ∇ X v S = 0 we get[ e ∇ X , σ ( v S )] = 0 . (4)A spin c connection e ∇ on Ψ ( k ) is uniquely determined by a choice of (hermitian)connection on L ( k ) . It follows that the set of all spin c connections is an affine spacemodelled on i Ω ( S ) (note that, L ( k ) has structure group U (1) with Lie algebra i R ).In particular, given e ∇ any other spin c connection on Ψ ( k ) can be written as e ∇ − iα for some α ∈ Ω ( S ).The curvature of the connection e ∇ can be viewed as the Hom(Ψ ( k ) ) valued 2-formgiven by e R ( X, Y ) ξ = e ∇ X e ∇ Y ξ − e ∇ Y e ∇ X ξ − e ∇ [ X,Y ] ξ for all X, Y ∈ T S and ξ ∈ Ψ ( k ) . The magnetic -form of e ∇ is then defined to be β = i Tr( e R ) ∈ Ω ( S ). The first Chern class of L ( k ) is the cohomology class of π β so Φ( β ) = 12 π Z S β = c ( L ( k ) )[ v S ] = k ;that is, the total flux of any magnetic 2-form on Ψ ( k ) must be equal to k . Thisflux condition is also sufficient for a 2-form to be the magnetic 2-form of a spin c ( k ) . More precisely if β ′ ∈ Ω ( S ) with Φ( β ′ ) = k then β ′ = β + dα for some α ∈ Ω ( S ) (this follows from the Hodge decomposition theorem and thefact that the harmonic 2-forms on S are simply the constant multiples of v S ). Astraightforward calculation then shows β ′ is the magnetic 2-form associated to thespin c connection e ∇ ′ = e ∇ − iα . The choice of α is only unique up to the addition ofa closed 1-form.The Dirac operator corresponding to a given a spin c connection e ∇ on Ψ ( k ) isdefined as D = − i Tr σ e ∇ . If { e , e } is a local orthonormal frame (of vector fields)with corresponding dual frame { θ , θ } (of 1-forms) we can equivalently write D = − iσ ( θ ) e ∇ e − iσ ( θ ) e ∇ e . The operator D maps Γ(Ψ ( k ) ) → Γ(Ψ ( k ) ). Taking closures D becomes a(n un-bounded) self-adjoint operator on the L sections of Ψ ( k ) ; we denote the latter by H . Since D is a first order elliptic differential operator on a compact manifold it hasa compact resolvent and discrete spectrum. Furthermore (3) and (4) give D ( σ ( v S ) · ) = − σ ( v S ) D , (5)so the spectrum of D is symmetric about 0. Combined with the Aharonov-Cashertheorem ([AC]; see [ES] for the S version) we then have the following. Theorem 2.1.
For any Dirac operator D on Ψ ( k ) we have dim Ker D = | k | , whilethe spectrum of D is symmetric about .Remark. For the decomposition Ψ ( k ) = L ( k )+ ⊕ L ( k ) − (induced by σ ( v S )) (5) leads to D = (cid:18) D − D + (cid:19) with D ± : Γ( L ( k ) ± ) → Γ( L ( k ) ∓ ). The Aharonov-Casher theorem can then be viewed asa combination of the Atiyah-Singer index theorem and a vanishing theorem for D ;the former gives dim Ker D + − dim Ker D − = c ( L ( k ) )[ v S ] = k, while the latter forces either Ker D + or Ker D − to be trivial.A straightforward calculation shows that the Dirac operator associated to thespin c connection e ∇ ′ = e ∇ − iα is D ′ = D − σ ( α ). Dirac operators also satisfy asimple gauge transformation rule; if ψ ∈ C ∞ ( S ) = Ω ( S ) then e iψ D ( e − iψ · ) = D − σ ( dψ ) , the Dirac operator corresponding to the spin c connection e ∇ − idψ . In particular theDirac operators corresponding to the spin c connections e ∇ and e ∇ − idψ are unitarily6quivalent and hence have the same spectrum. It follows that the spectrum of a Diracoperator on S is determined entirely by the magnetic 2-form of the correspondingspin c connection (note that H ( S ) = 0 so d Ω ( S ) is precisely the set of closed1-forms).Let e ∇ ( k ) denote a spin c connection on Ψ ( k ) corresponding to the “constant”magnetic 2-form k v S and let D ( k ) denote the corresponding Dirac operator. If β ∈ Ω ( S ) is any other 2-form with Φ( β ) = Φ( k v S ) = k we can find α ∈ Ω ( S )with β = k v S + dα (as above). The spin c connection e ∇ ( k ) − iα then has magnetic2-form β and corresponding Dirac operator D ( k ) α = D ( k ) − σ ( α ) . (6)This operator is uniquely determined by β up to gauge (and hence unitary) equiv-alence. We can view α as generating the “non-constant” part of β .The situation for Dirac operators on S is rather simpler. All Spin c bundles on S are isomorphic to the trivial bundle S × C , while any closed 2-form b ∈ Ω ( S ) givesrise to a self-adjoint Dirac operator D S ,b , which is unique up to unitary equivalence;see [ES] for further details. S β ∈ Ω ( S ) with Φ( β ) = 1. From the above discussion we can write β = v S + dα for some α ∈ Ω ( S ). Also set b = h ∗ β , the closed 2-form on S obtained by pullingback β using the Hopf fibration h : S → S . For t ∈ R the magnetic field tb isinvariant under rotations of S along the level sets of h . This symmetry is inheritedby the Dirac operator D S ,tb , which allows the spectrum of D S ,tb to be expressedin terms of the spectra of a family of Dirac operators on S . The following is arestatement of [ES, Theorem 8.1] (note that the metric h· , ·i S is used in [ES] soeigenvalues of Dirac operators on S must include an extra factor of 2 here). Theorem 3.1.
