Aharonov-Bohm effect on the Poincaré disk

Aharonov-Bohm effect on the Poincaré disk

Oleg Lisovyy^a兲

Laboratoire de Mathématiques et Physique Théorique, CNR/UMR 6083, Université de Tours, Parc de Grandmont, 37200 Tours, France

共Received 2 March 2007; accepted 18 April 2007; published online 31 May 2007兲 We consider formal quantum Hamiltonian of a charged particle on the Poincaré disk in the presence of an Aharonov-Bohm magnetic vortex and a uniform mag- netic field. It is shown that this Hamiltonian admits a four-parameter family of self-adjoint extensions. Its resolvent and the density of states are calculated for natural values of the extension parameters. ©2007 American Institute of Physics.

关DOI:10.1063/1.2738751兴

I. INTRODUCTION

Quantum dynamics on the Poincaré disk has long been a subject of theoretical interest, mainly because of the insights its study provides into the theory of quantum chaos. Analyzed examples include, for instance, the free motion under the action of constant magnetic fields,^8,10,24the Kepler problem,²⁵ the scattering by the Aharonov-Bohm^23,26 共AB兲 and Aharonov-Bohm-Coulomb²⁹ po- tentials, the study of point interactions,^3,5and quantum Hall effect.⁷

In the present paper, we consider the Hamiltonian of a charged spinless particle moving on the hyperbolic disk, pierced by an AB flux, in the presence of a uniform magnetic field. The first part of this work is rather standard: we determine the admissible boundary conditions on the wave functions using Krein’s theory of self-adjoint extensions 共SAEs兲.⁴ It turns out that in the most general case the formal Hamiltonian has deficiency indices共2, 2兲and thus admits a four-parameter family of SAEs. Let us remark that similar results on the plane have been found in Refs.2and11 in the case of a zero magnetic field, and in Ref. 21 for nonzero fields; SAEs of the Dirac Hamiltonian on the plane have been studied in Refs.16and22.

The rest of this paper is devoted to the study of a particular extension, corresponding to the choice of regular boundary conditions at the position of the AB flux. We start by constructing certain integral representations for common eigenstates of this Hamiltonian and the angular mo- mentum operator. These representations then allow us to sum up the contributions coming from different angular momenta to the resolvent kernel and to evaluate this kernel and the density of states共DOS兲in a closed form.

The material is organized as follows. In Sec. II, we introduce basic notations and study elementary solutions of the radial Schrödinger equation on the Poincaré disk. Self-adjointness of the full AB Hamiltonian is discussed in Sec. III. In Sec. IV we find a compact expression for the resolvent of the regular extension关formulas 共4.14兲and 共4.19兲–共4.22兲兴. These relations represent the main result of the present work. The DOS induced by the AB flux in the whole hyperbolic space 关see Eqs. 共5.8兲–共5.10兲兴 is obtained in Sec. V. Some technical results are relegated to the appendices.

II. FREE HAMILTONIAN ON THE POINCARÉ DISK A. Basic formulas

Let us identify the Poincaré disk D= SU共1 , 1兲/ SO共2兲 with the interior of the unit circle 兩z兩²

⬍1 in the complex plane, equipped with the metric

a兲Electronic mail: lisovyi@lmpt.univ-tours.fr

48, 052112-1

0022-2488/2007/48共5兲/052112/17/$23.00 © 2007 American Institute of Physics

ds²=gzz¯dzdz¯=R² dzdz¯

共1 −兩z兩²兲² 共2.1兲

of the constant Gaussian curvature −4 /R². We consider a spinless particle moving on the disk and interacting with a magnetic field. The latter can be introduced as a connection 1-form

A=A_zdz+A¯_zdz¯

on the trivial U共1兲-bundle overD. Quantum dynamics of a particle of unit charge is described by the Hamiltonian

Hˆ = − 2 g_zz¯

兵Dz,D¯z其, 共2.2兲

whereDz=⳵z+iAzandD¯z=⳵¯z+iA¯zare the usual covariant derivatives. To unburden formulas, we put the particle mass equal to 1 / 2 andប=c= 1 throughout the paper.

In the remainder of the present section, the following vector potential is considered:

A^共B兲= −iBR² 4

¯dzz −zdz¯

1 −兩z兩² . 共2.3兲

It generates a curvature 2-form F^共B兲 proportional to the invariant volume measure d␮

=共i/ 2兲gzz¯dz∧dz¯. Indeed, we have

F^共B兲= dA^共B兲=Bd␮.

