Magnetic Schr¨ odinger operators with discrete spectra on non-compact K¨ ahler manifolds

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Magnetic Schr¨ odinger operators with discrete spectra on non-compact K¨ ahler manifolds

Nicolae Anghel

Abstract

We identify a class of magnetic Schrödinger operators on Kähler manifolds which exhibit pure point spectrum. To this end we embed the Schrödinger problem into a Dirac-type problem via a parallel spinor and use a Bochner-Weitzenböck argument to prove our spectral discreteness criterion.

1. Introduction

Let (M, g) be a complete non-compact oriented Riemannian manifold of dimensionn≥2, with Riemannian metricg, and letabe areal1-form onM, of classC^∞. Thenainduces a metric connection∇^a on the trivial Hermitian bundleM ×C, identifiable to the first order differential operator

C^∞(M,C)3φ7−→d^aφ:=dφ+iφa∈C^∞(M, T^∗M⊗C), where d represents ordinary exterior differentiation andi =√

−1. As usual, the Riemannian metric allows one to consider pointwise Hermitian products h·,·ix, x ∈ M, in the complexified cotangent bundle T^∗M ⊗C and, via the volume form, global (integrated) Hermitian products (·,·), in the spaces C_cpt^∞(M,C) andC_cpt^∞(M,C⊗T^∗M). With respect to these products the formal adjoint (d^a)^∗ofd^a can be defined as a first order differential operator,

(d^a)^∗:C^∞(M,C⊗T^∗M)−→C^∞(M,C),

Key Words: Magnetic Schr¨odinger operator, Magnetic field, Discrete spectrum, Dirac operator, K¨ahler manifold.

2010 Mathematics Subject Classification: Primary: 35J10, 58J50. Secondary: 35P05, 47F05, 53C55, 81V10.

Received: August, 2011.

Accepted: February, 2012.

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and then the magnetic Schrödinger operator (magnetic bottle) with magnetic potential a is the second order differential operator Ha := (dâ)^∗dâ, viewed as an unbounded operator in L²(M,C). (see Section 2 for more details). It is known that regardless of a, Ha with domain C_cpt^∞(M,C) is an essentially self-adjoint operator inL²(M,C) [S1].

There is a great deal of work, especially on Euclidean spaces M = Rⁿ, dedicated to deciding which magnetic Schr¨odinger operatorsH_a have discrete spectrum, that is a spectrum consisting only in isolated eigenvalues of finite multiplicity [AHS, I, KS, A1]. Typically, these works provide sufficient conditions for spectral discreteness, in terms of the magnetic field B associated to a,B:=da.

The purpose of this note is to provide one more result along these lines, in the case M is a K¨ahler manifold with K¨ahler formω and Riemannian metric g naturally induced byω. This result can easily be seen to generalize that of [A1], whennis even.

Theorem. LetM be a non-compact Kähler manifold with Kähler formωand Riemannian metric induced byω. Assume thatHa is a magnetic Schrödinger operator onM associated to a real1-formaof classC^∞. ThenHahas discrete spectrum if the real-valued functionhB(x), ω(x)ionM, whereh·,·idenotes the natural pointwise inner product on2-forms, satisfies the condition

x→∞limhB(x), ω(x)i=−∞. (1) 2. Magnetic Schr¨odinger operators on manifolds

Let (M, g) be a complete non-compact oriented Riemannian (C^∞) manifold of dimensionn, equipped with the metricg. On the usual realC^∞-bundles ofp-forms onM, Λ^p(T^∗M), 0≤p≤n, consider the standard inner products h·,·ix, x ∈ M. Specifically, if (e1, e2, . . . , en) is an oriented local orthonormal frame in the tangent bundle T M, with local dual frame of 1-forms in the cotangent bundleT^∗M, (e^∗₁, e^∗₂, . . . , e^∗_n), then a local orthonormal basis of Λ^p(T^∗M) is{e^∗_J}_J,e^∗_J:=e^∗_j₁∧e^∗_j₂∧ · · · ∧e^∗_j_p, whereJ runs through the set of all multi-indices 1≤j1< j2<· · ·< jp≤n.

