A Quantum Dot with Impurity in the Lobachevsky Plane
aa r X i v : . [ m a t h - ph ] J a n A Quantum Dot with Impurity in the LobachevskyPlane
V. Geyler, P. ˇSˇtov´ıˇcek and M. Tuˇsek
Abstract.
The curvature effect on a quantum dot with impurity is investi-gated. The model is considered on the Lobachevsky plane. The confinementand impurity potentials are chosen so that the model is explicitly solvable.The Green function as well as the Krein Q -function are computed. Keywords. quantum dot, Lobachevsky plane, point interaction, spectrum.
1. Introduction
Physically, quantum dots are nanostructures with a charge carriers confinementin all space directions. They have an atom-like energy spectrum which can bemodified by adjusting geometric parameters of the dots as well as by the presenceof an impurity. Thus the study of these dependencies may be of interest from thepoint of view of the nanoscopic physics.A detailed analysis of three-dimensional quantum dots with a short-range im-purity in the Euclidean space can be found in [1]. Therein, the harmonic oscillatorpotential was used to introduce the confinement, and the impurity was modeled bya point interaction ( δ -potential). The starting point of the analysis was derivationof a formula for the Green function of the unperturbed Hamiltonian (i.e., in theimpurity free case), and application of the Krein resolvent formula jointly with thenotion of the Krein Q -function.The current paper is devoted to a similar model in the hyperbolic plane.The nontrivial hyperbolic geometry attracts regularly attention, and its influenceon the properties of quantum-mechanical systems has been studied on variousmodels (see, for example, [2, 3, 4]). Here we make use of the same method as in[1] to investigate a quantum dot with impurity in the Lobachevsky plane. We willintroduce an appropriate Hamiltonian in a manner quite analogous to that of [1]and derive an explicit formula for the corresponding Green function. In this sense, V. Geyler, P. ˇSˇtov´ıˇcek and M. Tuˇsekour model is solvable, and thus its properties may be of interest also from themathematical point of view.During the computations to follow, the spheroidal functions appear naturally.Unfortunately, the notation in the literature concerned with this type of specialfunctions is not yet uniform (see, e.g., [5] and [6]). This is why we supply, for thereader’s convenience, a short appendix comprising basic definitions and resultsrelated to spheroidal functions which are necessary for our approach.
2. A quantum dot with impurity in the Lobachevsky plane
Denote by ( ̺, φ ), 0 < ̺ < ∞ , 0 ≤ φ < π , the geodesic polar coordinates on theLobachevsky plane. Then the metric tensor is diagonal and reads( g ij ) = diag (cid:16) , a sinh ̺a (cid:17) where a , 0 < a < ∞ , denotes the so called curvature radius which is related to thescalar curvature by the formula R = − /a . Furthermore, the volume form equals dV = a sinh( ̺/a )d ̺ ∧ d φ . The Hamiltonian for a free particle of mass m = 1 / H = − (cid:18) ∆ LB + 14 a (cid:19) = − √ g ∂∂x i √ gg ij ∂∂x j − a where ∆ LB is the Laplace-Beltrami operator and g = det g ij . We have set ~ = 1.The choice of a potential modeling the confinement is ambiguous. We nat-urally require that the potential takes the standard form of the quantum dotpotential in the flat limit ( a → ∞ ). This is to say that, in the limiting case, itbecomes the potential of the isotropic harmonic oscillator V ∞ = ω ̺ . However,this condition clearly does not specify the potential uniquely. Having the freedomof choice let us discuss the following two possibilities:a) V a ( ̺ ) = a ω tanh ̺a , (2.1)b) U a ( ̺ ) = a ω sinh ̺a . (2.2)Potential V a is the same as that proposed in [7] for the classical harmonicoscillator on the Lobachevsky plane. With this choice, it has been demonstratedin [7] that the model is superintegrable, i.e., there exist three functionally inde-pendent constants of motion. Let us remark that this potential is bounded, andso it represents a bounded perturbation to the free Hamiltonian. On the otherhand, the potential U a is unbounded. Moreover, as shown below, the stationarySchr¨odinger equation for this potential leads, after the partial wave decomposition,to the differential equation of spheroidal functions. The current paper concentratesexclusively on case b).The impurity is modeled by a δ -potential which is introduced with the aidof self-adjoint extensions and is determined by boundary conditions at the base Quantum Dot with Impurity in the Lobachevsky Plane 3point. We restrict ourselves to the case when the impurity is located in the centerof the dot ( ̺ = 0). Thus we start from the following symmetric operator: H = − (cid:18) ∂ ∂̺ + 1 a coth (cid:16) ̺a (cid:17) ∂∂̺ + 1 a sinh − (cid:16) ̺a (cid:17) ∂ ∂φ + 14 a (cid:19) + 14 a ω sinh (cid:16) ̺a (cid:17) , Dom( H ) = C ∞ ((0 , ∞ ) × S ) ⊂ L (cid:16) (0 , ∞ ) × S , a sinh (cid:16) ̺a (cid:17) d ̺ d φ (cid:17) . (2.3) Substituting ξ = cosh( ̺/a ) we obtain H = 1 a (cid:20) (1 − ξ ) ∂ ∂ξ − ξ ∂∂ξ + (1 − ξ ) − ∂ ∂φ + a ω ξ − − (cid:21) =: 1 a ˜ H, Dom( H ) = C ∞ ((1 , ∞ ) × S ) ⊂ L (cid:0) (1 , ∞ ) × S , a d ξ d φ (cid:1) . (2.4)Using the rotational symmetry which amounts to a Fourier transform in the vari-able φ , ˜ H may be decomposed into a direct sum as follows˜ H = ∞ M m = −∞ ˜ H m , ˜ H m = − ∂∂ξ (cid:18) ( ξ − ∂∂ξ (cid:19) + m ξ − a ω ξ − − , Dom( ˜ H m ) = C ∞ (1 , ∞ ) ⊂ L ((1 , ∞ ) , d ξ ) . Note that ˜ H m is a Sturm-Liouville operator. Proposition 2.1. ˜ H m is essentially self-adjoint for m = 0 , ˜ H has deficiency indices (1 , .Proof. The operator ˜ H m is symmetric and semibounded, and so the deficiencyindices are equal. If we set µ = | m | , θ = − a ω , λ = − z − , then the eigenvalue equation (cid:18) − ∂∂ξ (cid:18) ( ξ − ∂∂ξ (cid:19) + m ξ − a ω ξ − − (cid:19) ψ = zψ (2.5)takes the standard form of the differential equation of spheroidal functions (A.1).According to chapter 3.12, Satz 5 in [6], for µ = m ∈ N a fundamental system { y I , y II } of solutions to equation (2.5) exists such that y I ( ξ ) = (1 − ξ ) m/ P (1 − ξ ) , P (0) = 1 ,y II ( ξ ) = (1 − ξ ) − m/ P (1 − ξ ) + A m y I ( ξ ) log (1 − ξ ) , V. Geyler, P. ˇSˇtov´ıˇcek and M. Tuˇsekwhere, for | ξ − | < P , P are analytic functions in ξ , λ , θ ; and A m is apolynomial in λ and θ of total order m with respect to λ and √ θ ; A = − / z ∈ C \ R . For m = 0, every solutions to (2.5) is square integrablenear 1; while for m = 0, y I is the only one solution, up to a factor, which is squareintegrable in a neighborhood of 1. On the other hand, by a classical analysis dueto Weyl, there exists exactly one linearly independent solution to (2.5) which issquare integrable in a neighborhood of ∞ , see Theorem XIII.6.14 in [8]. In thecase of m = 0 this obviously implies that the deficiency indices are (1 , m = 0then, by Theorem XIII.2.30 in [8], the operator ˜ H m is essentially self-adjoint. (cid:3) Define the maximal operator associated to the formal differential expression L = − ∂∂ξ (cid:18) ( ξ − ∂∂ξ (cid:19) + a ω ξ − − H max ) = (cid:26) f ∈ L ((1 , ∞ ) , d ξ ) : f, f ′ ∈ AC ((1 , ∞ )) , − ∂∂ξ (cid:18) ( ξ − ∂f∂ξ (cid:19) + a ω ξ − f ∈ L ((1 , ∞ ) , d ξ ) (cid:27) ,H max f = Lf.
