Open access peer-reviewed chapter

The Singular Values of the Logarithmic Potential Transform on Bound States Spaces of Landau Hamiltonians on the Poincaré Disk

Written By

M’hamed Elomari and Ali El Mfadel

Reviewed: 12 August 2022 Published: 21 December 2022

DOI: 10.5772/intechopen.107090

From the Edited Volume

Functional Calculus - Recent Advances and Development

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Abstract

In the present manuscript, we prove that the singular numbers of the Cauchy transform Lσfz=−1π∫Dfξξ−zlog1∣z−ξ∣1−ξξ¯σ−2dμξ (2) defined on the space L2,σD of complex-valued measurable functions, which are 1−ξξ¯σ−2dμξ-square integrable on D where σ>1 is a fixed parameter, are asymptotically ≈Ckm−4ν+1,ask→∞ where C is a constant.

Keywords

• the logarithmic potential transform
• the singular values
• Cauchy transform.

1. Introduction

Let D be the complex unit disk endowed with its Lebesgue measure μ and let ∂ID be its boundary. Denote by L2D the space of complex-valued measurable functions, which are square integrable on D. The logarithmic potential transform: L2DL2D is defined by

Lfz=1πDfξξzlog1zξξ.E1

This operator is very important as the transformed Cauchy and it often appears in Analysis [1].

The dimensional analysis [1, 2] and scaling arguments form an integral part of theoretical physics to solve some important problems without doing much calculation.

The logarithmic potential in physics forms an interesting one as it provides some unusual predictions about the system. Moreover, this potential can be used suitably to illustrate some of the important features of field theory such as dimensional regularization and renormalization. In most of our textbooks, this potential is not discussed in detail; although the calculations are quite simple to demonstrate some of its unique features. We have obtained the bound state energy of this logarithmic potential through uncertainty principle, phase space quantization, and the Hellmann-Feynman theorem.

In Ref. [3] the authors have been dealing with the restriction of L to the space La2D of analytic μ-square integrable on D. They precisely have considered the projection operator P0: L2DLa2D and they have proved that the singular values λk of LP0, (which turn out to be eigenvalues of the operator LP0LP0 behave like k1 as k goes to . They also concluded that LP0 belongs to the Schatten class S1,.

Now, consider the following weighted logarithmic potential transform

Lσfz=1πDfξξzlog1zξ1ξξ¯σ2ξ,E2

defined on the space L2,σD of complex-valued measurable functions, which are 1ξξ¯σ2ξ-square integrable on D where σ>1 is a fixed parameter. We observe that the subspace La2,σD of analytic functions on D and belonging to L2,σD coincides with the eigenspace

A0σDψL2,σDΔσψ=0,E3

of the second order differential operator

Δσ41zz¯1zz¯2zz¯σz¯z¯,E4

known as the σ-weight Maass Laplacian and its discrete eigenvalues are given by

εm4mσ1m,m=0,1,2,,σ1/2,E5

with their corresponding eigenspaces

AmσDψL2,σDandΔσψ=εmσψ,E6

are here called generalized Bergman spaces since …

After noticing that, we here deal with analogous questions as in Ref. [3] in the context of the weighted Cauchy transform (2) and for its restriction to the space AmσD. That is, we are concerned with the operator CσPmσ where Pmσ is the projection L2,σDAmσD. The results achieved are as follows:

Firstly, we find that the singular values of LσPmσ. For km, it can be expressed as

λk=J1+J2+J3,

where

J1=1+kmmm!km+12n=0AnΓ2n+2k2m+61Γ4ν2m1Γ2n+2k4m+4ν+6,J2=αkν,m2νm12n=0AnΓ4ν2m1Γ2n+2Γ2n+4ν2m+1,

and

J3=1+kmmαkν,mm!km+12νm1n=0AnΓkm+2Γ4ν2m1Γ4νk3m.