For any t ∈ R the spectrum of D S ,tb is [ k ∈ Z Σ k ∪ (cid:8) − + p λ + ( k − t ) , − − p λ + ( k − t ) : λ ∈ spec + (cid:0) D ( k ) tα (cid:1)(cid:9) where Σ k contains the number − − sgn( k ) ( k − t ) counted with multiplicity | k | (so Σ = ∅ ). The multiplicity of an eigenvalue of D S ,tb is equal to the number of timesit appears in the above list when the elements of Σ k and spec + ( D ( k ) tα ) are countedwith their relevant multiplicities. Set B = ( π − ) ∗ b = ( π − ) ∗ h ∗ β ∈ B ′ ES . From [ES, Theorem 8.7] we have thefollowing link between the Dirac operators D S ,tb and D R ,tB .7 heorem 3.2. For any t ∈ R we have dim Ker D R ,tB = dim Ker D S ,tb . Consider the disjoint partition of R given by the intervals e τ k = ( k − , k + ] if k > − , ] if k = 0,[ k − , k + ) if k < k ∈ Z . Also let τ k = ( k − / , k + 1 /
2) and τ k = [ k − / , k + 1 /
2] denote theinterior and closure of e τ k respectively. To identify the contribution to N B comingfrom t ∈ τ k and t ∈ e τ k set M ( k ) B = X t ∈ τ k dim Ker D R ,tB and N ( k ) B = X t ∈ e τ k dim Ker D R ,tB . From Theorems 3.1 and 3.2 it is clear that Ker D R ,tB is non-trivial preciselywhen there exists k ∈ Z such that either 0 ∈ Σ k or 4 λ + ( k − t ) = 1 / λ ∈ spec + ( D ( k ) tα ), with corresponding agreement of multiplicities. In the latter casewe have λ > k − t ) < / t ∈ τ k . It follows that M ( k ) B = (cid:8) ( t, λ ) : λ ∈ spec + (cid:0) D ( k ) tα (cid:1) and 4 λ + ( k − t ) = (cid:9) . We also know that 0 is contained in the spectrum of D ( k ) tα with multiplicity | k | for any t ∈ R (see Theorem 2.1), while 0 + ( k − t ) = 1 / t = k ± / D ( k ) tα is symmetric about 0. Combiningthese observations we get (cid:8) ( t, λ ) : λ ∈ spec (cid:0) D ( k ) tα (cid:1) and 4 λ + ( k − t ) = (cid:9) = 2 M ( k ) B + 2 | k | . On the other hand 0 ∈ Σ k , with multiplicity | k | , iff t ∈ e τ k \ τ k . It follows that N ( k ) B − M ( k ) B = | k | and so N ( k ) B = 12 (cid:8) ( t, λ ) : λ ∈ spec (cid:0) D ( k ) tα (cid:1) and 4 λ + ( k − t ) = (cid:9) . (7)Clearly in calculating the right hand side of (7) we need only consider t ∈ τ k andeigenvalues of D ( k ) tα in [ − / , / | k | (the Aharonov-Casher zero modes) there may be small non-zero eigenvalues (theapproximate zero modes). The total number of these eigenvalues can be determinedasymptotically in | k | (see Theorem 4.1) which ultimately leads to the following. Theorem 3.3.
We have N ( k ) B = Φ( | β | ) | k | + o ( | k | ) as | k | → ∞ . The lower bound for N ( k ) B contained in Theorem 3.3 was given in [Ta] and isincluded here for completeness (see Section 4). The justification of the upper boundfor N ( k ) B appears in Section 7.Given Theorem 3.3 the proof of our main result is now straightforward.8 roof of Theorem 1.1. We can extend the definition of N B ( T ) to cover T < T ≤ t ≤ β ) = 1 and prove N B ( T ) = Φ( | β | ) T + o ( T ) as T → ±∞ . (8)Now let T > k T ∈ Z with T ∈ e τ k T . Then S k T − k =1 e τ k ⊂ [0 , T ] ⊂ S k T k =0 e τ k so k T − X k =1 N ( k ) B ≤ N B ( T ) ≤ k T X k =0 N ( k ) B . Using Theorem 3.3 and the fact that | k T − T | ≤ / k T X k =0 N ( k ) B = k T X k =0 [Φ( | β | ) k + o ( k )] = Φ( | β | ) k T + o ( k T ) = Φ( | β | ) T + o ( T )as T → + ∞ . Since the removal of the first and last terms from the sum will notchange this asymptotic (8) for T >
T < Throughout the next four sections we consider a fixed β ∈ Ω ( S ) with Φ( β ) = 1and write β = v S + dα for some α ∈ Ω ( S ). For each k ∈ Z and ε, R > n ( ε ) = n k,α ( ε ) = (cid:8) λ ∈ spec (cid:0) D ( k ) kα (cid:1) : | λ | ≤ ε (cid:9) (counting according to multiplicity) and n ( ε, R ) = n k,α ( ε, R ) = ( n ( R ) − n ( ε ) if R ≥ ε ,0 if R < ε .Since D ( k ) kα has | k | zero modes (recall Theorem 2.1) we have n ( ε ) ≥ | k | = | Φ( kβ ) | ; astrict inequality (for suitable ε ) reflects the presence of approximate zero modes. Ingeneral there will be O ( | k | ) approximate zero modes whenever β has variable sign;more precisely we have the following. Theorem 4.1.
Suppose ε k = Ce − c | k | ρ for some C, c > and < ρ < , while R k = o ( | k | / ) as | k | → ∞ . Then lim inf | k |→∞ | k | n k,α ( ε k ) ≥ Φ( | β | ) and lim sup | k |→∞ | k | n k,α ( R k ) ≤ Φ( | β | ) . Consequently n k,α ( ε k ) = Φ( | β | ) | k | + o ( | k | ) and n k,α ( ε k , R k ) = o ( | k | ) as | k | → ∞ . R .From (6) we get D ( k ) tα = D ( k ) − tσ ( α ) . (9)It follows that t
7→ D ( k ) tα defines a self-adjoint holomorphic family of operators. Usingstandard perturbation theory (see [K]) we can then choose real-analytic functions µ n for n ∈ Z so that the full set of eigenvalues of D ( k ) tα (including multiplicities) is { µ n ( t ) : n ∈ Z } for any t ∈ R . We can now rewrite (7) as N ( k ) B = 12 (cid:8) ( n, t ) ∈ Z × R : 4 µ n ( t ) + ( t − k ) = (cid:9) . (10) Proof of lower bound in Theorem 3.3.
Fix ε ∈ (0 , /
4) and suppose | µ n ( k ) | ≤ ε forsome n ∈ Z . Then 4 µ n ( k ) + ( k − k ) ≤ ε < /
4. However µ n is continuous and4 µ n ( k ± /
2) + ( k ± / − k ) ≥ /
4, so there are at least two values of t with4 µ n ( t ) + ( t − k ) = 1 /
4. From (10) it follows that N ( k ) B ≥ (cid:8) n ∈ Z : | µ n ( k ) | ≤ ε (cid:9) = n k,α ( ε ) . (11)The lower bound in Theorem 3.3 now follows from Theorem 4.1.The complication with obtaining the upper bound in Theorem 3.3 is that, foreach n ∈ Z with µ n ( k ) < /
4, we need upper bounds on the number of values of t with 4 µ n ( t ) + ( t − k ) = 1 /
4; in general there is no reason why this can’t be morethan two. We need some information about how rapidly µ n ( t ) can change withrespect to t . Proposition 4.2.