According to Theorem 8.22 in [9], H max = ˜ H † . Proposition 2.2.
Let κ ∈ ( −∞ , ∞ ] . The operator ˜ H ( κ ) defined by the formulae Dom( ˜ H ( κ )) = { f ∈ Dom( H max ) : f = κf } , ˜ H ( κ ) f = H max f, where f := − πa lim ξ → f ( ξ )log(2 a ( ξ − , f := lim ξ → f ( ξ ) + 14 πa f log (cid:0) a ( ξ − (cid:1) ) , is a self-adjoint extension of ˜ H . There are no other self-adjoint extensions of ˜ H .Proof. The methods to treat δ like potentials are now well established [10]. Herewe follow an approach described in [11], and we refer to this source also for theterminology and notations. Near the point ξ = 1, each f ∈ Dom( H max ) has theasymptotic behavior f ( ξ ) = f F ( ξ,
1) + f + o (1) as ξ → f , f ∈ C and F ( ξ, ξ ′ ) is the divergent part of the Green function for theFriedrichs extension of ˜ H . By formula (2.11) which is derived below, F ( ξ,
1) = − / (4 πa ) log (cid:0) a ( ξ − (cid:1) . Proposition 1.37 in [11] states that ( C , Γ , Γ ), withΓ f = f and Γ f = f , is a boundary triple for H max .According to theorem 1.12 in [11], there is a one-to-one correspondence be-tween all self-adjoint linear relations κ in C and all self-adjoint extensions of ˜ H Quantum Dot with Impurity in the Lobachevsky Plane 5given by κ ←→ ˜ H ( κ ) where ˜ H ( κ ) is the restriction of H max to the domain ofvectors f ∈ Dom( H max ) satisfying(Γ f, Γ f ) ∈ κ. (2.6)Every self-adjoint relation in C is of the form κ = C v ⊂ C for some v ∈ R , v = 0.If (with some abuse of notation) v = (1 , κ ), κ ∈ R , then relation (2.6) means that f = κf . If v = (0 ,
1) then (2.6) means that f = 0 which may be identified withthe case κ = ∞ . (cid:3) Remark.
Let q be the closure of the quadratic form associated to the semiboundedsymmetric operator ˜ H . Only the self-adjoint extension ˜ H ( ∞ ) has the propertythat all functions from its domain have no singularity at the point ξ = 1 andbelong to the form domain of q . It follows that ˜ H ( ∞ ) is the Friedrichs extensionof ˜ H (see, for example, Theorem X.23 in [12] or Theorems 5.34 and 5.38 in [9]). Let us consider the Friedrichs extension of the operator ˜ H in L (cid:0) (1 , ∞ ) × S , d ξ d φ (cid:1) which was introduced in (2.4). The resulting self-adjoint operator is in fact theHamiltonian for the impurity free case. The corresponding Green function G z isthe generalized kernel of the Hamiltonian, and it should obey the equation( ˜ H − z ) G z ( ξ, φ ; ξ ′ , φ ′ ) = δ ( ξ − ξ ′ ) δ ( φ − φ ′ ) = 12 π ∞ X m = −∞ δ ( ξ − ξ ′ )e im ( φ − φ ′ ) . If we suppose G z to be of the form G z ( ξ, φ ; ξ ′ , φ ′ ) = 12 π ∞ X m = −∞ G mz ( ξ, ξ ′ )e im ( φ − φ ′ ) , (2.7)then, for all m ∈ Z , ( ˜ H m − z ) G mz ( ξ, ξ ′ ) = δ ( ξ − ξ ′ ) . (2.8)Let us consider an arbitrary fixed ξ ′ , and set µ = m, θ = − a ω , λ = − z − . Then for all ξ = ξ ′ equation (2.8) takes the standard form of the differentialequation of spheroidal functions (A.1). As one can see from (A.8), the solutionwhich is square integrable near infinity equals S | m | (3) ν ( ξ, − a ω / ξ = 1 equals P s | m | ν ( ξ, − a ω /