Therefore, the potential关Eq.共2.3兲兴describes a uniform magnetic field of intensityB. Introducing polar coordinatesz=re^i␸ and¯z=re^−i␸, one can write the corresponding Hamiltonian as

Hˆ共B兲= −共1 −r²兲²

R²

再

冎

Note that the domain of Hˆ共B兲 is not yet specified. It will be fixed in the next section by the requirement for the Hamiltonian to be a self-adjoint operator. According to Stone’s theorem, this condition ensures the existence of consistent dynamics.

B. Radial Hamiltonians

Formal HamiltonianHˆ共B兲commutes with the angular momentum operatorLˆ= −i⳵␸. Therefore, it leaves invariant the eigenspaces ofLˆ, spanned by the functionswl共r兲e^il␸共l苸Z兲. Being restricted to the eigenspace ofLˆ, characterized by the angular momentuml, the Hamiltonian acts as follows:

wl共r兲哫Hˆ

lwl共r兲, Hˆ

l= −共1 −r²兲²

R²

再

冎

Here, we have introduced a dimensionless parameterb=BR²/ 4, instead ofB.

It will be useful for us to let the parameterltake on not only an integer, but also arbitrary real values, and to study in some detail the properties of solutions of the radial Schrödinger equation

共Hˆ

l−k²兲wl= 0. 共2.6兲

In what follows it will be always assumed thatk²苸C\R⁺艛兵0其. It is also convenient to introduce a new variablet=r², instead ofr.

We are interested in the solutions of Eq.共2.6兲leading to square integrable共with the measure d␮兲 functions on D. These solutions should then be square integrable on the open interval I

=共0 , 1兲with the measure d␮t=R²dt/ 2共1 −t兲². For eachl苸Rthere exists only one solution of Eq.

共2.6兲, which is square integrable in the neighborhood of the pointt= 1. Its explicit form is w_l^共I兲共t兲=t^−l/2共1 −t兲^␹₂F₁共␹^−b,␹^+b−l,2␹^{,1 −}t兲=t^l/2共1 −t兲^␹₂F₁共␹^+b,␹^−b+l,2␹^{,1 −}t兲,

共2.7兲 where

␹^{= 1 +}

冑

and ₂F₁共␣,␤,␥,z兲 denotes a Gauss hypergeometric function. The branches of square roots are defined so that they take on real positive values for purely imaginaryk.

Similarly, for eachl苸共−⬁, −1兴艛关1 ,⬁兲there is only one solution of Eq.共2.6兲, which is square integrable with respect to d␮tnear the pointt= 0. The form of this solution depends on whether l艌1 orl艋−1. In the first case, i.e., forl艌1, it is given by

w_l^共II,+兲共t兲=t^l/2共1 −t兲^␹₂F₁共␹^+b,␹^{−b+l,1 +}l,t兲, 共2.8兲 while forl艋−1 this solution is written as follows:

w_l^共II,−兲共t兲=t^−l/2共1 −t兲^␹₂F₁共␹^−b,␹^{+b−l,1 −}l,t兲. 共2.9兲 Note that for兩l兩⬍1 both functionsw_l^共II,±兲共t兲 are square integrable in the vicinity of the pointt= 0 and solve the radial Schrödinger equation 关Eq. 共2.6兲兴. These solutions are linearly independent except for l= 0. However, in the latter case Eq.共2.6兲 still admits two distinct solutions that are square integrable ast→0,

w₀^共II兲共t兲=共1 −t兲^␹u共t兲, w˜₀^共II兲共t兲=共1 −t兲^␹v共t兲,

whereu andv are any two linearly independent solutions of the hypergeometric equation with parameters␣⁼␹^+b,␤⁼␹^{−b, and}␥^{= 1}共one can choose them, for instance, according to formulas 共15.5.16兲and共15.5.17兲of Ref.1兲.

Let us now show that the solutionsw_l^共I兲共t兲andw_l^共II,+兲共t兲are linearly independent forl⬎−1, and the solutionsw_l^共I兲共t兲andw_l^共II,−兲共t兲are linearly independent forl⬍1. This can be done by an explicit computation of their Wronskian

W共f1,f₂兲=f₁⳵tf₂−⳵tf₁f₂.