For k=m can be expressed as

λk2=αkν,m22νm18π2νm+1n=0Bnn+2νm,E7

where

Bn=n=0Γm+1Γ2νmΓ2νm+1n!Γ(2νmΓ2νm+2,αkν,m=Γ2Γ2mν+1Γm+1Γ2+m2ν.

Secondly, we show that these singular values behave like

where C is a constant.

The paper is organized as follows: In Section 2, we review the definition of the weighted logarithmic potential transform, as well as some of its needed properties. Section 3 deals with some basic facts on the spectral theory of Mass Laplacians on the Poincaré disk. In Section 4, a precise description of the generalized Bergmann spaces is reviewed. Section 5 is devoted to the computation of the singular values of the weighted logarithmic potential transform. The asymptotic behavior of these singular values is established in Section 6.

2. The weighted logarithmic potential transform Lν

2.1 The case ν=1

Let D the complex unit disk endowed with its Lebesgue measure μ and let ∂ID its boundary denote by L2D the space of complex-valued measurable functions on D with finite norm

f=Dfξ2ξ.E8

The Logarithmic Potential operator L:L2DL2D is defined by

Lfz=IDfξlog1ξzξ.E9

2.2 The case of ν≥1

We fix a real parameter ν such that 2ν>1 and we consider the following weighted logarithmic potential transform

Lνfz=Dfξlog1ξz1ξξ¯2ν2ξ,E10

defined on the space L2,νD complex-valued measurable functions are 1ξξ¯2ν2ξ-square integrable on D. As a convolution of L2,ν-functions with the compactly supported measure 1ξ2ν2ξ1DξLν:L2,νDL2,νD is obviously bounded. Moreover, it is not hard to show that Lν is in fact compact [4]. This raises a question concerning the spectral picture of Lν.

3. The Landau Hamiltonian Hν on the Poincaré disk D

Let D =zzz¯<1 be the complex unit disk with the Poincaré metric ds2=41zz¯2dzdz¯.D is a complete Riemannian manifold with all sectional curvature equal 1. It has an ideal boundary ∂ID identified with the circle ωωω¯=1. One refers to points ω∂ID as points at infinity. The geodesic distance between two points z and w is given by

coshdzw=1+2zwz¯w¯1zz¯1ww¯.E11

By Ref. [5] the Schrödinger operator on D with a constant magnetic field of strength proportional to ν>0 can be written as:

Lν1z222zz¯νz1z2z+νz¯1z2z¯+ν2z2.E12

which is also called Maass Laplacian on the disk. A slight modification of Lν is given by the operator

Hν4Lν4ν2E13

acting in the Hilbert space

L2,0Dφ:DDφz21z22z<+,E14

For our purpose, we shall consider the unitary equivalent realization H˜ν of the operator Hν in the Hilbert space

L2,νDφ:DCDφz21z22ν2z<+,E15

which is defined by

H˜νQν1HνQν,E16

where Qν:L2,νDL2,0D is the unitary transformation defined by the map φQνφ1z2νφ. Different aspects of the spectral analysis of the operator H˜ν have been studied by many authors. For instance, note that H˜ν is an elliptic densely defined operator on the Hilbert space L2,νD and admits a unique self-adjoint realization that we denote also by H˜ν. The spectrum of H˜ν in L2,νD consists of two parts: i a continuous part 1+, which corresponds to scattering states, ii a finite number of eigenvalues (hyperbolic Landau levels) of the form

εmν4νm1ν+m,m=0,1,2,,ν12E17

with infinite degeneracy, provided that 2ν>1. The eigenvalues in (17) correspond eigenfunctions, which are called bound states since the particle in such a state cannot leave the system without additional energy. A concrete description of these bound states spaces will be the goal of the next section.

4. The bound states spaces Aν,m2D

Here, we consider the eigenspace

Aν,m2DΦ:DCΦL2,νDandH˜νΦ=εmνΦ.E18

See Refs. [6, 7], for the following proposition.