16) asone may verify with the aid of the asymptotic formula P mν ( ξ ) ∼ Γ( ν + m + 1)2 m/ m ! Γ( ν − m + 1) ( ξ − m/ as ξ → , for m ∈ N . V. Geyler, P. ˇSˇtov´ıˇcek and M. TuˇsekWe conclude that the m th partial Green function equals G mz ( ξ, ξ ′ ) = − ξ − W ( P s | m | ν , S | m | (3) ν ) P s | m | ν (cid:18) ξ < , − a ω (cid:19) S | m | (3) ν (cid:18) ξ > , − a ω (cid:19) (2.9)where the symbol W ( P s | m | ν , S | m | (3) ν ) denotes the Wronskian, and ξ < , ξ > are re-spectively the smaller and the greater of ξ and ξ ′ . By the general Sturm-Liouvilletheory, the factor ( ξ − W ( P s | m | ν , S | m | (3) ν ) is constant. Since G mz = G − mz decom-position (2.7) may be simplified, G z ( ξ, φ ; ξ ′ , φ ′ ) = 12 π G z ( ξ, ξ ′ ) + 1 π ∞ X m =1 G mz ( ξ, ξ ′ ) cos [ m ( φ − φ ′ )] . (2.10) Q -function The Krein Q -function plays a crucial role in the spectral analysis of impurities. Itis defined at a point of the configuration space as the regularized Green functionevaluated at this point. Here we deal with the impurity located in the center ofthe dot ( ξ = 1, φ arbitrary), and so, by definition, Q ( z ) := G regz (1 ,
0; 1 , . Due to the rotational symmetry, G z ( ξ ) := G z ( ξ, φ ; 1 ,
0) = G z ( ξ, φ ; 1 , φ ) = G z ( ξ,
0; 1 ,
0) = 12 π G z ( ξ, , and hence ( ˜ H − z ) G z ( ξ ) = 0 , for ξ ∈ (1 , ∞ ) . Let us note that from the explicit formula (2.9), one can deduce that the coefficients G mz ( ξ,
1) in the series in (2.10) vanish for m = 1 , , , . . . . The solution to thisequation is G z ( ξ ) ∝ S ν (cid:18) ξ, − a ω (cid:19) . The constant of proportionality can be determined with the aid the followingtheorem which we reproduce from [13].
Theorem 2.3.
Let d ( x, y ) denote the geodesic distance between points x, y of atwo-dimensional manifold X of bounded geometry. Let U ∈ P ( X ) := n U : U + := max( U, ∈ L p loc ( X ) , U − := max( − U, ∈ n X i =1 L p i ( X ) o for an arbitrary n ∈ N and ≤ p i ≤ ∞ . Then the Green function G U of theSchr¨odinger operator H U = − ∆ LB + U has the same on-diagonal singularity asthat for the Laplace-Beltrami operator itself, i.e., G U ( ζ ; x, y ) = 12 π log 1 d ( x, y ) + G regU ( ζ ; x, y ) where G regU is continuous on X × X . Quantum Dot with Impurity in the Lobachevsky Plane 7Let us denote by G Hz and Q H ( z ) the Green function and the Krein Q -functionfor the Friedrichs extension of H , respectively. Since ˜ H = a H and ( ˜ H − z ) G z = δ ,we have G Hz ( ξ, φ ; ξ ′ , φ ′ ) = a G a z ( ξ, φ ; ξ ′ , φ ′ ) , Q H ( z ) = a Q ( a z ) . One may verify thatlog d ( ̺, ~
0) = log ̺ = log( a arg cosh ξ ) = 12 log (cid:0) a ( ξ − (cid:1) + O ( ξ − ̺ →
0+ or, equivalently, ξ → F ( ξ, ξ ′ ) := G z ( ξ, φ ; ξ ′ , φ ) − G regz ( ξ, φ ; ξ ′ , φ ) = G z ( ξ, ξ ′ , − G regz ( ξ, ξ ′ , G z we obtain the expression F ( ξ,