For j = 1 , suppose λ j is an eigenvalue of D ( k ) tα with normalisedeigenfunction ξ j . Then |h ξ , σ ( α ′ ) ξ i| ≤ π ( | λ | + | λ | ) k α ′ k L ∞ for any α ′ ∈ Ω ( S ) . The proof of this result is given in Section 9.
Remark. If ξ ∈ Ker D ( k ) tα Proposition 4.2 gives h ξ, σ ( α ′ ) ξ i = 0 for any α ′ ∈ Ω ( S ),which forces the value of ξ to lie in either L ( k )+ or L ( k ) − at each point of S . Thisresult can be viewed as a local version of the vanishing theorem underlying theAharonov-Casher theorem. Corollary 4.3.
Set a = 2 π k α k L ∞ . For any n ∈ Z and t ∈ R we have e − a | t − k | | µ n ( k ) | ≤ | µ n ( t ) | ≤ e a | t − k | | µ n ( k ) | . Proof.
Fix n . Since D ( k ) tα is a self-adjoint holomorphic family we can choose a nor-malised eigenfunction ξ ( t ) for µ n ( t ) which is real-analytic in t (see [K]). Applyingstandard first order perturbation theory to (9) then gives ddt µ n ( t ) = −h ξ ( t ) , σ ( α ) ξ ( t ) i . Thus | dµ n /dt | ≤ a | µ n | by Proposition 4.2. Integration completes the result.10et ε > | µ n ( k ) | ≤ ε (ultimately we will use ε to control the size ofthe approximate zero modes of D ( k ) kα ). For sufficiently small ε Corollary 4.3 providesenough control over the behaviour of µ n ( t ) when t ∈ τ k to ensure that there areprecisely two values of t with 4 µ n ( t ) + ( t − k ) = 1 /
4. Therefore the issue of extravalues of t can only arise when ε < | µ n ( k ) | ≤ /
4. For reasonable choices of ε Theorem 4.1 shows there are at most o ( | k | ) such eigenvalues; we need to show thatthese eigenvalues lead to at most o ( | k | ) extra values of t . Our aim (Proposition 5.1) is to re-express the quantity N ( k ) B as the number of eigen-values of some (compact non-self-adjoint) operator L within a prescribed set. Inessence this is achieved by using (9) and (10) to view N ( k ) B as the number of realeigenvalues of a quadratic spectral pencil and then linearising this pencil by movingto a suitably chosen 2 × s = t − k + 1. Then t ∈ τ k iff s ∈ J where J = [1 / , / D = D ( k )( k − α and A = σ ( α ) so (9) becomes D ( k ) tα = D − s A . Let I = I ⊗ I denote the identity on H = H ⊗ C . Introduce further operators P and Q = Q + Q on H where P = 2 D ⊗ σ + I ⊗ σ − I = (cid:18) D − I II − D − I (cid:19) , Q = I ⊗ σ = (cid:18) II (cid:19) and Q = 2 A ⊗ σ = (cid:18) A − A (cid:19) . In particular P − s Q = (cid:18) D − s A ) (1 − s ) I (1 − s ) I − D − s A ) (cid:19) − I . The operators P and Q are self-adjoint with P unbounded and Q bounded. Inparticular Dom P = (Dom D ) while P − s Q has a compact resolvent for any s ∈ R (as D − s A does). Also( P − s Q + I ) = (cid:2) D − s A ) + ( s − I (cid:3) ⊗ I . (12)Taking s = 0 we get ( P + I / ≥ I so | P | ≥ I / P = | P | U is the polardecomposition of P . It follows that | P | − / is an injective compact operator with k| P | − / k ≤ √
2. Define a further compact operator by L = U | P | − / Q | P | − / . Let C = 4 e a/ . For 0 ≤ ε ≤ /C set s ± ε = 1 ± (1 − C ε ) / J = [ s − , s +0 ]. Alsoset J + ε = [ s + ε , s +0 ] and J − ε = [ s − , s − ε ]. 11 roposition 5.1. We have N ( k ) B = 12 (cid:8) λ ∈ spec( L ) : λ − ∈ J (cid:9) . (13) Furthermore if < ε ≤ /C then (cid:8) λ ∈ spec( L ) : λ − ∈ J ± ε (cid:9) ≥ n ( ε ) . (14)Approximate zero modes correspond to the eigenvalues of L with reciprocals in J + ε and J − ε ; (14) is the corresponding restatement of (11). Proof.
From (10) and (12) we get N ( k ) B = 12 X s ∈ J (cid:8) n ∈ Z : 4 µ n ( s + k −
1) + ( s − = (cid:9) = 14 X s ∈ J dim Ker (cid:2) ( P − s Q + I ) − I (cid:3) . Now ( I ⊗ σ )( P − s Q )( I ⊗ σ ) = − ( P − s Q ) − I (note that σ = I while σ σ j σ = − σ j for j = 1 , (cid:2) ( P − s Q + I ) − I (cid:3) = dim Ker( P − s Q ) + dim Ker( P − s Q + I )= 2 dim Ker( P − s Q ) . However I − s L = U | P | − / ( P − s Q ) | P | − / so dim Ker( I − s L ) = dim Ker( P − s Q )for any s (recall that | P | − / is injective). Combining the above gives (13).Now | s ± ε − | = (1 − C ε ) / ≤ /
2. If | µ n ( k ) | ≤ ε ≤ /C for some n ∈ Z then | µ n ( k + s ± ε − | ≤ e a/ | µ n ( k ) | ≤ C ε using Corollary 4.3. It follows that4 µ n ( k + s ± ε −
1) + ( s ± ε − ≤ ( C ε ) + (1 − C ε ) ≤ . However 4 µ n ( k + s ± −
1) + ( s ± − ≥ / µ n is continuous. Thus there isat least one s ∈ J ± ε with 4 µ n ( k + s −
1) + ( s − = 1 /
4. Since n ( ε ) = { n ∈ Z : | µ n ( k ) | ≤ ε } estimate (14) now follows. Define compact self-adjoint operators by K j = | P | − / Q j | P | − / , j = 0 , K = K + K . Then L = UK so L ∗ L = K ; in particular, the singular valuesof L are simply the moduli of the eigenvalues of K . In order to study the latter wetreat K as a perturbation of K ; in turn, the spectrum of K can be determinedfrom that of D .For any d ∈ R let X d denote the symmetric 2 × X d = (cid:18) d − − d − (cid:19) . The eigenvalues of X d are − / − / − ∆ where ∆ = √ d + 1 ≥
1. Thus k| X d | − / k = (cid:0) ∆ − (cid:1) − / ≤ min {√ , | d | − / } . (15)Define a quadratic polynomial by p d ( λ ) = λ + ∆ − λ + − ∆ . Then p d (0) ≤ − / p d has one root of each sign; let κ ± ( d ) denote the reciprocalof the root with sign ±