Namely, using the connection and analytic continuation formulas for hypergeometric functions,¹ one obtains

W共wl共I兲共t兲,wl共II,±兲共t兲兲=共tC_k,l^±兲⁻¹, 共2.10兲 with

C_k,l^± =⌫共␹^±b兲⌫共␹⫿b±l兲

⌫共2␹兲⌫共1 ±l兲 . 共2.11兲

Therefore, for k²苸C\R⁺艛兵0其 and兩l兩艌1 Eq. 共2.6兲 has no square integrable solutions 共with the measure d␮t兲on the whole intervalI. This is true, in particular, for all radial HamiltoniansHˆ

l苸Zof the free particle in a uniform magnetic field, except for the s-wave HamiltonianHˆ

0. In the case 兩l兩⬍1 Eq.共2.6兲has exactly one square integrable solution, given by formula共2.7兲.

Let us now restrict the domain of Hˆ_l toD共Hˆ_l兲=C₀^⬁共I兲, i.e., to smooth compactly supported functions. Then, the above remarks imply that

• Hˆ

lis essentially self-adjoint for兩l兩艌1 and

• for兩l兩⬍1 the operatorHˆ_lhas deficiency indices共1, 1兲and thus admits a one-parameter family of SAEs.

Different extensions Hˆ

l

共␥兲 共兩l兩⬍1兲 are in one-to-one correspondence with the isometries between the deficiency subspacesKl±= ker共Hˆ_l⫿i␧兲, where␧苸R⁺ may be chosen arbitrarily. They can be labeled by a real parameter␥苸关0 , 2␲兲and characterized by the domains

D共Hˆ

l

共␥兲兲=兵f+c共wl

++ e^i␥w_l⁻兲兩f苸C₀^⬁共I兲,c苸C其, where the functionsw_l^±共t兲may be chosen as follows:

w_l^±共t兲=兩wl共I兲共t兲兩k²=±i␧. 共2.12兲

Remark:For a particular value of␥the domainD共Hˆ

l

共␥兲兲is composed of functions regular at t= 0. The corresponding SAE ofHˆ

lwill be denoted byHˆ

l reg. C. Resolvent

The kernelG_k,l共t,t⬘^兲 of the resolvent of the radial HamiltonianHˆ

lsatisfies the equation 共Hˆ

l共t兲−k²兲Gk,l共t,t⬘^{兲=2共1 −t兲2}

R² ␦共t−t⬘^{兲. 共2.13兲}

It basically means that if共Hˆ

l−k²兲 u=vfor someu苸D共Hˆ

l兲, then u共t兲=

冕

G_k,l共t,t⬘兲v共t⬘兲d␮t⬘.

In order to find the solution of Eq.共2.13兲, consider the following ansatz:

G_k,l共t,t⬘兲=

再

冎

where the signs “⫹” and “⫺” should be chosen forl艌0 andl⬍0, respectively. It is clear that the function, defined by Eq.共2.14兲, solves Eq.共2.13兲fort⫽t⬘and satisfies the boundary conditions of square integrability at the pointst= 0 andt= 1.共In the case where兩l兩⬍1, the requirement of square integrability at the boundary points is not sufficient to make the operatorHˆ

lself-adjoint; however, for such l, the ansatz 关Eq. 共2.14兲兴 also satisfies the regularity condition at t= 0 and thus corre- sponds to the resolvent of the extensionHˆ

l reg.兲

Taking into account the explicit form of the operator Hˆ

l, one may show that the required singular behavior of the Green’s function at the pointt=t⬘is guaranteed, provided the condition

兩⳵t⬘G_k,l共t⬘^,t兲兩t−0t+0= − 1 2t holds. Using Eq.共2.14兲, one can rewrite this condition as

2tC˜

k,l

±W共w_l^共I兲共t兲,w_l^共II,±兲共t兲兲= 1.

It follows from Eq.共2.10兲that the last relation is satisfied if we chooseC˜

k,l

± =C_k,l^± / 2. Substituting this expression into Eq. 共2.14兲, one finds a representation for the radial Green’s functions G_k,l共t,t⬘兲.

III. HAMILTONIAN IN THE PRESENCE OF A MAGNETIC VORTEX A. Radial Hamiltonians

Let us now add to the Hamiltonian the field of an AB magnetic flux ⌽= 2␲␯, centered atz

= 0,

A^共v兲= −i␯

2

冉

冊

This choice of the flux position involves no loss of generality since we have a well-known transitive SU共1, 1兲action onD, which preserves the metric关Eq. 共2.1兲兴,

z哫zg=␣z+␤

␤

¯ z+␣^{¯, g=}

冉

冊

Any gauge field configuration corresponding to a single vortex and a uniform magnetic field can be reduced toA=A^共B兲+A^共v兲 using transformation共3.2兲combined with a gauge change.