1) = − πa log (cid:0) a ( ξ − (cid:1) . (2.11)From the above discussion, it follows that the Krein Q -function depends on thecoefficients α , β in the asymptotic expansion S ν (cid:18) ξ, − a ω (cid:19) = α log( ξ −
1) + β + o (1) as ξ → , (2.12)and equals Q ( z ) = − β πa α + log(2 a )4 πa . (2.13)To determine α , β we need relation (A.10) for the radial spheroidal functionof the third kind. For ν and ν + 1 / S ν ( ξ, θ ) = sin( νπ ) π e − iπ ( ν +1) K ν ( θ ) Qs − ν − ( ξ, θ ) ,S − ν − ( ξ, θ ) = sin( νπ ) π e iπν K − ν − ( θ ) Qs ν ( ξ, θ ) . (2.14)Applying the symmetry relation (A.5) for expansion coefficients, we derive that Qs − ν − ( ξ, θ ) = ∞ X r = −∞ ( − r a − ν − ,r ( θ ) Q − ν − r ( ξ )= ∞ X r = −∞ ( − r a ν,r ( θ ) Q − ν − − r ( ξ ) . Using the asymptotic formulae (see [5]) Q ν ( ξ ) = −
12 log ξ −
12 + Ψ(1) − Ψ( ν + 1) + O (( ξ −
1) log( ξ − , V. Geyler, P. ˇSˇtov´ıˇcek and M. Tuˇsekthe series expansion in (A.11) and formulae (2.14), we deduce that, as ξ → S ν ( ξ, θ ) ∼ − sin( νπ ) π e − iπ ( ν +1) K ν ( θ ) × (cid:20) s ν ( θ ) − (cid:18)
12 log ξ − − Ψ(1) + π cot( νπ ) (cid:19) + Ψ s ν ( θ ) (cid:21) ,S − ν − ( ξ, θ ) ∼ − sin( νπ ) π e iπν K − ν − ( θ ) × (cid:20) s ν ( θ ) − (cid:18)
12 log ξ − − Ψ(1) (cid:19) + Ψ s ν ( θ ) (cid:21) , where the coefficients s µn ( θ ) are introduced in (A.7),Ψ s ν ( θ ) := ∞ X r = −∞ ( − r a ν,r ( θ )Ψ( ν + 1 + 2 r ) , and where we have made use of the following relation for the digamma function:Ψ( − z ) = Ψ( z + 1) + π cot( πz ).We conclude that S ν ( ξ, θ ) ∼ α log( ξ −
1) + β + O (( ξ −
1) log( ξ − ξ → , where α = i tan( νπ )2 πs ν ( θ ) (cid:16) e iπν K − ν − ( θ ) − e − iπ (2 ν +3 / K ν ( θ ) (cid:17) ,β = α (cid:0) − log 2 − s ν ( θ ) s ν ( θ ) (cid:1) + e − iπν s ν ( θ ) − K ν ( θ ) . The substitution for α , β into (2.13) yields Q ( z ) = − πa (cid:18) − log 2 − s ν (cid:18) − a ω (cid:19) s ν (cid:18) − a ω (cid:19)(cid:19) + 12 a tan( νπ ) e iπ (3 ν +3 / K − ν − ( − a ω ) K ν ( − a ω ) − ! − + log(2 a )4 πa (2.15)where ν is chosen so that λ ν (cid:18) − a ω (cid:19) = − z − . (2.16)For ν = n being an integer, we can immediately use the known asymptoticformulae for spheroidal functions (see Section 16.12 in [5]) which yield S n ( ξ, θ ) = is n ( θ )4 √ θK n ( θ ) log( ξ − − is n ( θ ) log 24 √ θK n ( θ )+ is n ( θ ) √ θK n ( θ ) X r ≥− n ( − r a n,r ( θ ) h n +2 r + K n ( θ ) s n ( θ ) + O ( ξ − , as ξ → h = 1 , h k = 1 / / . . . + 1 /k . By (2.13), one can calculatethe Q -function in this case, too. Quantum Dot with Impurity in the Lobachevsky Plane 9 The Green function of the Hamiltonian describing a quantum dot with impurityis given by the Krein resolvent formula G H ( χ ) z ( ξ, φ ; ξ ′ , φ ′ ) = G Hz ( ξ, φ ; ξ ′ , φ ′ ) − Q H ( z ) − χ G Hz ( ξ,
0; 1 , G Hz (1 , ξ ′ , G Hz ( ξ, φ ; 1 ,
0) = G Hz ( ξ,