Proposition 4.1.Let2ν>1andm=0,1,2,,ν12.Then, we have.

ian orthogonal basis of Aν,m2Dis given by the functions

ϕkν,mzzmk1z2meimkargz×2F1m+mk+mk22νm+mkm+k21+mkz2E19

k=0,1,2,,in terms of a terminating2F1Gauss hypergeometric function.

iithe norm square ofϕkν,minL2,νDis given by

ϕkν,m2=πΓ1+mk22νm1Γmmk+mk2+1Γ2νmmk+mk2Γm+mkm+k2+1Γ2νm+mkm+k2.E20

Corollary 4.1.The functionsΦkν,m,k=0,1,2,,given by

Φkν,mz1k2νm1π12k!Γ2νm+mm!Γ2νm+k12E21
×1z2mz¯mkPkmk2νm112zz¯,E22

in terms of Jacobi polynomials constitute an orthonormal basis of Am2,νD.

Proof. Write the connection between the 2F1-sum and the Jacobi polynomial

Pkα,βu=1+αkk!.2F1k1+α+β+k1+α1u2,

then the functions

ϕkν,mz=1minmk1z2mzmkeimkargzPminmkmk2νm112zz¯,E23

constitute an orthonormal basis of Aν,m2. The norm square of ϕkν,m in L2,νD is given by

ϕkν,m2=π2νm1mk!Γ2νm+mkmk!Γ2νm+mk.E24

Here, mkminmk and mkmaxmk. Thus, the set of functions

Φkν,mϕkν,mϕkν,m,k=0,1,2,E25

is an orthonormal basis of Aν,m2D and can be rewritten as.

Φkν,mz=1k2νm1π12k!Γ2νm+mm!Γ2νm+k12E26
×1z2mz¯mkPkmk2νm112zz¯E27

by making appeal to the identity Sp.63:

Γm+1Γms+1Pmsαu=Γm+α+1Γms+α+1u12sPmssαu,1smE28

for s=mk,t=12z2, and α=2νm1..

Corollary 4.2.TheL2eigenspaceAν,02D, corresponding tom=0in3.1and associated with the bottom energyε0ν=0in2.6,reduces further to the weighted Bergman space consisting of holomorphic functionsϕ:DCsuch that

Dϕz21z22ν2z<+.E29

5. Computation of the singular values λk

Elements of this basis are given in terms of Jacobi polynomials as

ϕkν,mz=1minmk1z2mzmkeimkargzPminmkmk2νm112zz¯.E30

The norm square of ϕkν,m in L2,νD is given by

ρkν,m=π2νm1mk!Γ2νm+mkmk!Γ2νm+mk.E31

Here, mkminmk and mkmaxmk. Let us introduce the notation. The set of functions

γkν,m1mkρkν,m,k=0,1,2,E32

So that we consider the elements

Φkν,mzγkν,m11zz¯mzmkeimkargzPminmkmk2νm112zz¯.E33

5.1 The action Lν

Lemma 5.1.We setz=ρeit, andI=02πeikmθlogzre2π, we have

I=logρrk=m,I=eikmt2mkrρmkrρmkkm,E34

Proof. By ref. [3], it remains to prove that this lemma for k<m.

We have

02πeikmθlogρeitre=02πeimkθlogreitρeiθdθE35

The function θeimkθlogreitρeiθ is a periodic mapping with the period equal 2π, then

02πeikmθlogρeitre=02πeimkθlogreitρeiθdθ=eikmt2mk×rρmkrρmk.

Lemma 5.2.For allλD.Lνcommutes with the rotationsRλ, where

Rλfz=fλz.

Proof. We observe that

Rλϕkν,mz=λkmϕkν,mz,km.

Corollary 5.1.Lνϕkν,mk=0are orthonormal inL2,νD.