The Hamiltonian关Eq. 共2.2兲兴in the presence of a vortex has thus the following form:

Hˆ

v= −共1 −r²兲²

R²

再

冎

This Hamiltonian still commutes with the angular momentum operator Lˆ. Radial Hamiltonians Hˆ

v,l are obtained by the restriction ofHˆ

v to the eigenspaces of Lˆ with fixed angular momental 苸Z. Namely, one obtainsHˆ

v,l=Hˆ

l+␯, where the operatorsHˆ

␣苸Rare defined as in Eq.共2.5兲. Thus, the only effect the AB vortex has on the formal Hamiltonians is the shift of the angular momentum variable by␯. This observation allows us to considerably simplify the derivation of many results using the calculations from the previous section.

Remark:As usual, for integer flux values some further simplifications occur. The Hamilto- niansHˆ共B兲 andHˆ

v are related by a gauge transformation Hˆ

v=UHˆ共B兲U^†, U:w哫e^−i␯␸w, 共3.4兲

which is globally well defined for␯苸Z. The kernels of the resolvents ofHˆ共B兲andHˆ

v in this case differ only by a factor of e^{i␯共␸−␸⬘兲}, and this change has no effect on the observable quantities.

B. Self-adjointness

From now on it will be assumed that −1⬍␯艋0共it is clear from the above discussion that this involves no loss of generality兲. Let us consider the full HamiltonianHˆ

vand restrict its domain to functions with compact support on the punctured disk: D共Hˆ

v兲=C₀^⬁共D\兵0其兲. It was shown in the previous section that for 兩l兩艌1 the operator Hˆ

l is essentially self-adjoint, and for 兩l兩⬍1 it has deficiency indices共1, 1兲. One should then distinguish two cases:

␯⁼0. In this caseHˆ

vhas deficiency indices共1, 1兲and admits a one-parameter family of SAEs Hˆ

v

共␥兲with␥苸关0 , 2␲兲and D共Hˆ

v共␥兲兲=兵f+c共w₀⁺+ e^i␥w₀⁻兲兩f苸C₀^⬁共D\兵0其兲,c苸C其.

These Hamiltonians describe a purely contact共nonmagnetic兲interaction of a particle with the AB solenoid. They have already been considered in Ref.3. So, we will not pursue their study.

−1⬍␯⬍0. For such␯the deficiency subspaces K^±of the full HamiltonianHˆ

vare generated by those of the operators Hˆ

␯ and Hˆ

1+␯. Thus, Hˆ

v has deficiency indices 共2, 2兲 and admits a four-parameter family of SAEs. Different extensions can be labeled by a unitary 2⫻2 matrixU and characterized by the domains

D共Hˆ

v

U兲=

再

^兺

^冉

^兺

^冊

冎

wherew_1,2^± are orthonormal elements of the bases ofK^±, w₁^±共t,␸兲= w_␯^±共t兲

储w_␯^±共t兲储, w₂^±共t,␸兲= w_1+␯^± 共t兲 储w_1+␯^± 共t兲储e^i␸, and储·储denotes theL²norm onIwith respect to the measure d␮t.

Note that the diagonal matrixUdescribes magnetic point interactions acting separately in the schannel共l= 0兲and thep channel共l= 1兲. NondiagonalUintroduces a coupling between the two modes so that the Hamiltonian no longer commutes with the angular momentum.

Further analysis of spectral properties ofH_v^Uis a bit cumbersome in the general case共see, for example, Refs.21,2, and11, where such an analysis has been performed for the AB effect on the plane with and without magnetic field兲. We remark, however, that there exists a distinguished SAE ofHˆ

v, whose domain consists of functions vanishing fort→0. This extension will be denoted by Hˆ

v

reg. The next section is devoted to the calculation of its resolvent共Hˆ

v

reg−k²兲⁻¹. The resolvent of any other SAE can be obtained from the latter using Krein’s formula.⁴

IV. ONE-VORTEX RESOLVENT

A. Contour integral representations of the radial waves

The main technical difficulty in the calculation of the resolvent kernelGk共z,z⬘^兲of the Hamil- tonianHˆ

v

reg is the summation of radial contributions coming from different angular momenta,