1. Note that κ + (0) = 2 and κ − (0) = − / Lemma 6.1.
The eigenvalues of the × matrix | X d | − / σ | X d | − / are κ + ( d ) and κ − ( d ) . Furthermore ± κ ± ( d ) ≤ min { , | d | − } and | κ ± ( d ) − κ ± (0) | ≤ d . Let x ± d ∈ C denote a normalised eigenvector of | X d | − / σ | X d | − / correspondingto κ ± ( d ). Proof.
We have det( | X d | σ ) = −| det X d | = − ∆ while 2∆ | X d | + X d = (2∆ − ) I so Tr(2∆ | X d | σ ) = − Tr( X d σ ) = −
2. Thus p d is the characteristic polynomial of | X d | σ and hence | X d | / σ | X d | / . The first part of the result follows as σ − = σ ,while the second part can then be obtained from (15) and the fact that k σ k = 1.Let χ ± d = 1 /κ ± ( d ) denote the roots of p d ; in particular | χ ± d | ≥ /
2. Now p d ( λ ) isdecreasing in d for fixed λ > χ + d ≥ χ +0 = 1 /
2. Also p d (cid:0) ∆ − (cid:1) ≥ ∆ − (1 + ∆ − ) ≥ ≥
1) so χ + d ≤ ∆ − /
2. Thus 0 ≤ χ + d − χ +0 ≤ ∆ − d . On theother hand χ + d + χ − d = − ∆ − for any d so( χ + d − χ +0 ) + ( χ − d − χ − ) = 1 − ∆ − ∈ [0 , d ] . It follows that | χ − d − χ − | ≤ d . Combined we then get | κ ± ( d ) − κ ± (0) | = | χ ± d − χ ± || χ ± d | | χ ± | ≤ d ( )( ) = 16 d , completing the result. 13et ν n = µ n ( k −
1) for n ∈ Z (the eigenvalues of D ). By Corollary 4.3 we have e − a | µ n ( k ) | ≤ | ν n | ≤ e a | µ n ( k ) | . (16)Choose an orthonormal basis { ξ n : n ∈ Z } of H with D ξ n = ν n ξ n . For each n ∈ Z set κ ± n = κ ± ( ν n ) and u ± n = ξ n ⊗ x ± ν n ∈ H . The definitions of K and X d lead to K u ± n = κ ± n u ± n so, in particular, { u + n , u − n : n ∈ Z } is an eigenbasis for K .Given ε, R > M ε = { n ∈ Z : | µ n ( k ) | ≤ ε } and M ′ R = { n ∈ Z : | µ n ( k ) | > R } .Let Π ± ε , Π ′ R and Π ε,R denote the (orthogonal) spectral projections of K withRan Π ± ε = Sp { u ± n : n ∈ M ε } , Ran Π ′ R = Sp { u + n , u − n : n ∈ M ′ R } and Π ε,R = I − Π + ε − Π − ε − Π ′ R . Clearlydim Ran Π ± ε = M ε = n ( ε ) and dim Ran Π ε,R = 2 n ( ε, R ) . Lemma 6.2.
Let ε, R > . Then ± K Π ± ε ≥ while (cid:13)(cid:13) [ K − κ ± (0) I ] Π ± ε (cid:13)(cid:13) ≤ C , ε and k K Π ′ R k ≤ C , R − for some constants C , and C , .Proof. We have ± κ ± n > n while Lemma 6.1 and (16) give | κ ± n − κ ± (0) | ≤ ν n ≤ e a ε for n ∈ M ε , and | κ ± n | ≤ | ν n | − ≤ e a R − for n ∈ M ′ R . The result follows (with C , = 16 e a and C , = e a ).Next we consider K ; we begin with estimates for K restricted to certain spectralsubspaces of K . Lemma 6.3.
Suppose ε, R > and π , π ∈ { + , −} . Then k Π π ε K Π π ε k ≤ C , ε n ( ε ) and k K Π ′ R k ≤ C , R − / for some constants C , and C , .Proof. Since { u ± n : n ∈ M ε } is an orthonormal basis for Ran Π ± ε we have k Π π ε K Π π ε k ≤ X m,n ∈ M ε |h u π m , K u π n i| . (17)Now the definitions of K and Q give h u π m , K u π n i = (cid:10) | P | − / u π m , Q | P | − / u π n (cid:11) = 2 h ξ m , A ξ n i (cid:10) | X ν m | − / x π ν m , σ | X ν n | − / x π ν n (cid:11) . Note that k σ k = 1 so |h| X ν m | − / x π ν m , σ | X ν n | − / x π ν n i| ≤ m, n ∈ M ε Proposition 4.2 and (16) give |h ξ m , A ξ n i| = |h ξ m , σ ( α ) ξ n i| ≤ π ( | ν m | + | ν n | ) k α k L ∞ ≤ ae a ε. |h u π m , K u π n i| ≤ C , ε with C , = 4 ae a . Since M ε = n ( ε ) the first partof the result now follows from (17).Now let u ∈ Ran Π ′ R . Then u = P n ∈ M ′ R ξ n ⊗ z n for some z n ∈ C , so | P | − / u = X n ∈ M ′ R ξ n ⊗ | X ν n | − / z n . For n ∈ M ′ R (15) and (16) lead to k| X ν n | − / z n k ≤ | ν n | − k z n k ≤ e a R − k z n k . Since { ξ n : n ∈ M ′ R } is an orthonormal set (in H ) it follows that k| P | − / u k = X n ∈ M ′ R k| X ν n | − / z n k ≤ e a R − X n ∈ M ′ R k z n k = e a R − k u k . Therefore k| P | − / Π ′ R k ≤ e a/ R − / . Since k| P | − / k ≤ √ k Q k = 2 kAk =2 k α k L ∞ the required estimate for k K Π ′ R k follows with C , = 2 √ e a/ k α k L ∞ .For ε, R > δ ( ε, R ) = C , ε + 4 C , ε n ( ε ) + C , R − + 2 C , R − / . (18)Let { λ + n : n ∈ N } and { λ − n : n ∈ N } denote the sets of positive and negativeeigenvalues of K = K + K , enumerated to include multiplicities and ordered sothat λ − ≤ λ − ≤ · · · < < · · · ≤ λ +2 ≤ λ +1 . Proposition 6.4.