Proof. As Rλ is an isometry of L2,νD,

Lνϕkν,mLνϕjν,m=RλLνϕkν,mRλLνϕjν,m=LνRλϕkν,mLνRλϕjν,m=λjk¯Lνϕkν,mLνϕjν,m,ifm>k,

or

=λkjLνϕkν,mLνϕjν,m,ifm<k

For all λD, since λ0, we have

Lνϕkν,mLνϕjν,m=0ifjk.

Lemma 5.3.If we denote1ϕkν,mz,ifk>mand2ϕkν,mz,ifk<m, we have

Lν1ϕkν,mz=Γk+1Γ2νmΓm+1Γ2νkLν2ϕkν,mz.

Proof. Just use

Γm+1Γms+1Pmsαu=Γm+α+1Γms+α+1u12sPmssαu,1sm.E36

Proposition 5.1.The action of the operatorLon a basis elementϕkν,mis of the form:

If k=m, We put z=ρe then

Lνϕkν,mz=αkν,m22νm+12νm1π1ρ22νm13F2m+1,2νm,2νm+12νm,2νm+21ρ2.E37

Ifkmthen

Lνϕkν,mz=πγkν,meikmt2kmI3+I4,

where

I3=1+kmmm!km+1ρkm+21ρ22νm12F1m+1,2νm+k2+kmρ2,

and

I4=αkν,m2νm11ρ22νm12F1m+1,2νm12νm,ρ2.

Proof. For k=m, we have

Lνϕkν,mz=1mπ2νm1πD1ξ22νm2Pm02νm112ξ2logzξξ=1m2νm1π011r22νm2Pm02νm112r2logρrdr2=1m22νm1π011t2νm2Pm02νm112tlogρ2tdt=1m22νm1πI1+I2.

where

I1=0ρ21t2νm2Pm02νm112tlogρ2tdt,

and

I2=ρ211t2νm2Pm02νm112tlogtdt.

Calculus of I1.

I1=logρ2ρ211t2νm2Pm02νm112tdt.

We use the formula

Pkαβu=1+αkk!2F1k,1+α+β+k1+α1u2.

We have

I1=logρ20ρ21t2νm22F1m,2νm1tdt.

By Ref. [8], we have

xc11xbc12F1a,bcxdx=1cxc1xbc2F1a+1,bc+1x,

implies that

I1=logρ2ρ21ρ22νm12F1m+1,2νm2ρ2.

Calculus of I2.

I2=ρ211t2νm2Pm02νm112tlogtdt.

Use the previous formula in Ref. [8] and the integration by part gives

I2=t1t2νm12F1m+1,2νm2tlogtρ21ρ211t2νm2F1m+1,2νm2tdt=ρ2logρ21ρ22νm12F1m+1,2νm2ρ2ρ211t2νm2F1m+1,2νm2tdt.

Calculus of

ρ211t2νm2F1m+1,2νm2tdt.

Use the following formula, which has place in [9]

2F1a,bct=ΓcΓcabΓcaΓcb2F1a,ba+bc+11t+ΓcΓa+bcΓaΓb1tcab2F1a,ba+bc+11t.

We put a=1m, b=2νm, c=2 and use the formula Boher-Mollerup, for zIR+,

Γz=eγzzn=11+zn1ezn,

which implies 1Γ1m=0, then

2F1m+1,2νm2t=2Γ2mν+1m!Γ2+m2ν2F1m+1,2νm2νm1t,

implies that

ρ211t2νm2F1m+1,2νm2tdt=2Γ2mν+1m!Γ2+m2νρ211t2νm2F1m+1,2νm2νm1tdt.

By the change 1t=s, we get

ρ211t2νm2F1m+1,2νm2tdt=2Γ2mν+1m!Γ2+m2ν01ρ2t2νm2F1m+1,2νm2νmtdt.