Gk共z,z⬘兲= 1 2␲_l

兺

苸Z

G_k,l+␯共t,t⬘兲e^{il共␸−␸⬘兲}. 共4.1兲

In order to address this problem, it is useful to introduce the functions depending on bothtand␸^, instead of the radial waves关Eqs. 共2.7兲–共2.9兲兴,

w_l^共I兲共z兲=⌫共␹⁺b兲⌫共␹⁻b兲

⌫共2␹兲 e^{il共␸+␲兲}w_l^共I兲共t兲, 共4.2兲

wˆ_l^共I兲共z兲=⌫共␹⁺b兲⌫共␹⁻b兲

⌫共2␹兲 e^{−il共␸+␲兲}w_l^共I兲共t兲, 共4.3兲

w_l^共II,±兲共z兲= 2␲^{i ⌫共}␹⫿b±l兲

⌫共␹⫿b兲⌫共1 ±l兲e^{il共␸+␲兲}w_l^共II,±兲共t兲, 共4.4兲

wˆ_l^共II,±兲共z兲= 2␲^{i ⌫共}␹⫿b±l兲

⌫共␹⫿b兲⌫共1 ±l兲e^{−il共␸+␲兲}w_l^共II,±兲共t兲. 共4.5兲 Combining these formulas with relations共2.14兲and共2.11兲, one can rewrite the Green’s function 关Eq.共4.1兲兴in the following way:

Gk共z,z⬘^{兲= e−i␯共␸−␸⬘兲} 8i␲^{2 共Gk共}

+兲共z,z⬘^兲+Gk共−兲共z,z⬘^{兲兲, 共4.6兲} where the functionsG_k^共±兲共z,z⬘^兲are given by

Gk共±兲共z,z⬘兲=_l苸Z+␯,l

兺

0

w_l^共I兲共z兲wˆ_l^共II,±兲共z⬘兲 for兩z兩⬎兩z⬘兩, 共4.7兲

Gk共±兲共z,z⬘兲=_l

兺

苸Z+␯,l0

w_l^共II,±兲共z兲wˆ_l^共I兲共z⬘兲 for兩z兩⬍兩z⬘兩. 共4.8兲 The sums关Eqs.共4.7兲and共4.8兲兴can be computed using a special set of solutions of stationary Schrödinger equation without an AB flux, known as horocyclic waves.¹²These solutions have the form

⌿±共z,␪兲= 共1 −兩z兩²兲^␹±

共1 +ze^−␪兲^␹±−b共1 +¯ez ^␪兲^␹±+b, 共4.9兲 where

␹±=1

2±

冉

冊

and␪ is an arbitrary complex parameter. Being considered as functions of␪, horocyclic waves

⌿±共z,␪兲 have an infinite number of branch points located at ␪^{= ± lnr+}i共␸⁺␲^{+ 2}␲Z兲. Let us introduce a system of branch cuts in the ␪ plane, as shown in Fig. 1. The sheets of Riemann surfaces of the functions ⌿±共z,␪兲 are fixed by the requirement that the arguments of both 1 +ze^−␪ and 1 +z¯e^␪ are equal to zero on the line Im␪⁼␸^.

Recall that the Hamiltonians Hˆ共B兲 and Hˆ

v are related by the gauge transformation 共3.4兲. Although this transformation is singular for noninteger values of the flux, one can still relate any solution of the equation 共Hˆ

v−k²兲w= 0 to a solution of the same equation without an AB field, 共Hˆ共B兲−k²兲␺= 0. However, since we have w= e^−i␯␸␺, the function ␺ should be branched with the monodromy e^2␲i␯ at the point z= 0. Motivated by this well-known fact, we will try to represent radial wave functions关Eqs.共4.2兲–共4.5兲兴as superpositions of elementary solutions关Eq.共4.9兲兴,

w共z兲=

冕

⌿±共z,␪兲␳共␪兲d␪,

whereCis an integration contour and␳共␪兲is an appropriately chosen weight function. There will be three types of contours that will be important to us共see also Fig.1兲:

FIG. 1. Contours of integration in the␪^plane.

• ContourC₊共z兲starts at −⬁+i␣, surrounds the branch cutb₊=共−⬁+i共␸⁺␲兲, lnr+i共␸⁺␲兲兴in a counter-clockwise manner, and goes to −⬁+i共␣^{+ 2}␲兲.

• ContourC₋共z兲starts at⬁+i共␣^{+ 2}␲兲, then goes counter-clockwise around the branch cut b₋

=关−lnr+i共␸⁺␲兲,⬁+i共␸⁺␲兲兲, and finally travels to ⬁+i␣along the ray parallel to the real axis.

• ContourC₀共z兲joins two branch points:␪1= lnr+i共␸⁺␲兲and␪2= −lnr+i共␸⁺␲兲.

Real parameters ␣ ^and ␥ can be chosen arbitrarily; the only conditions they should satisfy are given by

兩␸⁻␣兩⬍␲^{, 0}艋␥⬍− lnr.