Suppose ε, R > . Then (cid:8) n ∈ N : | λ ± n | > ± κ ± (0) + δ ( ε, R ) (cid:9) ≤ n ( ε, R ) (19) and (cid:8) n ∈ N : | λ ± n | > δ ( ε, R ) (cid:9) ≤ n ( ε ) + 2 n ( ε, R ) . (20)The basic argument is a variational one with Lemmas 6.2 and 6.3 providing therelevant information about K and K respectively. Proof.
Set M = dim Ran Π ε,R = 2 n ( ε, R ) and let H ≤ H with dim H = M + 1.Choose u ∈ H with k u k = 1 and Π ε,R u = 0. Then u = (Π + ε + Π − ε + Π ′ R ) u so h u, K u i = h u, K Π + ε u i + h u, K Π − ε u i + h u, K Π ′ R u i ≤ k K Π + ε k + k K Π ′ R k (since K Π − ε ≤ h u, K u i = h u, (Π + ε + Π − ε ) K (Π + ε + Π − ε ) u i + h u, K Π ′ R u i + h K Π ′ R u, (Π + ε + Π − ε ) u i≤ k (Π + ε + Π − ε ) K (Π + ε + Π − ε ) k + 2 k K Π ′ R k . λ + M +1 ≤ k K Π + ε k + k K Π ′ R k + k (Π + ε + Π − ε ) K (Π + ε + Π − ε ) k + 2 k K Π ′ R k . Lemmas 6.2 and 6.3 then give λ + M +1 ≤ κ + (0) + δ ( ε, R ). The case of the upper signin (19) clearly follows. The lower sign can be obtained by a similar argument.Now set M = dim Ran(Π + ε + Π ε,R ) = n ( ε ) + 2 n ( ε, R ). A slightly simpler versionof the above argument leads to λ + M +1 ≤ k K Π ′ R k + k Π − ε K Π − ε k + 2 k K Π ′ R k ≤ C , R − + C , ε n ( ε ) + 2 C , R − / . Since the right hand side is clearly bounded above by δ ( ε, R ) (20) now follows. The upper bound in Theorem 3.3 follows from Theorem 4.1 if we can show that N ( k ) B − n k,α ( ε k ) is bounded from above by o ( | k | ) for suitably chosen ε k . We firstly estimatethis difference using Propositions 5.1 and 6.4 together with Weyl’s inequality. Lemma 7.1.
Suppose < ε ≤ /C and R > . Then (cid:2) − δ ( ε, R ) (cid:3)(cid:2) N ( k ) B − n k,α ( ε ) (cid:3) ≤ [ δ ( ε, R ) + C ε ] n k,α ( ε ) + 2 k K k n k,α ( ε, R ) . Proof.
Put N = 2 N ( k ) B and M = N − n ( ε ). Let Λ = (cid:8) λ ∈ spec( L ) : λ − ∈ J (cid:9) so N by Proposition 5.1. Also let K ⊂ spec( K ) denote the collection of the N eigenvalues of K with largest moduli. Since L ∗ L = K the singular values of L areprecisely the moduli of the eigenvalues of K . Weyl’s inequality ([W]) then gives X λ ∈ Λ λ ≤ X λ ∈ K | λ | . (21)For any λ ∈ Λ we have λ − ∈ J = [1 / , /
2] so λ ≥ /
3. If λ − ∈ J − ε then λ ≥ s − ε = 21 + C ε ≥ − C ε ) . From Proposition 5.1 it follows that X λ ∈ Λ λ ≥ − C ε ) n ( ε ) + [ N − n ( ε )] = 2 (cid:0) − C ε (cid:1) n ( ε ) + M. (22)Write δ = δ ( ε, R ). Proposition 6.4 shows that K has at most 2 n ( ε, R ) eigenvaluesin each of the intervals ( −∞ , − / − δ ) and (2 + δ, ∞ ), and at most n ( ε ) + 2 n ( ε, R )eigenvalues in each of the intervals ( −∞ , − δ ) and ( δ, ∞ ). Furthermore, the spectralradius of K is k K k while − (2 n ( ε ) + 4 n ( ε, R )) = M − n ( ε, R ) ≤ M . Therefore X λ ∈ K | λ | ≤ k K k n ( ε, R ) + (cid:0) + δ (cid:1) n ( ε ) + (2 + δ ) n ( ε ) + δM. (23)The result now follows when we combine (21), (22) and (23).16 emark. Key to our argument is the identification of those eigenvalues and singularvalues of L which arise from the Aharonov-Casher and approximate zero modes.These contribute n ( ε ) to each side of (21), the cancellation of which allows thequantity N ( k ) B − n k,α ( ε ) to be estimated with sufficient precision.Since k| P | − / k ≤ √ Q give k K k ≤ k Q + Q k = 2(1 + 4 k α k L ∞ ) / . (24) Proof of upper bound in Theorem 3.3.
