Spectral transitions for Aharonov-Bohm Laplacians on conical layers

# Spectral transitions for Aharonov-Bohm Laplacians on conical layers

D. Krejčiřík Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 250 68, Řež near Prague, Czech Republic V. Lotoreichik Department of Theoretical Physics, Nuclear Physics Institute, Czech Academy of Sciences, 250 68, Řež near Prague, Czech Republic  and  T. Ourmières-Bonafos BCAM - Basque Center for Applied Mathematics, Alameda de Mazarredo, 14 E48009 Bilbao, Basque Country - Spain
###### Abstract.

We consider the Laplace operator in a tubular neighbourhood of a conical surface of revolution, subject to an Aharonov-Bohm magnetic field supported on the axis of symmetry and Dirichlet boundary conditions on the boundary of the domain. We show that there exists a critical total magnetic flux depending on the aperture of the conical surface for which the system undergoes an abrupt spectral transition from infinitely many eigenvalues below the essential spectrum to an empty discrete spectrum. For the critical flux we establish a Hardy-type inequality. In the regime with infinite discrete spectrum we obtain sharp spectral asymptotics with refined estimate of the remainder and investigate the dependence of the eigenvalues on the aperture of the surface and the flux of the magnetic field.

###### Key words and phrases:
Schrödinger operator, quantum layers, existence of bound states, spectral asymptotics, conical geometries
###### 2010 Mathematics Subject Classification:
Primary 35P20; Secondary 35P15, 35Q40, 35Q60, 35J10
\pdfstringdefDisableCommands

## 1. Introduction

### 1.1. Motivation and state of the art

Various physical properties of quantum systems can be explained through a careful spectral analysis of the underlying Hamiltonian. In this paper we consider the Hamiltonian of a quantum particle constrained to a tubular neighbourhood of a conical surface by hard-wall boundary conditions and subjected to an external Aharonov-Bohm magnetic field supported on the axis of symmetry. It turns out that the system exhibits a spectral transition: depending on the geometric aperture of the conical surface, there exists a critical total magnetic flux which suddenly switches from infinitely many bound states to an empty discrete spectrum.

The choice of such a system requires some comments. First, the existence of infinitely many bound states below the threshold of the essential spectrum is a common property shared by Laplacians on various conical structures. This was first found in [DEK01, CEK04], revisited in [ET10], and further analysed in [DOR15] for the Dirichlet Laplacian in the tubular neighbourhood of the conical surface. In agreement with these pioneering works, in this paper we use the term layer to denote the tubular neighbourhood. Later, the same effect was observed for other realisations of Laplacians on conical structures [BEL14, BDPR15, BR15, BPP16, LO16, P15]. Second, the motivation for combining Dirichlet Laplacians on conical layers with magnetic fields has a clear physical importance in quantum mechanics [SST69]. Informally speaking, magnetic fields act as “repulsive” interactions whereas the specific geometry of the layer acts as an “attractive” interaction. Therefore, one expects that if a magnetic field is not too strong to change the essential spectrum but strong enough to compensate the binding effect of the geometry, the number of eigenvalues can become finite or the discrete can even fully disappear.

Our main goal is to demonstrate this effect for an idealised situation of an infinitely thin and long solenoid put along the axis of symmetry of the conical layer, which is conventionally realised by a singular Aharonov-Bohm-type magnetic potential. First of all, we prove that the essential spectrum is stable under the geometric and magnetic perturbations considered in this paper. As the main result, we establish the occurrence of an abrupt spectral transition regarding the existence and number of discrete eigenvalues. In the sub-critical regime, when the magnetic field is weak, we prove the existence of infinitely many bound states below the essential spectrum and obtain a precise accumulation rate of the eigenvalues with refined estimate of the remainder. The method of this proof is inspired by [DOR15], see also [LO16]. In the case of the critical magnetic flux we obtain a global Hardy inequality which, in particular, implies that there are no bound states in the sup-critical regime.