There is a Levi-Civit`a metric connection∇^LCon Λ^p(T^∗M), extending naturally the Levi-Civit`a connection on T^∗M, the exterior product connection;

For a local vector fieldeinT M and local formsv^∗inT^∗M andφin Λ^p(T^∗M),

∇^LC_e (v^∗∧φ) =∇^LC_e v^∗∧φ+v^∗∧ ∇^LC_e φ. (2) Denote now by Ω^p(M,C) :=C^∞(M,Λ^p(T^∗M)⊗C) the Hermitian vector space ofC^∞complex globalp-forms and by

d: Ω^p(M,C)−→Ω^p+1(M,C)

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the usual exterior differential. In terms of the complexified Levi-Civit`a metric connection∇^LC on Λ^p(T^∗M)⊗C,dcan be written locally as

d=

n

X

j=1

e^∗_j∧ ∇^LC_e

j .

Fix now a ∈ Ω¹(M,R) a real global 1-form. Then the twisted differential d^a:=d+ia∧, defined on Ω^p(M,C) by

Ω^p(M,C)3φ7−→d^aφ=dφ+ia∧φ∈Ω^p+1(M,C), has the local frame counterpart

d^a =

n

X

j=1

e^∗_j ∧ ∇^LC,a_e_j ,

where ∇^LC,a is the twisted metric connection on Λ^p(T^∗M)⊗Cdefined by

∇^LC,a_v φ=∇^LC_v φ+ia(v)φ, vglobal vector field inT M, φ∈Ω^p(M,C). (3) Forφ∈Ω^p(M,C) andψ∈Ω^p_cpt(M,C) the global Hermitian product (φ, ψ) :=

R

Mhφ, ψidvol induces the formal adjoint (dâ)^∗ ofdâ, (dâ)^∗: Ω^p+1(M,C)−→Ω^p(M,C), subject to

((d^a)^∗φ, ψ) = (φ, d^aψ), φ∈Ω^p+1(M,C), ψ∈Ω^p_cpt(M,C).

It follows that locally

(d^a)^∗=−

n

X

j=1

e_jy∇^LC,a_e

j ,

wheree_jydenotes interior multiplication (contraction) by the local vector field e_j.

Making in the above discussion p = 0 we get a second order differential operator

H_a := (d^a)^∗d^a:C^∞(M,C)−→C^∞(M,C).

Seen as an unbounded operator inL²(M,C), the completion ofC_cpt^∞(M,C) with respect to (·,·), H_a is called the (scalar) magnetic Schr¨odinger operator generated by the potential a. It is then a nice exercise to see that in a local frame,

Ha=−

n

X

j=1

(ej+ia(ej))²+

n

X

j=1

∇^LC_e_j ej+ia(∇^LC_e_j ej) .

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Ha with domainC_cpt^∞(M,C) can be closed in only one way in L²(M,C), i.e., Ha is an essentially self-adjoint operator [S1].

In this note we will be interested in reasonably simple conditions on M and a which would ensure that Ha has pure point spectrum. We therefore conclude this section with a general criterion for spectral discreteness.

Proposition 1. H_abeing defined as above, if there is a functionf ∈C⁰(M,R), lim_x→∞f(x) =∞, such that

(H_aφ, φ)≥(f φ, φ), φ∈C_cpt^∞(M,C), (4) thenHa has discrete spectrum.

Proof. We will supply a somewhat less traditional proof to this proposition.

To this end, letW²(M, a) be the domain of the unique closed extension ofHa

from C_cpt^∞(M,C) into L²(M,C). W²(M, a) is the completion of C_cpt^∞(M,C) with respect to the Sobolev inner product (·,·)2 := (·,·) + (Ha·, Ha·). Since Ha :W²(M, a)−→L²(M,C) is self-adjoint, its spectrum is contained in the real line.

To prove the proposition it suffices to show that for every λ ∈ R the operatorH_a−λwith domain W²(M, a) is Fredholm, since for any Fredholm operator 0 is an isolated point of its spectrum, and in fact an eigenvalue with finite multiplicity.

Fix now a number λ∈R. The assumption on the function f provides a compact subsetK ofM such thatf(x)≥λ+ 1, ifx∈M\K. The hypothesis (4) and the density ofC_cpt^∞(M,C) inW²(M, a) imply that

((H_a−λ)φ, φ)−((f−λ)φ, φ)_K ≥(φ, φ)_M\K, φ∈W²(M, a), (5) where for a subsetU ofM, (·,·)U indicates integration is carried out only over U.

As in [A2], Ha−λ will be a Fredholm operator if we can show that any sequence{φn}n from W²(M, a), which isL²-bounded and for which {(Ha− λ)φn}n isL²-convergent, admits aL²-convergent subsequence.