In [8], p. 44,

xα12F1a,bctdx=xαα3F2a,b,αc,α+1t+ΓαΓaαΓbαΓcΓaΓbΓcα

Since a=1m, b=2νm, c=2νm, and α=2νm+1 we have

ΓαΓaαΓbαΓcΓaΓbΓcα=0

and by the change t=s

01ρ2t2νm+12F1m+1,2νm2νmtdt=1m0ρ2t2νm2F1m+1,2νm2νmtdt=1mρ212νm+12νm+13F2m+1,2νm,2νm+12νm,2νm+21ρ2.

we set αkν,m=2Γ2mν+1m!Γ2+m2ν. We get

I2=ρ2logρ21ρ22νm12F1m+1,2νm2ρ2+1mαkν,m1ρ22νm12νm+13F2m+1,2νm,2νm+12νm,2νm+21ρ2.

Finally

Lνϕkν,mz=αkν,m22νm+12νm1π1ρ22νm13F2m+1,2νm,2νm+12νm,2νm+21ρ2.

Now if k>m, set z=ρeit.

Lνϕkν,mz=γkν,mD1ξ22νm2ξkmlog1zξPmkm2νm112ξ2ξ=γkν,m011r22νm2rkm+1Pmkm2νm112r202πeikmθlog1zrdθdr=πγkν,meikmt2km011r22νm2rkmPmkm2νm112r2rρkmρrkmdr2=πγkν,meikmt2km0ρ1r22νm2rkmPmkm2νm112r2rρkmρrkmdr2+ρ11r22νm2rkmPmkm2νm112r2rρkmρrkmdr2.

We set

I3=0ρ1r22νm2rkmPmkm2νm112r2rρkmρrkmdr2.

and

I4=ρ11r22νm2rkmPmkm2νm112r2rρkmρrkmdr2.

Calculus of I3.

I3=ρmk1+kmmm!0ρ2tkm1t2νm22F1m,2νm+k1+kmtdt.

By the formula

xc11xbc12F1a,bcxdx=1cxc1xbc2F1a+1,bc+1x,

we have

I3=1+kmmm!km+1ρkm+21ρ22νm12F1m+1,2νm+k2+kmρ2

Calculus of I4.

I4=ρ11r22νm2rkmPmkm2νm112r2rρkmρrkmdr2=ρkm1+kmm2m!ρ211t2νm22F1m,2νm+k1+kmtdt.

As the previous

ρ211t2νm22F1m,2νm+k1+kmtdt=αkν,mρ211t2νm22F1m+1,2νm2νm1tdt
=1mαkν,mρ210t2νm22F1m+1,2νm2νmtdt
=αkν,m2νm11ρ22νm13F2m+1,2νm,2νm12νm,2νm1ρ2

also

3F2m+1,2νm,2νm12νm,2νm1ρ2=2F1m+1,2νm12νm1ρ2

Now if k<m. We have

ϕkν,mz=1k2νm1πk!Γ2νm+mm!Γ2νm+k1z2mz¯mkPkmk2νm112z2.

By the formula

Γm+1Γms+1Pmsαu=Γm+α+1Γms+α+1u12sPmssαu,1sm,E38

and put s=mk and α=2νm1, we have

Pkmk2νm112z2=m!Γk+α+1k!Γm+α+1Pmkm2νm112z2,

substituting in the expression of ϕkν,mz, we get

ϕkν,mz=1m2νm1πm!Γ2νm+kk!Γ2νm+m1z2mzkmPmkm2νm112z2,

it is the same formula for k>m, which proves the same formula of Lνϕkν,mz if k>m.

Remark 5.1.By the previous formula in [9], we have

2F1m+1,2νm+k2νmρ2=k!Γ2+kmΓ12νm2F1m+1,2νm+k2νm1ρ2.