A similar phenomenon is observed in [NR16] where it is shown that a sufficiently strong Aharonov-Bohm point interaction can remove finitely many bound states in the model of a quantum waveguide laterally coupled through a window [ESTV96, P99]. There are also many other models where a sort of competition between binding and repulsion caused by different mechanism occurs. For example, bending of a quantum waveguide acts as an attractive interaction [DE95, CDFK05] whereas twisting of it acts as a repulsive interaction [EKK08, K08]. Thus, bound states in such a waveguide exist only if the bending is in a certain sense stronger than twisting. It is also conjectured in [S00, Sec. IX] (but not proven so far) that a similar effect can arise for atomic many-body Hamiltonians at specific critical values of the nucleus charge. Here, both binding and repulsive forces are played by Coulombic interactions.

### 1.2. Aharonov-Bohm magnetic Dirichlet Laplacian on a conical layer

Given an angle , our configuration space is a -tubular neighbourhood of a conical surface of opening angle . Such a domain will be denoted here by and called a conical layer. Because of the rotational symmetry, it is best described in cylindrical coordinates.

To this purpose, let be the Cartesian coordinates on the Euclidean space and be the positive half-plane . We consider cylindrical coordinates defined via the following standard relations

 (1.1) x1=rcosϕ,x2=rsinϕ,x3=z.

For further use, we also introduce the axis of symmetry . We abbreviate by the moving frame

 er:=(cosϕ,sinϕ,0),eϕ:=(−sinϕ,cosϕ,0),ez:=(0,0,1),

associated with the cylindrical coordinates .

To introduce the conical layer with half-opening angle , we first define its meridian domain (see Figure 1.1) by

 (1.2) Gui(θ)={(r,z)∈R2+:−πsinθ

Then the conical layer associated with is defined in cylindrical coordinates (1.1) by

 (1.3) Lay(θ):=Gui(θ)×S1.

The layer can be seen as a sub-domain of constructed via rotation of the meridian domain around the axis .

For later purposes we split the boundary of into two parts defined as

 ∂0Gui(θ):={(0,z):−π

The distance between the two connected components of is said to be the width of the layer . We point out that the meridian domain is normalised so that the width of equals for any value of . This normalization simplifies notations significantly and it also preserves all possible spectral features without loss of generality, because the problem with an arbitrary width is related to the present setting by a simple scaling.

In order to define the Aharonov-Bohm magnetic field (AB-field) we are interested in, we introduce a real-valued function and the vector potential by

 (1.4) Aω(r,z,ϕ):=ω(ϕ)reϕ.

This vector potential is naturally associated with the singular AB-field

 (1.5) Bω=∇×Aω=2πΦωδΓez,

where is the -distribution supported on and is the magnetic flux

 Φω:=12π∫2π0ω(ϕ)dϕ.

Note that to check identity (1.5) it suffices to compute in the distributional sense [M, Chap. 3].

We introduce the usual cylindrical -spaces on and on

For further use, we also introduce the cylindrical Sobolev space defined as

 H1cyl(Lay(θ)):={u∈L2cyl(Lay(θ)):∫Lay(θ)(|∂ru|2+|∂zu|2+|∂ϕu|2r2)rdrdzdϕ<+∞}.

The space is endowed with the norm defined, for all , by

 ∥u∥2H1cyl(Lay(θ))=∥u∥2L2cyl(Lay(θ))+∫Lay(θ)(|∂ru|2+|∂zu|2+|∂ϕu|2r2)rdrdzdϕ.

Now, we define the non-negative symmetric densely defined quadratic form on the Hilbert space by

 (1.6) Qω,θ,0[u]:=∥(i∇−Aω)u∥2L2cyl(Lay(θ)),domQω,θ,0:=C∞0(Lay(θ)).

The quadratic form is closable by [K, Thm. VI.1.27], because it can be written via integration by parts as

where the operator with is non-negative, symmetric, and densely defined in . In the sequel, it is convenient to have a special notation for the closure of

 (1.7) Qω,θ:=¯¯¯¯¯¯¯¯¯¯¯¯¯Qω,θ,0.

Now we are in a position to introduce the main object of this paper.

###### Definition 1.1.

The self-adjoint operator in associated with the form via the first representation theorem [K, Thm. VI.2.1] is regarded as the Aharonov-Bohm magnetic Dirichlet Laplacian on the conical layer .

The Hamiltonian can be seen as an idealization for a more physically realistic self-adjoint Hamiltonian associated with the closure of the quadratic form

 u∈C∞0(R2+×S1)↦∥(i∇−Aω)u∥2L2cyl(R3)+(Wu,u)L2cyl(R3)

where the potential is a piecewise constant function given by

 W(r,z,ϕ)={0,(r,z,ϕ)∈Lay(θ),W0,(r,z,ϕ)∉Lay(θ).