Assuming that Rek²⬍0, one may now write a number of contour integral representations for the radial waves关Eqs.共4.2兲–共4.5兲兴,

w_l^共I兲共z兲=

冕

冕

−共z,␪兲e^−l␪d␪^, 共4.11兲

w_l^共II,±兲共z兲= ±

冕

wˆ_l^共II,±兲共z兲= ⫿

冕

+共z,␪兲e^−l␪d␪^, 共4.13兲

where the functions⌿ˆ

±共z,␪兲are obtained from⌿±共z,␪兲by replacingb→−b. Although the valid- ity of representations共4.10兲–共4.13兲can be checked directly, their general structure may also bea posteriori understood as follows. Consider, for instance, the functions w_l^共I兲共z兲 and w_l^共II,±兲共z兲 as defined by Eqs. 共4.10兲 and 共4.12兲. Continuation of these functions along a counter-clockwise circuit enclosing the pointz= 0 amounts to a simultaneous shift of the branch cuts and integration contours upwards by 2␲^{in the}␪ plane. This shift is in turn equivalent to simple multiplication of both functions by e^2␲il. Moreover, elementary solutions关Eq. 共4.9兲兴satisfy the relation

Lˆ⌿±共z,␪兲=共z⳵z−¯z⳵¯z兲⌿±共z,␪兲= −⳵␪⌿±共z,␪兲,

which means that right-hand sides of Eqs.共4.10兲and共4.12兲are common共multivalued兲eigenfunc- tions ofHˆ共B兲 andLˆ, their angular momenta being equal to l. The first function is regular for t

→1 since in this case the branch cuts pinch the imaginary axis. Similarly, the second function is regular fort→0. This implies共modulo constant factors that have to be found by a direct calcu- lation兲relations共4.2兲and共4.4兲.

B. Summation

Let us now turn to the calculation of the sums 关Eqs.共4.7兲and共4.8兲兴. For simplicity the case 兩z兩⬎兩z⬘兩 is treated in detail and we only indicate the changes needed to handle another case.

Substituting contour representations共4.2兲and共4.5兲into relation共4.7兲, one obtains

Gk共±兲共z,z⬘^{兲= ⫿}

兺

l苸Z+␯,l0

冕

d␪1

冕

+共z⬘^,␪2兲e^{l共␪1−␪2兲}.

Since兩z兩⬎兩z⬘兩, one may choose the contoursC_±共z⬘^兲in such a way that␥z⬘⬎−lnr. Consequently, we have Re共␪1−␪2兲⬍0 for all ␪1苸C₀共z兲, ␪2苸C₋共z⬘^{兲 and Re共␪}1−␪2兲⬎0 for all ␪1苸C₀共z兲, ␪2

苸C₊共z⬘兲. Then, it becomes possible to perform the summation inside the integrals, and one finds Gk共+兲共z,z⬘兲+Gk共−兲共z,z⬘兲=

冕

d␪1

冕

+共z⬘^,␪2兲e^{共1+␯兲共␪1−␪2兲} e^␪1−␪2− 1 .

We would like to deform the contoursC_±共z⬘^兲in the last integral over␪2so that their vertical parts compensate one another. Then,C₊共z⬘^兲艛C−共z⬘^兲transforms into two horizontal lines, but one also earns a pole contribution coming from e^␪2= e^␪1. Next, if we assume that␸⁻␸⬘^⫽±␲, then the two lines can be deformed into Im␪2=␸⬘ using quasiperiodicity in␪2. Together with Eq. 共4.6兲, this leads to the following representation for the Green’s function:

G_k共z,z⬘兲=

冦

冧

with

G_k^共0兲共z,z⬘兲= 1 4␲

冕

d␪⌿−共z,␪兲⌿ˆ

+共z⬘^,␪兲, 共4.15兲

⌬k共z,z⬘兲=1 − e^−2␲i␯

8i␲^{2 e−i␯共␸−␸⬘兲}

冕

d␪1

冕

+共z⬘^,␪2兲e^{共1+␯兲共␪1−␪2兲} e^␪1−␪2− 1 .

共4.16兲 Similarly, assuming that 兩z兩⬍兩z⬘兩, one obtains an integral representation of the Green’s func- tion which has exactly the same form as Eq. 共4.14兲, except that the functions G_k^共0兲共z,z⬘^{兲 and}

⌬k共z,z⬘^兲are now given by

G_k^共0兲共z,z⬘兲= 1

4␲

冕

−共z⬘^,␪兲⌿+共z,␪兲, 共4.17兲

⌬k共z,z⬘兲=e^2␲i␯− 1

8i␲^{2 e−i␯共␸−␸⬘兲}

冕

冕

d␪2⌿ˆ

−共z⬘^,␪1兲⌿+共z,␪2兲e^{共1+␯兲共␪2−␪1兲} e^␪2−␪1− 1 .