0; 1 , χ := a κ ∈ ( −∞ , ∞ ] determines the corresponding self-adjoint exten-sion H ( χ ) of H . In the physical interpretation, this parameter is related to thestrength of the δ interaction. Recall that the value χ = ∞ corresponds to theFriedrichs extension of H representing the case with no impurity. This fact is alsoapparent from the Krein resolvent formula.The unperturbed Hamiltonian H ( ∞ ) describes a harmonic oscillator on theLobachevsky plane. As is well known (see, for example, [14]), for the confinementpotential tends to infinity as ̺ → ∞ , the resolvent of H ( ∞ ) is compact, and thespectrum of H ( ∞ ) is discrete and semibounded. The eigenvalues of H ( ∞ ) aresolutions of a scalar equation whose introduction also relies heavily on the theoryof spheroidal functions. We are sceptic about the possibility of deriving an explicitformula for the eigenvalues. But the equation turned out to be convenient enoughto allow for numerical solutions. A more detailed discussion jointly with a basicnumerical analysis is provided in a separate paper [15].A similar observation about the basic spectral properties (discreteness andsemiboundedness) is also true for the operators H ( χ ) for any χ ∈ R since, by theKrein resolvent formula, the resolvents for H ( χ ) and H ( ∞ ) differ by a rank oneoperator. Moreover, the multiplicities of eigenvalues of H ( χ ) and H ( ∞ ) may differat most by ± σ the set of poles of the function Q H ( z )depending on the spectral parameter z . Note that σ is a subset of spec( H ( ∞ )).Consider the equation Q H ( z ) = χ. (2.17) Theorem 2.4.
The spectrum of H ( χ ) is discrete and consists of four nonintersectingparts S , S , S , S described as follows: S is the set of all solutions to equation (2.17) which do not belong to thespectrum of H ( ∞ ) . The multiplicity of all these eigenvalues in the spectrumof H ( χ ) equals 1. S is the set of all λ ∈ σ that are multiple eigenvalues of H ( ∞ ) . If the mul-tiplicity of such an eigenvalue λ in spec( H ( ∞ )) equals k then its multiplicityin the spectrum of H ( χ ) equals k − . S consists of all λ ∈ spec( H ( ∞ )) \ σ that are not solutions to equation (2.17).the multiplicities of such an eigenvalue λ in spec( H ( ∞ )) and spec( H ( χ )) areequal. S consists of all λ ∈ spec( H ( ∞ )) \ σ that are solutions to equation (2.17).If the multiplicity of such an eigenvalue λ in spec( H ( ∞ )) equals k then itsmultiplicity in the spectrum of H ( χ ) equals k + 1 . Hence the eigenvalues of H ( χ ), χ ∈ R , different from those of the unperturbedHamiltonian H ( ∞ ) are solutions to (2.17). As far as we see it, this equation canbe solved only numerically. We have postponed a systematic numerical analysis ofequation (2.17) to a subsequent work. Note that the Krein Q -function (2.15) is infact a function of ν , and hence dependence (2.16) of the spectral parameter z on ν is fundamental. In this context, it is quite useful to know for which values of ν the spectral parameter z is real. A partial answer is given by Proposition A.1.