The strong resolvent convergence of to in the limit follows from the monotone convergence for quadratic forms [RS-I, §VIII.7].

Before going any further, we remark that with can alternatively be seen as a constant real-function in and that

 (1.8) AΦω+k−Aω=∇VwithV(ϕ):=(Φω+k)ϕ−∫ϕ0ω(ξ)dξ.

The gauge transform is defined as

 (1.9) GV:L2cyl(Lay(θ))→L2cyl(Lay(θ)),GVu:=eiVu.

Clearly, the operator is unitary. By Proposition A.1 proven in Appendix A the operators and are unitarily equivalent via the transform . Therefore, taking we can reduce the case of general to constant . For symmetry reasons is unitary equivalent to for any . Thus, the case of constant is further reduced to .

When , we remark that the quadratic form coincides with the quadratic form of a Dirichlet Laplacian in cylindrical coordinates. Moreover, we have

 ¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯C∞0(Lay(θ))∥⋅∥H1cyl(Lay(θ))=¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯¯C∞0(Lay0(θ))∥⋅∥H1cyl(Lay(θ)),

where . Consequently, the case reduces to the one analysed in [DEK01, DOR15, ET10] and we exclude it from our considerations. From now on, we assume that is a constant, without loss of generality.

 Qω,θ[u]=∫Lay(θ)(|∂ru|2+|∂zu|2+|i∂ϕu−ωu|2r2)rdrdzdϕ.

Following the strategy of [K13, §3.4.1], we consider on the Hilbert space the ordinary differential self-adjoint operator

 (1.10) hωv:=iv′−ωv,domhω:={v∈H1(S1):v(0)=v(2π)}.

The eigenvalues of are associated with the orthonormal basis of given by

 (1.11) vm(ϕ)=(2π)−1/2eimϕ,m∈Z.

For any and , we introduce the projector

 (1.12) (π[m]u)(r,z)=⟨u(r,z,ϕ),vm(ϕ)⟩L2(S1).

According to the approach of [RS78, §XIII.16], see also [DOR15, LO16] for related considerations, we can decompose , with respect to this basis, as

 (1.13) Hω,θ≅⨁m∈ZF[m]ω,θ,

where the symbol stands for the unitary equivalence relation and, for all , the operators acting on are the fibers of . They are associated through the first representation theorem with the closed, densely defined, symmetric non-negative quadratic forms

 (1.14) f[m]ω,θ[u]:=∫Gui(θ)(|∂ru|2+|∂zu|2+(m−ω)2r2|u|2)rdrdz,domf[m]ω,θ:=π[m](domQω,θ).

The domain of the operator can be deduced from the form in the standard way via the first representation theorem.

Finally, we introduce the unitary operator , . This unitary operator allows to transform the quadratic forms into other ones expressed in a flat metric. Indeed, the quadratic form is unitarily equivalent via to the form on the Hilbert space defined as

 (1.15) q[m]ω,θ[u]:=∫Gui(θ)(|∂ru|2+|∂zu|2+(m−ω)2−1/4r2|u|2)drdz,domq[m]ω,θ:=U(domf[m]ω,θ).

In fact, one can prove that is a form core for and that its form domain satisfies

 (1.16) domq[m]ω,θ=H10(Gui(θ)).

We refer to Appendix B for a justification of (1.16) and we would like to emphasise that (1.16) does not hold for but we excluded this case from our considerations.

It will be handy in what follows to drop the superscript for and to set

 (1.17) Fω,θ:=F[0]ω,θ,fω,θ:=f[0]ω,θ,qω,θ:=q[0]ω,θ.

### 1.3. Main results

We introduce a few notation before stating the main results of this paper. The set of positive integers is denoted by and the set of natural integers is denoted by . Let be a semi-bounded self-adjoint operator associated with the quadratic form . We denote by and the essential and the discrete spectrum of , respectively. By , we denote the spectrum of (i.e.).

Let and be two quadratic forms of domains and , respectively. We say that we have the form ordering if

 dom(t2)⊂dom(t1)andt1[u]≤t2[u], for% all u∈dom(t2).