共4.18兲 After some computations 共technical details are outlined in Appendix A兲, one may show that both representations coincide. Moreover, the integrals共4.15兲and共4.17兲can be carried out explic- itly,

G_k^共0兲共z,z⬘^兲=

冉

冊

where u共z,z⬘兲=兩共z⬘^−z兲/共1 −z¯z⬘兲兩² has a simple relation with the geodesic distance between the pointsz andz⬘, and the function␨共u兲 is given by

␨共u兲= 1 4␲

⌫共␹⁺b兲⌫共␹⁻b兲

⌫共2␹兲 共1 −u兲^␹₂F₁共␹^+b,␹^−b,2␹^{,1 −}u兲. 共4.20兲 Note that G_k^共0兲共z,z⬘兲 coincides with the well-known expression for the resolvent kernel of the Hamiltonian without an AB field.^8,20 This can also be seen directly from representation 共4.14兲 since⌬k共z,z⬘兲in Eq.共4.16兲or共4.18兲obviously vanishes for␯= 0. The function⌬k共z,z⬘兲may also be written in a symmetric form,

⌬k共z,z⬘^{兲= sin}_␲^␲␯

冕

冉

冊

with

v共r,r⬘^,␪兲= r²+r⬘^{2+ 2rr}⬘^cosh␪

1 +r²r⬘^{2+ 2rr}⬘^cosh␪^{. 共4.22兲} In our opinion, representation 共4.14兲 and formulas 共4.19兲–共4.22兲 constitute the most interesting results of the present paper. It is instructive to compare them with the known results in the flat space关cf. relations共2.25兲and共2.26兲in Ref.30or formula共5.10兲from Ref.27兴. Notice that the

“free” part of the Green’s function is manifestly separated in Eq.共4.14兲from the vortex-dependent contribution⌬k共z,z⬘^兲.

V. SPECTRUM AND DENSITY OF STATES The spectrum of the regular extensionHˆ

v

regconsists of three parts:

• a continuous spectrumE苸关共1 + 4b²兲/R²,⬁兲;

• a finite number of infinitely degenerate eigenvalues, which coincide with the usual Landau levels on the hyperbolic disk^8,23 in the absence of the AB field 关These levels are explicitly given by

E_n^共0兲= 1

R²

冋

^冉

^冊

册

wheren= 0 , 1 , . . . ,n_max⬍兩b兩− 1 / 2. Corresponding common eigenfunctions of the Hamil- tonianHˆ

v

regand the angular momentum operatorLˆ can be expressed in terms of Jacobi’s polynomials关cf. relation 共13兲 in Ref.23兴,

⌿_n,l^共0兲共t,␸兲 ⬃t^{兩l+␯兩/2}共1 −t兲^兩b兩−nP_n^{共2兩b兩−2n−1,兩l+␯兩兲}共2t− 1兲e^il␸. Here, one should takel= 0 , −1 , −2 , . . .共for b⬎0兲andl= 1 , 2 , . . .共for b⬍0兲.兴

• a finite number of bound states E_n^共␯兲 with finite degeneracy关The form of these eigenvalues depends on the sign of the magnetic field. Namely, forb⬎0 one has

E_n^共␯,+兲= 1

R²

冋

^冉

^冊

册

wheren= 0 , 1 , . . . ,n_max⬘ ⬍b−共␯+ 1兲− 1 / 2. In the caseb⬍0, the eigenvalues may be written as

E_n^{共␯,−兲}= 1

R²

冋

冉

冊

册

withn= 0 , 1 , . . . ,n_max⬙ ⬍兩b兩+␯− 1 / 2. Common eigenstates ofHˆ

v

regandLˆ are again given by Jacobi’s polynomials,

b⬎0:⌿_n,l^共␯,+兲共t,␸兲 ⬃t^{共l+␯兲/2}共1 −t兲^{b−n−共␯+1兲}P_n共2b−2n−2共␯+1兲−1,l+␯兲共2t− 1兲e^il␸,

b⬍0:⌿_n,l^{共␯,−兲}共t,␸兲 ⬃t^{兩l+␯兩/2}共1 −t兲^{兩b兩−n+␯}P_n^共2兩b兩−2n+2␯−1,兩l+␯兩兲共2t− 1兲e^il␸.

For given radial quantum numbern the allowed eigenvalues of the angular momentum are l= 1 , 2 , . . . ,n+ 1共forb⬎0兲andl= 0 , −1 , . . . , −n 共forb⬍0兲.兴

Remark:The above expressions共5.1兲–共5.3兲for the energy levels can also be extracted from Ref.6. It is worthwhile to emphasize that the discrete spectrum is absent for兩b兩⬍1 / 2.