3. Conclusion
We have proposed a Hamiltonian describing a quantum dot in the Lobachevskyplane to which we added an impurity modeled by a δ potential. Formulas for thecorresponding Q - and Green functions have been derived. Further analysis of theenergy spectrum may be accomplished for some concrete values of the involvedparameters (by which we mean the curvature a and the oscillator frequency ω )with the aid of numerical methods. Appendix: Spheroidal functions
Here we follow the source [5]. Spheroidal functions are solutions to the equation(1 − ξ ) ∂ ψ∂ξ − ξ ∂ψ∂ξ + (cid:2) λ + 4 θ (1 − ξ ) − µ (1 − ξ ) − (cid:3) ψ = 0 , (A.1)where all parameters are in general complex numbers. There are two solutionsthat behave like ξ ν times a single-valued function and ξ − ν − times a single-valuedfunction at ∞ . The exponent ν is a function of λ , θ , µ , and is called the charac-teristic exponent. Usually, it is more convenient to regard λ as a function of ν , µ and θ . We shall write λ = λ µν ( θ ). If ν or µ is an integer we denote it by n or m ,respectively. The functions λ µν ( θ ) obey the symmetry relations λ µν ( θ ) = λ − µν ( θ ) = λ µ − ν − ( θ ) = λ − µ − ν − ( θ ) . (A.2)A first group of solutions (radial spheroidal functions) is obtained as expan-sions in series of Bessel functions, S µ ( j ) ν ( ξ, θ ) = (1 − ξ − ) − µ/ s µν ( θ ) ∞ X r = −∞ a µν,r ψ ( j ) ν +2 r (2 θ / ξ ) , (A.3) Quantum Dot with Impurity in the Lobachevsky Plane 11 j = 1 , , ,
4, where the factors s µν ( θ ) are determined below and ψ (1) ν ( ζ ) = r π ζ J ν +1 / ( ζ ) , ψ (2) ν ( ζ ) = r π ζ Y ν +1 / ( ζ ) ,ψ (3) ν ( ζ ) = r π ζ H (1) ν +1 / ( ζ ) , ψ (4) ν ( ζ ) = r π ζ H (2) ν +1 / ( ζ ) . The coefficients a µν,r ( θ ) (denoted only a r for the sake of simplicity) satisfy a threeterm recurrence relation( ν + 2 r − µ )( ν + 2 r − µ − ν + 2 r − / ν + 2 r − / θa r − + ( ν + 2 r + µ + 2)( ν + 2 r + µ + 1)( ν + 2 r + 3 / ν + 2 r + 5 / θa r +1 + (cid:20) λ µν ( θ ) − ( ν + 2 r )( ν + 2 r + 1) + ( ν + 2 r )( ν + 2 r + 1) + µ − ν + 2 r − / ν + 2 r + 3 /
2) 2 θ (cid:21) a r = 0 . (A.4)Here and in what follows we assume that ν +1 / a µν,r ( θ ) may be chosen such that a µν, ( θ ) = a µ − ν − , ( θ ) = a − µν, ( θ ) , and so (see (A.2)) a µν,r ( θ ) = a µ − ν − , − r ( θ ) = ( ν − µ + 1) r ( ν + µ + 1) r a − µν,r ( θ ) (A.5)where ( a ) r := a ( a + 1)( a + 2) . . . ( a + r −
1) = Γ( a + r ) / Γ( a ), ( a ) := 1. Equation(A.4) leads to a convergent infinite continued fraction and this way one can provethat lim r →∞ r a r a r − = lim r →−∞ r a r a r +1 = θ . (A.6)From (A.6) and the asymptotic formulae for Bessel functions, it follows that (A.3)converges if | ξ | > s µν ( θ ) = " ∞ X r = −∞ ( − r a µν,r ( θ ) − (A.7)then S µ ( j ) ν ( ξ, θ ) ∼ ψ ( j ) ν (2 θ / ξ ) , for | arg( θ / ξ ) | < π as ξ → ∞ . We have the asymptotic forms, valid as ξ → ∞ , S µ (3) ν ( ξ, θ ) = 12 θ − / ξ − e i (2 θ / ξ − νπ/ − π/ [1 + O ( | ξ | − )] , for − π < arg( θ / ξ ) < π, (A.8)2 V. Geyler, P. ˇSˇtov´ıˇcek and M. Tuˇsekand S µ (4) ν ( ξ, θ ) = 12 θ − / ξ − e − i (2 θ / ξ − νπ/ − π/ [1 + O ( | ξ | − )] , for − π < arg( θ / ξ ) < π. (A.9)The radial spheroidal functions satisfy the relation S µ (3) ν = 1 i cos( νπ ) (cid:16) S µ (1) − ν − + i e − iπν S µ (1) ν (cid:17) . (A.10)The radial spheroidal functions are especially useful for large ξ ; the larger isthe ξ the better is the convergence of the expansion. To obtain solutions useful near ±
1, and even on the segment ( − , P s µν ( ξ, θ ) = ∞ X r = −∞ ( − r a µν,r ( θ ) P µν +2 r ( ξ ) ,Qs µν ( ξ, θ ) = ∞ X r = −∞ ( − r a µν,r ( θ ) Q µν +2 r ( ξ ) . (A.11)These solutions are called the angular spheroidal functions and are related to theradial spheroidal functions by the following formulae: S µ (1) ν ( ξ, θ ) = π − sin[( ν − µ ) π ]e − iπ ( ν + µ +1) K µν ( θ ) Qs µ − ν − ( ξ, θ ) ,S m (1) n ( ξ, θ ) = K mn ( θ ) P s mn ( ξ, θ ) , (A.12)where K µν ( θ ) can be expressed as a series in coefficients a µν,r ( θ ), and sometimes itis called the joining factor. In more detail, for any k ∈ Z it holds true that K µν ( θ ) = 12 (cid:18) θ (cid:19) ν/ k Γ(1 + ν − µ + 2 k ) e ( ν + k ) πi s µν ( θ ) × k X r = −∞ ( − r a µν,r ( θ )( k − r )! Γ( ν + k + r + 3 / ∞ X r = k ( − r a µν,r ( θ )( r − k )! Γ(1 / − ν − k − r ) . Proposition A.1.