We set and, for , denotes the -th Rayleigh quotient of , defined as

 Ek(T)=supu1,…,uk−1∈domtinfu∈span(u1,…,uk−1)⊥u∈domt∖{0}t[u]∥u∥2.

From the min-max principle (see e.g.[RS78, Chap. XIII]), we know that if , the -th Rayleigh quotient is a discrete eigenvalue of finite multiplicity. Especially, we have the following description of the discrete spectrum below

Consequently, if , it is the -th eigenvalue with multiplicity taken into account. We define the counting function of as

 NE(T):=#{k∈N:Ek(T)

When working with the quadratic form , we use the notations , , , , and instead.

Our first result gives the description of the essential spectrum of .

###### Theorem 1.2.

Let and . There holds,

 σess(Hω,θ)=[1,+∞).

The minimum at of the essential spectrum is a consequence of the normalisation of the width of to . The method of the proof of Theorem 1.2 relies on a construction of singular sequences as well as on form decomposition techniques. A similar approach is used e.g. in [CEK04, DEK01, ET10] for Dirichlet conical layers without magnetic fields and in [BEL14] for Schrödinger operators with -interactions supported on conical surfaces. In this paper we simplify the argument by constructing singular sequences in the generalized sense [KL14] on the level of quadratic forms.

Now we state a proposition that gives a lower bound on the spectra of the fibers with .

###### Proposition 1.3.

Let and . There holds

 infσ(F[m]ω,θ)≥1,∀m≠0.

Relying on this proposition and on Theorem 1.2, we see that the investigation of the discrete spectrum of reduces to the axisymmetric fiber of decomposition (1.13). When there is no magnetic field () this result can be found in [ET10, Prop. 3.1]. An analogous statement holds also for -interactions supported on conical surfaces [LO16, Prop. 2.5].

Now, we formulate a result on the ordering between Rayleigh quotients.

###### Proposition 1.4.

Let , , and . Then

 Ek(Fω1,θ1)≤Ek(Fω2,θ2).

holds for all .

If the Rayleigh quotients in Proposition 1.4 are indeed eigenvalues, we get immediately an ordering of the eigenvalues for different apertures and values of . In particular, if , we obtain that the Rayleigh quotients are non-decreasing functions of the aperture . The latter property is reminiscent of analogous results for broken waveguides [DLR12, Prop. 3.1] and for Dirichlet conical layers without magnetic fields [DOR15, Prop. 1.2]. A similar claim also holds for -interactions supported on broken lines [EN03, Prop. 5.12] and on conical surfaces [LO16, Prop. 1.3]. The new aspect of Proposition 1.4 is that we obtain a monotonicity result with respect to two parameters. Proposition 1.4 implies that the eigenvalues are non-decreasing if we weaken the magnetic field and compensate by making the aperture of the conical layer smaller and vice versa.

The next theorem is the first main result of this paper.

###### Theorem 1.5.

Let and . The following statements hold.

• For , .

• For , and

 N1−E(Fω,θ)=√cos2θ−4ω24πsinθ|lnE|+O(1),E→0+.

For a fixed , Theorem 1.5 yields the existence of a critical flux

 (1.18) ωcr=ωcr(θ):=cosθ2

at which the number of eigenvalues undergoes an abrupt transition from infinity to zero. This is, to our knowledge, the first example of a geometrically non-trivial model that exhibits such a behaviour. In comparison, in the special case , this phenomenon arises at which is geometrically simple because the domain can be seen in the Cartesian coordinates as the layer between two parallel planes at distance .

The spectral asymptotics proven in Theorem 1.5 (ii) is reminiscent of [DOR15, Thm. 1.4]. However, it can be seen that the magnetic field enters the coefficient in front of the main term. As a slight improvement upon [DOR15, Thm. 1.4], in Theorem 1.5 we explicitly state that the remainder in this asymptotics is just . The main new feature in Theorem 1.5, compared to the previous publications on the subject, is the absence of discrete spectrum for strong magnetic fields stated in Theorem 1.5 (i). This result is achieved by proving a Hardy-type inequality for the quadratic form . This inequality is the second main result of this paper. It is also of independent interest in view of potential applications in the context of the associated heat semigroup, cf. [K13, CK14].