Let us now consider the DOS on the hyperbolic disk. It can be obtained from the boundary values of the resolvent kernel on the real axis in the complex energy plane using the following formula:

␳共E兲=

冏

冏

, E苸R.

Both terms in representation 共4.14兲of the Green’s function contribute to the DOS. The con- tribution of the free-resolvent kernelG_k^共0兲共z,z⬘^兲has been first calculated by Comtet.⁸His results 共supplemented by an additional term,⁷ coming from the discrete spectrum兲 give the following expression for the DOS:

␳^共0兲共E,z兲=

冏

冏

= 1 4␲

sinh 2␲␭

cosh 2␲␭+ cos 2␲^b⌰

冉

冊

+ 2

␲^R2

兺

n=0 n_max

冉

冊

Here,⌰共x兲denotes Heaviside function and

␭=1

2

冑

One cannot expect that the DOS per unit area, induced by the AB field, will also be constant onD. However, it should depend only on the geodesic distance between a given point on the disk and the flux position. Indeed, since the function ⌬共z,z⬘^兲 is nonsingular for z→z⬘, the vortex- dependent part of the DOS is given by

␳^共␯兲共E,z兲=

冏

冏

, 共5.5兲

where the function⌬k共t兲 is obtained from⌬共z,z⬘^{兲 by setting}␸⁼␸⬘^andr2=r⬘^2=t,

⌬k共t兲=sin␲␯

␲

冕

冉

冊

冉

冊

As it stands, representation共5.6兲is valid in the left half-plane Rek²⬍0, where the function⌬k共t兲 is analytic. However, the DOS is determined by the singularities of⌬k共t兲that occur on the positive part of the real axis 共we may expect there a finite number of poles and the branch cut 关共1 + 4b²兲/R²,⬁兲, corresponding to the continuous part of the spectrum ofHˆ

v

reg兲. One could try to construct the appropriate analytic continuation of⌬k共t兲, considering Eq.共5.6兲as a contour integral and then suitably deforming the contour. It seems, however, that this approach does not lead to any satisfactory result because of the complicated singularity structure of the function under the inte- gral sign in the␪ ^plane.

An alternative method consists in the following. Remark that the vortex-dependent contribu- tion to the DOS in the whole hyperbolic space

␳^共␯兲共E兲=

冕

d␮␳^共␯兲共E,z兲 共5.7兲

has a finite value since

冉

冊

冉

冊

=

冦

4␲

⌫共␹⁺b兲⌫共␹⁻b兲

⌫共2␹兲

共1 −t兲^2␹

共1 + e^␪兲^␹+b共1 + e^−␪兲^␹−b+o共共1 −t兲^2␹兲 fort→1.

冧

If one now integrates ⌬k共t兲 over spatial coordinates 共see Appendix B兲 and then considers the analytic continuation of the result to the complex energy plane, the following expression for␳^共␯兲

⫻共E兲 can be obtained:

␳^共␯兲共E兲= − R² 4␲ ^Im

1

2␹^{− 1兩兵共}␹^−b+␯兲关␺共␹⁻b兲−␺共␹^−b+␯+ 1兲兴+共␹^+b−␯− 1兲 关␺共␹⁺b兲

−␺共␹^+b−␯− 1兲兴其兩k²=E+i0=␳d共␯兲共E兲+␳c共␯兲共E兲, 共5.8兲 where the contributions of the discrete and continuous part of the spectrum are given by

␳d共␯兲共E兲=

冦

^兺

^兺

^兺

^兺

冧

␳c共␯兲共E兲= −R²

8␭⌰

冉

冊 再

^共

^兲

−␭sinh 2␲␭+

共

兲

cosh 2␲␭+ cos 2␲^b

冎

and the parameter␭is defined as in Eq. 共5.4兲.

At last we add a comment concerning the flat space limit 共R→⬁兲 at a zero magnetic field 共b= 0兲. In this case representation 共5.8兲for the vortex-dependent DOS transforms into

␳^共␯兲共E兲——→

R→⬁

冏

␯共␯+ 1兲 2k²

冏

= −␯共␯+ 1兲

2 ␦共E兲. 共5.11兲

This result has been first obtained in Ref.9, and it has important consequences in the theory of disordered magnetic systems 共see, for example Refs. 13–15兲. Obtaining relation共5.11兲 directly from共5.10兲is more subtle; one should consider␳c

共␯兲共E兲as a distribution and supply it with a proper regularization at the edge of the spectrum, i.e., as␭→+ 0.