5.2 The spectrum of Lν

Proposition 5.2.Ifkm, then

λk=J1+J2+J3.

where

J1=1+kmmm!km+12n=0AnΓ2n+2k2m+61Γ4ν2m1Γ2n+2k4m+4ν+6,J2=αkν,m2νm12n=0AnΓ4ν2m1Γ2n+2Γ2n+4ν2m+1.

and

J3=1+kmmαkν,mm!km+12νm1n=0AnΓkm+2Γ4ν2m1Γ4νk3m,

Ifk=mthen

λk2=αkν,m22νm18π2νm+1n=0Bnn+2νm.E39

where

Bn=n=0Γm+1Γ2νmΓ2νm+1n!Γ(2νmΓ2νm+2.

Proof. If km. We have

Lνϕkν,mz=πγkν,mI3+I42kmeikmt.

We set H=L2D1ξ22ν2ξ, I3=I3ρ, and I4=I4ρ we have

λk2=Lνϕkν,mLνϕkν,mH=π2γkν,mkm01I3ρ+I4ρ2ρdρ.

Calculus of 01I3ρ2ρdρ.

I3ρ=1+kmmm!km+1ρkm+21ρ22νm12F1m+1,2νm+k2+kmρ2.

Since

2F1m+1,2νm+k2+kmρ2=n=0m+1n2νm+kn2+kmnρ2nn!,

then

I3ρ2=1+kmmm!km+12n=0Anρ2n1ρ24ν2m2.

where

An=1n!i=0nm+1im+1ni2νm+ki2νm+kni2νmi2νmni.

Thus

J1=01I3ρ2ρdρ=1+kmmm!km+12n=0An01ρ2n+2k2m+611ρ24ν2m11.

Use the fact that

01tα11tβ1dt=ΓαΓβΓα+β,

implies

01I3ρ2ρdρ=1+kmmm!km+12n=0AnΓ2n+2k2m+61Γ4ν2m1Γ2n+2k4m+4ν+6.E40

Calculus of 01I4ρ2ρdρ.

In the same,

J2=01I4ρ2ρdρ=αkν,m2νm12n=0AnΓ4ν2m1Γ2n+2Γ2n+4ν2m+1.E41

Calculus of 201I3ρI4ρρdρ.

J3=201I3ρI4ρρdρ=1+kmmαkν,mm!km+12νm1n=0AnΓkm+2Γ4ν2m1Γ4νk3m.E42

If k=m.

Since

3F2m+1,2νm,2νm+12νm,2νm+21ρ22=n=0Γm+1Γ2νmΓ2νm+1n!Γ(2νmΓ2νm+21ρ2n.E43
λk2=αkν,m22νm18π2νm+1n=0Bn011ρ2n+2νm1=αkν,m22νm18π2νm+1n=0Bnn+2νm,E44

where

Bn=n=0Γm+1Γ2νmΓ2νm+1n!Γ(2νmΓ2νm+2.

6. Asymptotic behavior of singular values λk as k→∞

Proposition 6.1.

where C is a constant.

Proof. If k>m, then

λk=J1+J2+J3,

where

J1=1+kmmm!km+12n=0AnΓ2n+2k2m+61Γ4ν2m1Γ2n+2k4m+4ν+6,J2=αkν,m2νm12n=0AnΓ4ν2m1Γ2n+2Γ2n+4ν2m+1

and

J3=1+kmmαkν,mm!km+12νm1n=0AnΓkm+2Γ4ν2m1Γ4νk3m.

The limit of λk as k.

We use the formula

Γk+ak=bkab

we have

J1k1mm!2n=0AnΓ4ν2m12k2m4ν1k4ν122m4ν1Γ4ν2m1n=0Anm!.E45
J2=Ok1E46

In the same

J3km4ν+1αkν,mΓ4ν2m1m!2νm1n=0An.E47

Therefore

λkCkm4ν+1,

where C is a constant.

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Written By

M’hamed Elomari and Ali El Mfadel

Reviewed: 12 August 2022 Published: 21 December 2022