###### Theorem 1.6 (Hardy-type inequality).

Let . There exists such that

 (1.19) qθ[u]−∥u∥2L2(Gui(θ))≥c∫Gui(θ)(rcosθ−zsinθ)31+r2sin2θln2(rcosθ2rcosθ−zsinθ)|u|2drdz

holds for any .

Finally, we point out that Theorem 1.6 implies that for any

 (1.20) #σdisc(Hωcr,θ−μV)=0

holds for all sufficiently small . This observation can be extended to some potentials , but we can not derive (1.20) for any from Theorem 1.6, because the weight on the right-hand side of (1.19) vanishes on the part of satisfying . It is an open question whether a global Hardy inequality with weight non-vanishing on the whole can be proven.

### 1.4. Structure of the paper

In Section 2 we prove Theorem 1.2 about the structure of the essential spectrum. In Section 3 we reduce the analysis of the discrete spectrum of to the discrete spectrum of its axisymmetric fiber, prove Proposition 1.4 about inequalities between the Rayleigh quotients, and Theorem 1.5 (ii) on infiniteness of the discrete spectrum and its spectral asymptotics. Theorem 1.5 (i) on absence of discrete spectrum and Theorem 1.6 on a Hardy-type inequality are proven in Section 4. Some technical arguments are gathered into Appendices A and B.

## 2. Essential spectrum

In this section we prove Theorem 1.2 on the structure of the essential spectrum of . Observe that for any the form ordering follows directly from (1.14). Hence, according to decomposition (1.13), to prove Theorem 1.2 it suffices only to verify which is equivalent to checking that .

To simplify the argument we reformulate the problem in another set of coordinates performing the rotation

 (2.1) s=zcosθ+rsinθ,t=−zsinθ+rcosθ,

that transforms the meridian domain into the half-strip with corner (see Figure 2.1) defined by

 (2.2) Ωθ={(s,t)∈R×(0,π):s>−tcotθ}.

In the sequel of this subsection, and denote the inner product and the norm on , respectively.

Rotation (2.1) naturally defines a unitary operator

 (2.3) Uθ:L2(Ωθ)→L2(Gui(θ)),(Uθu)(r,z):=u(zcosθ+rsinθ,−zsinθ+rcosθ),

and induces a new quadratic form

 (2.4a) hω,θ[u]:=qω,θ[Uθu]=∫Ωθ(|∂su|2+|∂tu|2−γ|u|2(s+tcotθ)2)dsdt,domhω,θ:=H10(Ωθ), (2.4b) whereγ=γ(ω,θ):=1/4−ω2sin2θ.

Since the form is unitarily equivalent to , proving Theorem 1.2 is equivalent to showing that . We split this verification into checking the two inclusions.

### 2.1. The inclusion σess(hω,θ)⊃[1,+∞)

We verify this inclusion by constructing singular sequences for in the generalized sense [KL14, App. A] for every point of the interval . Let us start by fixing a function such that . For all , we define the functions , , as

 (2.5) un,p(s,t):=(1√nχ(sn)exp(ips))(√2πsin(t)).

According to (1.16) it is not difficult to check that . It is also convenient to introduce the associated functions , , as

 vn,p(s,t):=(1n3/2χ′(sn)exp(ips))(√2πsin(t)),wn,p(s,t):=(1√nχ(sn)exp(ips))(√2πcos(t)).

First, we get

 (2.6) ∥un,p∥2 =2π∫π0∫2nn1n∣∣χ(sn)∣∣2sin2(t)dsdt=1. (2.7) ∥vn,p∥2 =2π1n2∫π0∫2nn1n∣∣χ′(sn)∣∣2sin2(t)dsdt=1n2∥χ′∥2L2(1,2)→0,n→∞.

Further, we compute the partial derivatives and

 (2.8) (∂sun,p)(s,t)=ipun,p(s,t)+vn,p(s,t),(∂tun,p)(s,t)=wn,p(s,t),

and we define an auxiliary potential by

 (2.9) Vω,θ(s,t):=γ(ω,θ)(s+tcotθ)2.