Set ε k = e −| k | / and R k = | k | / for all k ∈ Z .As | k | → ∞ we clearly have ε k = o ( | k | − ) and R k → ∞ , while Theorem 4.1 gives n ( ε k ) = Φ( | β | ) | k | + o ( | k | ) and n ( ε k , R k ) = o ( | k | ). It follows that δ ( ε k , R k ) = o (1)(recall (18)) and so N ( k ) B − n k,α ( ε k ) = o ( | k | ) by Lemma 7.1 and (24). S S (respectively S − ) denote the sphere with the south (respectively north) poleremoved; if we view S as the unit sphere in R then S ± = S \ { (0 , , ∓ } . Let z ± : S ± → R denote stereographic projection, given by z ± ( x ) = 11 ± x ( x , x ) , x = ( x , x , x ) ∈ S ± . Set e Ω( x ) = 2(1 + | x | ) − for x ∈ R , and Ω ± = e Ω ◦ z ± . It is straightforward tocheck that the map z ± is an isometry if R is given the conformal metric e Ω h· , ·i R (where h· , ·i R is the usual Euclidean metric on R ). Hence z ∗± ( e Ω v R ) = v S (where v R = dx ∧ dx is the usual volume form on R ).For any δ ∈ [0 ,
1] set S δ, ± = S ∩ {± x < δ } ; in particular S , ± = S ± while S , + and S , − are the north and south hemispheres. It is easy to check that z ± ( S δ, ± ) = D r δ where r δ = (1 + δ ) / (1 − δ ), while we have the bounds1 − δ < e Ω( x ) ≤ , x ∈ D r δ . (25)Using the isometry z − ± we can pull-back the (restricted) spin c bundle Ψ ( k ) from S ± to get a spin c bundle on R . Since R is contractible the latter is isomorphicto the trivial bundle R × C , so sections of this bundle (spinors) can be identifiedwith maps R → C . For ξ ∈ Γ(Ψ ( k ) ) with supp( ξ ) ⊂ S ± let η = ξ ◦ z − ± denote thecorresponding map in C ∞ ( R , C ). Then k ξ k L ( S ) = Z S ± | ξ | ( k ) v S = Z R | ξ ◦ z ± | e Ω v R = k e Ω η k L ( R ) . (26)Using the isometry z ± and the above identification of spin c bundles any Dirac op-erator on S can be restricted to S ± and then considered as a Dirac operator on R e Ω h· , ·i R . Conformal mapping properties of Dirac opera-tors (see [H, Section 1.4] or [ES, Theorem 4.3]) mean the latter is simply related to aDirac operator on R with the usual metric. Under the above identification of spin c bundles a Dirac operator on R becomes a Weyl-Dirac operator corresponding to apotential A ′ = A ′ dx + A ′ dx on R ; that is, an operator given by the 2-dimensionalversion of (1). More precisely let α ∈ Ω ( S ) and consider the Dirac operator D ( k ) α on Ψ ( k ) . Then we can find A ± ∈ Ω ( R ) so that (cid:0) Ω / ± D ( k ) α Ω − / ± (cid:1) ( η ◦ z ± ) = ( D R ,A ± η ) ◦ z ± (27)for all η : R → C (note that η ◦ z ± ∈ Γ(Ψ ( k ) ± ), where Ψ ( k ) ± is the restriction ofΨ ( k ) to S ± ). Furthermore the magnetic field corresponding to D R ,A ± is simply thepull-back of that corresponding to D ( k ) α under the map z − ± ; if the latter is β = f v S then the former will be given by β ± = dA ± = ( f ◦ z − ± ) e Ω v R . In particular for anyopen subset U ⊆ R we have Z U β ± = Z z − ± ( U ) β. (28)For A ′ ∈ Ω ( R ) and r > P D r ,A ′ denote the Pauli operator on D r withmagnetic potential A ′ and Dirichlet boundary conditions; this can be defined as thenon-negative self-adjoint operator associated to the closure of the quadratic formgiven by η
7→ kD R ,A ′ η k L ( R ) for η ∈ C ∞ ( D r , C ).For the next result let D ( k ) α denote a Dirac operator on Ψ ( k ) and let A ± denotethe corresponding 1-forms on R as discussed above. Proposition 8.1.
There exists C > so that for any µ > and δ ∈ (0 , we have { λ ∈ spec( D ( k ) α ) : | λ | ≤ µ }≥ (cid:8) λ ∈ spec( P D ,A + ) : λ ≤ µ (cid:9) + (cid:8) λ ∈ spec( P D ,A − ) : λ ≤ µ (cid:9) (29) and (cid:8) λ ∈ spec( D ( k ) α ) : λ ≤ µ − C δ − (cid:9) ≤ (cid:8) λ ∈ spec( P D rδ ,A + ) : λ ≤ (4 µ ) (cid:9) + (cid:8) λ ∈ spec( P D rδ ,A − ) : λ ≤ (4 µ ) (cid:9) . (30) Proof.
Let η ± ∈ C ∞ ( D , C ). Set ξ ± = ( e Ω − / η ± ) ◦ z − ± giving ξ ± ∈ Γ(Ψ ( k ) ± ) withsupp( ξ ± ) ⊆ S , ± . Extend ξ ± by 0 and set ξ = ξ + + ξ − ∈ Γ(Ψ ( k ) ). From (25) we have e Ω ≥ D . Together with (26) and (27) we then get k ξ ± k L ( S ± ) = (cid:13)(cid:13)e Ω / η ± (cid:13)(cid:13) L ( D ) ≥ k η ± k L ( D ) kD ( k ) α ξ ± k L ( S ± ) = (cid:13)(cid:13)e Ω − / D R ,A ± η ± (cid:13)(cid:13) L ( D ) ≤ kD D ,A ± η ± k L ( D ) . Since ξ + and ξ − have disjoint support it follows that k ξ k L ( S ) = k ξ + k L ( S ) + k ξ − k L ( S − ) ≥ k η + k L ( D ) + k η − k L ( D ) and kD ( k ) α ξ k L ( S ) = kD ( k ) α ξ + k L ( S ) + kD ( k ) α ξ − k L ( S − ) ≤ kD D ,A + η + k L ( D ) + kD D ,A − η − k L ( D ) . A standard variational argument then leads to (29).Now choose non-negative functions χ δ, ± ∈ C ∞ ( S δ, ± ) so that χ δ, + + χ δ, − = 1and | dχ δ, ± | ≤ C , δ − on S , where C , is independent of