Since{φn}nis bounded in the Sobolev norm|| · ||2, by Rellich’s lemma [S2]

the sequence{φn|K}nhas a convergent subsequence inL²(K,C) (assumed to be the sequence itself).

The property (5) applied now to the differences {φm−φn}m,nshows that {φn|_M\K}n is a Cauchy sequence inL²(M \K,C). We conclude that{φn}n

converges in theL²-norm, since its restrictions toK andM \K do so.

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3. Generalized Dirac operators

As mentioned in the introduction, our spectral discreteness analysis will come about by embedding the magnetic Schr¨odinger operator formalism into a Dirac-type framework. It is then desirable to briefly review here the concept of generalized Dirac bundle with its associated Dirac operator [GL].

If (M, g) is, as before, a complete non-compact oriented Riemannian manifold of dimensionn, letCl(M) be the real Clifford bundle of algebras induced by the tangent bundleT M and the Riemannian metricg. There is a canonical embedding T M ⊂Cl(M), and then the Riemannian metric and Levi-Civit`a connection extend fromT M toCl(M) in such a way that the connection∇^LC ofCl(M) preserves the metric and acts as a derivation.

A complex bundle of left modules over the bundle of algebrasCl(M), say S −→M, will be called a (generalized) Dirac bundle ifS is furnished with a Hermitian metrich·,·iand a metric connection∇^S such that

i) The action onSby unit vectors inT M ⊂Cl(M) is a pointwise isometry.

ii) The connection ∇^S is compatible with the Clifford multiplication, in the sense that for local sections einT M,φinCl(M), andsinS, we have

∇^S_e(φ·s) = ∇^LC_e φ

·s+φ· ∇^S_es .

Above, the “·” indicates the action ofCl(M) onS, while the multiplication in Cl(M) will be simply represented by juxtaposition. Since T M generates Cl(M), the action·ofCl(M) onS is completely determined by its restriction to T M.

There are several fundamental examples and constructs of Dirac bundles associated to M, which are relevant to us:

a)S=Cl(M)⊗C. In this caseCl(M) acts onSby left algebra multiplication and∇^S is the complexification of∇^LC.

b)S= Λ(T^∗M)⊗C. This case, where Λ(T^∗M) represents the real bundle of exterior forms on M, is relevant to our concept of magnetic Schr¨odinger operator, in the sense that the scalar concept we work with admits an extension to a concept of exterior form magnetic Schr¨odinger operator.

If (e1, e2, . . . , en) is a local frame in T M then the action · of ej on S is given by ej· = e^∗_j ∧ −ejy. ∇^S is the exterior form extension of the Levi- Civit`a connection∇^LC onT^∗M, cf. (2). In fact caseb) coincides with casea) under the canonical vector bundle linear isometry Λ(T^∗M)'Cl(M),e^∗_J 7−→

ej₁ej₂. . . ej_p. This is a vector bundle isomorphism which also preserves the Levi-Civit`a connections, but of course not an algebra bundle isomorphism.

c) For a K¨ahler manifold M of complex dimension m [GH] let ω be the K¨ahler 2-form and letgbe the Riemannian metric naturally induced onT Mby ω. Then the integrable complex structure J in the tangent bundleT M makes

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(T M, g) a Hermitian bundle, and there is a complex linear isometry between (T M,J) and the Hermitian bundle of (0,1)-formsT^∗0,1M ⊂T^∗M ⊗C. Since M is K¨ahler this isometry takes the Levi-Civit`a connection of T M to the unique anti-holomorphic Hermitian connection ∇z on T^∗0,1M. Then S :=

Λ(T^∗0,1M) is a Dirac bundle, when endowed with a Clifford multiplication similar to that of caseb), via the above-said complex isometry, and with the exterior product connection induced by, and extending,∇z[B].

d) IfM is aspinmanifold [LM] then S can be taken to be the spinor bundle Σ(M) of M. To be more specific, for a spin manifold the principal SO(n)- bundle P_SO(M) of oriented frames in T M lifts to a principal Spin-bundle PSpin(M), equivariantly with respect to the 2-cover map Spin(n)−→SO(n).

The spinor bundle Σ(M) is then the fiber product Σ(M) :=PSpin(M)×µ∆, where ∆ is an irreducible representation of the Euclidean Clifford algebra onn generatorsCln⊗Candµis the unitary representationµ: Spin(n)−→U(∆) induced by the left multiplication with elements of Spin(n)⊂Cln⊗C. We get then the compatible connection∇^Spin of Σ(M) by lifting the Riemannian connection onPSO(M) toPSpin(M), via the Lie algebra isomorphism so(n)' spin(n).

e) IfS is a Dirac bundle andF is any Hermitian bundle overM, equipped with a metric connection ∇^F, then the twisted bundle S⊗F is naturally a Dirac bundle, with Clifford multiplication induced by that ofSand connection

∇^S⊗F :=∇^S⊗Id+Id⊗ ∇^E.