For any we have

Integrating by parts and applying the Cauchy-Schwarz inequality we obtain

 |Jn,p(ϕ)|=∣∣−⟨ϕ,ip∂sun,p+∂twn,p⟩−(1+p2)⟨ϕ,un,p⟩∣∣=∣∣⟨ϕ,p2un,p+un,p⟩−(1+p2)⟨ϕ,un,p⟩−⟨ϕ,ipvn,p⟩∣∣=|⟨ϕ,ipvn,p⟩|≤p∥ϕ∥∥vn,p∥.

Applying the Cauchy-Schwarz inequality once again and using (2.6) and (2.8) we get

 |Kn,p(ϕ)|≤∥ϕ∥sup(s,t)∈(n,2n)×(0,π)|Vω,θ(s,t)|+∥∇ϕ∥∥vn,p∥=γn2∥ϕ∥+∥∇ϕ∥∥vn,p∥.

Let us define the norm as

 ∥ϕ∥2+1:=hω,θ[ϕ]+∥ϕ∥2,ϕ∈domhω,θ.

Clearly, and, moreover, for sufficiently small , it holds

 ω(ε):=√1/4+(1−ε)−1(ω2−1/4)∈(0,1/2]

and

 ∥ϕ∥2+1≥hω,θ[ϕ]=ε∥∇ϕ∥2+(1−ε)hω(ε),θ[ϕ]≥ε∥∇ϕ∥2,

where we used in the last step. Therefore, for any , , we have by (2.7)

 (2.10) |In,p(ϕ)|∥ϕ∥+1≤|Jn,p(ϕ)|∥ϕ∥+1+|Kn,p(ϕ)|∥ϕ∥+1≤p∥vn,p∥+γn2+ε−1/2∥vn,p∥→0,n→∞.

Here, the upper bound on is given by a vanishing sequence which is independent of .

Since the supports of and with are disjoint, the sequence converges weakly to zero. Hence, (2.6) and (2.10) imply that is a singular sequence in the generalized sense [KL14, App. A] for corresponding to the point . Therefore, by [KL14, Thm. 5], for all and it follows that .

### 2.2. The inclusion σess(hω,θ)⊂[1,+∞)

We check this inclusion using the form decomposition method. For we define two subsets of

 (2.11) Ω+n:={(s,t)∈Ωθ:sn},

as shown in Figure 2.2. For the sake of simplicity we do not indicate dependence of on . We also introduce

 Λn:={(s,t)∈Ωθ:s=n}.

For we set . Further, we introduce the Sobolev-type spaces

 (2.12) H10,N(Ω±n):={u∈H1(Ω±n):u|∂Ω±n∖Λn=0}

and consider the following quadratic forms

 (2.13) h±ω,θ,n[u]:=∫Ω±n(|∂su±|2+|∂tu±|2−Vω,θ|u±|2)dsdt,domh±ω,θ,n:=H10,N(Ω±n),

where is as in (2.9).

One can verify that the form is closed, densely defined, symmetric and semibounded from below in .

Due to the compact embedding of into the spectrum of is purely discrete. The spectrum of can be estimated from below as follows

 (2.14) infσ(h−ω,θ,n)≥1−sup(s,t)∈Ω−nVω,θ(s,t)=1−γn2.

The discreteness of the spectrum for and the estimate (2.14) imply that

 infσess(h+ω,θ,n⊕h−ω,θ,n)≥1−γn2.

Notice that the ordering holds. Hence, by the min-max principle we have

 infσess(hω,θ)≥infσess(h+ω,θ,n⊕h−ω,θ,n)≥1−γn2,

and passing to the limit we get .

## 3. Discrete spectrum

The aim of this section is to discuss properties of the discrete spectrum of , which has the physical meaning of quantum bound states. In subsection 3.1 we reduce the study of the discrete spectrum of to its axisymmetric fiber introduced in (1.17). Then, in subsection 3.2, we prove Proposition 1.4 about the ordering of the Rayleigh quotients. Finally, in subsection 3.3, we are interested in the asymptotics of the counting function in the regime and we give a proof of Theorem 1.5 (ii).

### 3.1. Reduction to the axisymmetric operator

The goal of this subsection is to prove Proposition 1.3. In the proof we use the strategy developed in [DOR15, ET10] for Dirichlet conical layers without magnetic fields.

Consider the quadratic forms in the flat metric given in (1.15). For all and , we have . Consequently, for any , we get

 (3.1) q[m]ω,θ[u]≥