δ . Let ξ ∈ Γ(Ψ ( k ) )and define compactly supported sections of Ψ ( k ) ± by setting ξ δ, ± = χ δ, ± ξ . Also set η δ, ± = e Ω / ξ δ, ± ◦ z ± giving η δ, ± ∈ C ∞ ( D r δ , C ). Then (26), (the upper bound in)(25) and (27) give k ξ δ, ± k L ( S ± ) = (cid:13)(cid:13)e Ω / η δ, ± (cid:13)(cid:13) L ( R ) ≤ k η δ, ± k L ( D rδ ) so k ξ k L ( S ) = k ξ δ, + k L ( S ) + k ξ δ, − k L ( S − ) ≤ (cid:2) k η δ, + k L ( D rδ ) + k η δ, − k L ( D rδ ) (cid:3) . Similarly kD ( k ) α ξ δ, ± k L ( S ± ) = (cid:13)(cid:13)e Ω − / D R ,A ± η δ, ± (cid:13)(cid:13) L ( R ) ≥ kD D rδ ,A ± η δ, ± k L ( D rδ ) while kD ( k ) α ξ k L ( S ) = k χ δ, + D ( k ) α ξ k L ( S ) + k χ δ, − D ( k ) α ξ k L ( S ) = (cid:13)(cid:13) D ( k ) α ξ δ, + − iσ ( dχ δ, + ) ξ (cid:13)(cid:13) L ( S ) + (cid:13)(cid:13) D ( k ) α ξ δ, − − iσ ( dχ δ, − ) ξ (cid:13)(cid:13) L ( S ) ≥ (cid:2) kD ( k ) α ξ δ, + k L ( S ) + kD ( k ) α ξ δ, − k L ( S − ) (cid:3) − C , δ − k ξ k L ( S ) . Therefore kD ( k ) α ξ k L ( S ) + 2 C , δ − k ξ k L ( S ) ≥ (cid:2) kD D rδ ,A + η δ, + k L ( D rδ ) + kD D rδ ,A − η δ, − k L ( D rδ ) (cid:3) . A standard variational argument now gives (30) (with C = 2 C , ; note that Γ(Ψ ( k ) )is a core for D ( k ) α ).We can use (27) to transfer results about approximate zero modes on R to S ;information about the former was obtained in [E3].19 roof of Theorem 4.1. For each k ∈ Z we have a Dirac operator D ( k ) kα on Ψ ( k ) withmagnetic 2-form k (cid:0) v S + dα (cid:1) . Pulling this back to R using z ± as discussed above,we can arrange so that the corresponding 1-forms on R are simply kA ± for fixed( k independent) 1-forms A ± . The corresponding field is kβ ± where β ± = dA ± . By(28) we have Z D rδ | β ± | = Z S δ, ± | β | , δ ∈ [0 , . (31)From [E3, Theorem 1.2] and (31) we getlim inf | k |→∞ | k | (cid:8) λ ∈ spec( P D ,kA ± ) : λ ≤ ε k (cid:9) ≥ π Z D | β ± | = 12 π Z S , ± | β | . Combined with Proposition 8.1 we then havelim inf | k |→∞ | k | n k,α ( ε k ) ≥ π Z S , + | β | + 12 π Z S , − | β | = Φ( | β | ) . Now let δ > e R k = 16( R k + C δ − ) for k ∈ Z . Then e R k = o ( | k | ) as | k | → ∞ , so [E3, Theorem 1.1] and (31) givelim sup | k |→∞ | k | (cid:8) λ ∈ spec( P D rδ ,kA ± ) : λ ≤ e R k (cid:9) ≤ π Z D rδ | β ± | = 12 π Z S δ, ± | β | . Combined with Proposition 8.1 we then havelim sup | k |→∞ | k | n k,α ( R k ) ≤ π Z S δ, + | β | + 12 π Z S δ, − | β | = Φ( | β | ) + O ( δ )as δ → + (note that β is bounded while | S δ, + ∩ S δ, − | = O ( δ )). Taking δ → + leadsto the stated upper bound for n k,α ( R k ). S n let d : Ω n ( S ) → Ω n − ( S ) and δ : Ω n ( S ) → Ω n − ( S ) denote theexterior derivative and its adjoint with respect to the Hodge ∗ operator. We have ∗ : Ω n ( S ) → Ω − n ( S ) with ∗∗ = ( − n and δ = − ∗ d ∗ . Also ∗ v S = 1.The expression dδ + δd defines the Laplace-de Rham operator on n -forms. For n = 0 this reduces to δd = − ∆, the negative of the Laplace-Beltrami operator on(scalar) functions. The Green’s function for the latter is given in terms of log(1 − x.y )(where the dot product is defined by viewing S as the unit sphere in R ); moreprecisely for any f ∈ C ∞ ( S ) with R S f v S = 0 we have f ( x ) = 14 π Z S log(1 − x.y ) ∆ f ( y ) v S ( y ) (32)20or all x ∈ S (see [FS, Theorem 4.15]). From this we can obtain a related integralrepresentation for 1-forms. Firstly for any y ∈ R let ρ y ∈ Ω ( S ) denote the exteriorderivative of x x.y . Proposition 9.1.
For any ω ∈ Ω ( S ) and x ∈ S we have ω ( x ) = 14 π Z S ρ y ( x )1 − x.y δω ( y ) v S ( y ) − π Z S ( ∗ ρ y )( x )1 − x.y dω ( y ) . Proof.
Suppose f ∈ C ∞ ( S ) satisfies R S f v S = 0. Taking the exterior derivative of(32) with respect to x gives df ( x ) = 14 π Z S ρ y ( x )1 − x.y δdf ( y ) v S ( y ) . (33)Now suppose ν ∈ Ω ( S ) with R S ν = 0. Set g = ∗ ν ∈ C ∞ ( S ) so δν = − ∗ dg and( δdg ) v S = dδν . Applying the Hodge ∗ to (33) then leads to δν ( x ) = − π Z S ( ∗ ρ y )( x )1 − x.y dδν ( y ) . (34)Finally suppose ω ∈ Ω ( S ). Since H ( S ) = 0 the Hodge decomposition theoremgives f ∈ C ∞ ( S ) and ν ∈ Ω ( S ) such that ω = df + δν . Since d δ v S wemay assume R S f v S = 0 = R S ν . The result now follows from (33) and (34).For any x, y ∈ S it is easy to check | ρ y ( x ) | S = | ( ∗ ρ y )( x ) | S = 1 − ( x.y ) . Astraightforward calculation then gives Z S (cid:12)(cid:12)(cid:12)(cid:12) ρ y ( x )1 − x.y (cid:12)(cid:12)(cid:12)(cid:12) S v S ( x ) = 2 π = Z S (cid:12)(cid:12)(cid:12)(cid:12) ( ∗ ρ y )( x )1 − x.y (cid:12)(cid:12)(cid:12)(cid:12) S v S ( x ) . Coupled with Proposition 9.1 we immediately get the following estimate for 1-forms.
Corollary 9.2.