Any Dirac bundle S generates a distinguished differential operator D_S : C^∞(M, S)−→C^∞(M, S), the generalized Dirac operator, defined as follows:

If m:T^∗M ⊗S −→ S denotes the restriction to T^∗M (metrically identified withT M) of the Clifford action·ofCl(M) onS, thenDS =m◦ ∇^S. Locally, DS admits the representation

D_S=

n

X

j=1

e_j·∇^S_e_j,

where as usual (e1, e2, . . . , en) is a local orthonormal frame inT M.

Since M is complete, DS with domain C_cpt^∞(M, S) is an essentially self- adjoint first order elliptic differential operator inL²(M, S) [GL].

Clearly, the Dirac operator associated toS= Λ(T^∗M)⊗C(caseb) above) is d+d^∗, where d is the exterior differential andd^∗ its formal adjoint, as in section 2.

In case c), when M is a K¨ahler manifold and S = Λ(T^∗0,1M) the Dirac operator becomes √

2(∂+∂^∗), where∂ is the Dolbeault operator and ∂^∗ its formal adjoint [B].

On a spin manifoldM the Dirac operator associated to the spinor bundle Σ(M) of cased) is called theclassicalDirac operator.

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For the square of a generalized Dirac operatorDS the following Bochner- Witzenb¨ock formula holds true [GL],

D²_S = ∇^S∗

∇^S+R^S,

whereR^S is the Hermitian curvature bundle morphism acting onS according to the formula

R^S=X

j<k

e_j·e_k·R^S_e

j,e_k, R^S_e

j,e_k = [∇^S_e_j,∇^S_e_k]− ∇^S_[e_j_,e_k_]. In caseb),R^Λ(T^∗^M)⊗C preserves Λ^p(T^∗M)⊗Cand evidently, R^Λ(T^∗^M)⊗C|_Λ0(T^∗M)⊗C= 0.

In case d), R^Σ(M) = k/4, where k is the scalar curvature of the spin manifoldM (Lichnerowicz’s theorem [LM]).

In casee),R^S⊗F can be written as R^S⊗F =R^S⊗Id+X

j<k

e_j·e_k·⊗R^F_e_j_,e_k. (6)

If F = Ca, the trivial bundle M ×C equipped with the metric connection

∇â associated to some real 1-forma∈Ω¹(M,R), as in the introduction, then S⊗Ca =S, and so (6) becomesR^S⊗Câ =R^S+iρâ·, whereρâ is the global section ofCl(M) given by

ρ^a=X

j<k

R^a_e

j,e_ke_je_k, R^a_e

j,e_k =e_j(a(e_k))−e_k(a(e_j))−a([e_j, e_k]). (7) It is elementary to see that under the linear isometry Λ(T^∗M)'Cl(M) explained at caseb) above,ρ^a∈C^∞(M, Cl(M)) is the image of the real 2-form B =da∈Ω²(M,R).

Finally, ifS= Λ(T^∗M)⊗CandF =Ca, then∇^(Λ(T^∗^M)⊗C)⊗C^a=∇^LC,a, in the notation of section 2, cf. (3). The connection Laplacian (∇^LC,a)^∗∇^LC,a can then be called an exterior form magnetic Schr¨odinger operator, since it restricts toH_a on Ω⁰(M,C).

4. Our results

We are now ready to state and prove an abstract discreteness criterion for certain Ha’s and, as an application, supply a proof to the theorem given in the introduction.

Proposition 2. Suppose that are given a non-compact Riemannian manifold (M, g), a real1-form a∈Ω¹(M,R)with associated scalar Schr¨odinger opera- torHa, and a generalized Dirac bundleS overM with Clifford multiplication

·, compatible connection∇^S, and Dirac operatorDS.

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In addition, suppose that there exists a ∇^S-parallel global section σ ∈ C^∞(M, S)such that

x→∞limhiρâ·σ, σi=−∞, (8) where ρâ is the global section of Cl(M) given by (7). Then the magnetic Schrödinger operator Ha has discrete spectrum.

Proof. Consider the twisted Dirac bundle S ⊗Ca and its Dirac operator D_S⊗C_a. We have the Bochner-Weitzenb¨ock formula

D²_S⊗C_a= ∇^S⊗C^a^∗

∇^S⊗C^a+R^S+iρ^a·,

which will be applied to sections of type φσ=σ⊗φ∈C_cpt^∞(M, S⊗Ca), for arbitraryφ∈C_cpt^∞(M,C).

Therefore, D_S⊗C²

a(φσ), φσ

= ∇^S⊗C^aσ⊗φ,∇^S⊗C^aσ⊗φ

+ φR^Sσ, ρσ

+(iφρâ·σ, φσ). (9) However,∇^S⊗Câσ⊗φ=∇^Sσ⊗φ+σ⊗dâφ=σ⊗dâφ, sinceσis∇^S-parallel.

For the same reason, R^Sσ = 0. By the hypothesis (8),σ is non-trivial, and since∇^S is a metric connection,hσ, σiis a (positive) constant function onM. By scalingσappropriately we can assume thathσ, σi= 1.

Consequently, ∇^S⊗C^aσ⊗φ,∇^S⊗C^aσ⊗φ

= (σ⊗d^aφ, σ⊗d^aφ) = R

Mhσ, σihd^aφ, d^aφidvol=R

Mhd^aφ, d^aφidvol= (Haφ, φ).

Equation (9) now becomes

||D_S⊗C_a(φσ)||²= (Haφ, φ) + (hiρ^a·σ, σiφ, φ), which implies

(Haφ, φ)≥(−hiρ^a·σ, σiφ, φ).

The result follows by applying Proposition 1 to the function f =−ihρ^a· σ, σi, in the presence of the hypothesis (8).

A successful application of the above proposition rests obviously on the ability of finding Dirac bundles with non-trivial parallel sectionsσ for which hρ^a·σ, σican be effectively computed. This is indeed the case with the theorem stated in the introduction.

Proof of the Theorem. For a K¨ahler manifold of complex dimension m, n = 2m. Ifω is the K¨ahler form inducing the Riemannian metricg and if J is the integrable complex structure on T M then there is a local orthonormal frame (e1,Je1, e2,Je2, . . . , em,Jem) inT M such thatω=e^∗₁∧(Je1)^∗+e^∗₂∧(Je2)^∗+

· · ·+e^∗_m∧(Jem)^∗. Expanding on the discussion on K¨ahler manifolds initiated in

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section 3, casec),T^∗0,1M is the space dual toT^0,1M :={v∈T^∗M⊗C|Jv=

−iv}. Since a local orthonormal basis of T^0,1M is {1, 2, . . . , m}, j :=

√1

2(ej+iJej), a local orthonormal basis ofT^∗0,1M will be{1∗, 2∗, . . . , m∗}, with j∗ := ^√¹

2 e^∗_j−i(Jej)^∗

. So, for the Dirac bundle Λ(T^∗0,1M) a local orthonormal basis for Λ^p(T^∗0,1M) is{J∗}_J,J∗=j₁∗∧j₂∗∧. . . j_p∗,J = (j1, j2, . . . , jp)p-multi-index.

The Clifford multiplication in Λ(T^∗0,1M) is then implemented by setting e_j·=_j^∗∧ −_jy , (Je_j)·=i(_j^∗∧+_jy), j= 1,2, . . . , m. (10) In preparation for applying proposition 2 notice thatσ:= 1∈

C^∞(M,Λ⁰(T^∗0,1M)) is a parallel section of Λ(T^∗0,1M)). An elementary cal- culation based on (10) and (7) shows now that

hiρ^a·σ, σi=

m

X

j=1

R_e^a

j,Je_j.

The theorem follows from proposition 2 and the hypothesis (1), sincea= Pm

j=1a(ej)e^∗_j+Pm

j=1a(Jej)(Jej)^∗implieshda, ωi=Pm j=1R^a_e

j,Je_j =hiρ^a·σ, σi.

Acknowledgment. Part of this work was completed while the author visited the Simion Stoilow Mathematical Institute of the Romanian Academy (IMAR) on a Bitdefender Professorship. The author would like to thank IMAR for hospitality and support.

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[S1] M. Shubin, Essential Self-Adjointness for Magnetic Schrödinger Oper- ators on Non-Compact Manifolds, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, (1999).

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Nicolae ANGHEL,

Department of Mathematics, University of North Texas, Denton, TX 76203, USA.

Email: [email protected]