For any ω ∈ Ω ( S ) we have k ω k L ≤ π ( k δω k L + k dω k L ) . When needed { e , e } denotes an orthonormal frame (of local vector fields) while { θ , θ } denotes the corresponding orthonormal dual frame (of local 1-forms). Weassume { e , e } is positively oriented so v S = θ ∧ θ . Also ∗ θ = θ and ∗ θ = − θ .For any ω ∈ Ω ( S ) we have the local expression δω = − tr ∇ ω = − (cid:2) ( ∇ e ω )( e ) + ( ∇ e ω )( e ) (cid:3) , (35)where ∇ denotes the Levi-Civita connection (on 1-forms; see [GHL, Lemma 4.8]).For any spinors ξ, η ∈ Γ(Ψ ( k ) ) let ω ξ,η ∈ Ω ( S ) be the unique 1-form satisfying h ω ξ,η , ρ i S = h ξ, σ ( ρ ) η i Ψ ( k ) for all ρ ∈ Ω ( S ). In terms of a local orthonormal frame we can write ω ξ,η = h ξ, σ ( θ ) η i Ψ ( k ) θ + h ξ, σ ( θ ) η i Ψ ( k ) θ . emma 9.3. Let e ∇ be a spin c connection on Ψ ( k ) . If ξ, η ∈ Γ(Ψ ( k ) ) and X ∈ Γ( T S ) then ∇ X ω ξ,η = ω e ∇ X ξ,η + ω ξ, e ∇ X η .Proof. We have X h ω ξ,η , ρ i S = h∇ X ω ξ,η , ρ i S + h ω ξ,η , ∇ X ρ i S while X h ξ, σ ( ρ ) η i Ψ ( k ) = h ξ, e ∇ X σ ( ρ ) η i Ψ ( k ) + h ξ, e ∇ X ( σ ( ρ ) η ) i Ψ ( k ) = h ξ, e ∇ X σ ( ρ ) η i Ψ ( k ) + h ξ, σ ( ρ ) e ∇ X σ ( ρ ) η i Ψ ( k ) + h ξ, σ ( ∇ X ρ ) η i Ψ ( k ) . The result now follows from the definition of ω ξ,η .Recall that Clifford multiplication extends naturally to 2-forms; in particular σ ( v S ) = σ ( θ ) σ ( θ ) while for any 1-form ρσ ( ρ ) σ ( v S ) = − σ ( ∗ ρ ) . (36) Proposition 9.4.
Let D be a Dirac operator on Ψ ( k ) . If ξ, η ∈ Γ(Ψ ( k ) ) then δω ξ,η = i hD ξ, η i Ψ ( k ) − i h ξ, D η i Ψ ( k ) (37) and dω ξ,η = − i (cid:2) hD ξ, σ ( v S ) η i Ψ ( k ) + h ξ, σ ( v S ) D η i Ψ ( k ) (cid:3) v S . (38) Proof.
Let e ∇ denote the spin c connection defining D . By (35) and Lemma 9.3 δω ξ,η = −∇ e ω ξ,η ( e ) − ∇ e ω ξ,η ( e )= − ω e ∇ e ξ,η ( e ) − ω e ∇ e ξ,η ( e ) − ω ξ, e ∇ e η ( e ) − ω ξ, e ∇ e η ( e )= − (cid:10)(cid:2) σ ( θ ) e ∇ e + σ ( θ ) e ∇ e (cid:3) ξ, η (cid:11) Ψ ( k ) − (cid:10) ξ, (cid:2) σ ( θ ) e ∇ e + σ ( θ ) e ∇ e (cid:3) η (cid:11) Ψ ( k ) = −h i D ξ, η i Ψ ( k ) − h ξ, i D η i Ψ ( k ) . On the other hand working in a local orthonormal frame and applying (36) gives ∗ ω ξ,η = h ξ, σ ( −∗ θ ) η i Ψ ( k ) ∗ θ + h ξ, σ ( ∗ θ ) η i Ψ ( k ) ∗ θ = −h ξ, σ ( θ ) σ ( v S ) η i Ψ ( k ) θ + h ξ, σ ( θ ) σ ( v S ) η i Ψ ( k ) ( − θ ) = ω ξ,σ ( v S ) η . Together with (5) and (37) we get δ ∗ ω ξ,η = − δω ξ,σ ( v S ) η = i hD ξ, σ ( v S ) η i Ψ ( k ) − i h ξ, − σ ( v S ) D η i Ψ ( k ) . However d = − ∗ δ ∗ and ∗ v S so (38) follows. Proof of Proposition 4.2.
Define a vector field X ′ on S by α ′ = h X ′ , ·i S . Then | X ′ | S = | α ′ | S while h ξ , σ ( α ′ ) ξ i Ψ ( k ) = ω ξ ,ξ ( X ′ ). Hence |h ξ , σ ( α ′ ) ξ i| ≤ Z S (cid:12)(cid:12) h ξ , σ ( α ′ ) ξ i Ψ ( k ) (cid:12)(cid:12) v S ≤ Z S | X ′ | S | ω ξ ,ξ | S v S ≤ k α ′ k L ∞ ( S ) k ω ξ ,ξ k L ( S ) ≤ π k α ′ k L ∞ ( S ) (cid:2) k δω ξ ,ξ k L ( S ) + k dω ξ ,ξ k L ( S ) (cid:3)
22y Corollary 9.2. On the other hand Proposition 9.4 leads to | δω ξ ,ξ | S , | dω ξ ,ξ | S ≤ (cid:12)(cid:12) D ( k ) tα ξ (cid:12)(cid:12) Ψ ( k ) | ξ | Ψ ( k ) + | ξ | Ψ ( k ) (cid:12)(cid:12) D ( k ) tα ξ (cid:12)(cid:12) Ψ ( k ) = ( | λ | + | λ | ) | ξ | Ψ ( k ) | ξ | Ψ ( k ) (note that σ ( v S ) is a unitary operator in the fibres of Ψ ( k ) ). However2 Z S | ξ | Ψ ( k ) | ξ | Ψ ( k ) v S ≤ Z S (cid:2) | ξ | ( k ) + | ξ | ( k ) (cid:3) v S = k ξ k L ( S ) + k ξ k L ( S ) = 2 . The result follows.
Acknowledgements
The author wishes to thank I. Sorrell and D. Vassiliev for several useful discussions.This research was supported by EPSRC under grant EP/E037410/1. The authoralso acknowledges the hospitality of the Isaac Newton Institute for MathematicalSciences in Cambridge, where this work was completed during the programme Pe-riodic and Ergodic Spectral Problems.
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