Let ν, θ ∈ R and set µ = 0 . Then λ ν ( θ ) ∈ R .Proof. To simplify the notation we denote, in (A.4), α µ,νr ( θ ) = ( ν + 2 r + µ + 2)( ν + 2 r + µ + 1)( ν + 2 r + 3 / ν + 2 r + 5 / θ,β µ,νr ( θ ) = − ( ν + 2 r )( ν + 2 r + 1) + ( ν + 2 r )( ν + 2 r + 1) + µ − ν + 2 r − / ν + 2 r + 3 /
2) 2 θ,γ µ,νr ( θ ) = ( ν + 2 r − µ )( ν + 2 r − µ − ν + 2 r − / ν + 2 r − / θ. Quantum Dot with Impurity in the Lobachevsky Plane 13The resulting formula may be written in the matrix form, . . . γ − β − α − γ β α γ β α . . . ... a − a a ... = − λ ... a − a a ... (A.13)where we have omitted the fixed indices.As one can see, γ ,νr +1 ( θ ) = ν + 2 r + 5 / ν + 2 r + 1 / α ,νr ( θ )and so ν + 2 r + 1 / ν + 2 r − / α ,νr − ( θ ) a ν,r − ( θ ) + β ,νr ( θ ) a ν,r ( θ ) + α ,νr ( θ ) a ν,r +1 ( θ ) = − λ ν ( θ ) a ν,r ( θ ) . Substitution a ν,r = L r ( ν )˜ a ν,r , where L r ( ν ) are non-zero constants, yields ν + 2 r + 1 / ν + 2 r − / α ,νr − ( θ )˜ a ν,r − ( θ ) L r − ( ν ) L r ( ν ) + β ,νr ( θ )˜ a ν,r ( θ ) + α ,νr ( θ )˜ a ν,r +1 ( θ ) L r +1 ( ν ) L r ( ν )= − λ ν ( θ )˜ a ν,r ( θ ) . We require the matrix in (A.13) to be symmetric in the new coordinates { ˜ a r } .This implies that ν + 2 r + 1 / ν + 2 r − / L r − ( ν ) L r ( ν ) = L r ( ν ) L r − ( ν ) . For r / ∈ ( − ν/ − / , − ν/ / L r ( ν ) = p | ν + 2 r + 1 / | . For r ≡ r ∈ ( − ν/ − / , − ν/ / L r ( ν ) = p | ν + 2 r + 1 / | and make another transformation of coordinates:˜˜ a r = ( − ˜ a r for r = r − (2 k − , k ∈ N , ˜ a r for all other r. Relation (A.13) can be viewed as an eigenvalue equation with a symmetricmatrix in the coordinate system (cid:8) ˜˜ a k (cid:9) , hence λ ν ( θ ) must be real. (cid:3) References [1] J. Br¨uning, V. Geyler, and I. Lobanov,
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On the Harmonic Oscillator on the Lobachevsky Plane ,Russian J. Math. Phys. (2007), 401-405. Acknowledgments
The authors wish to acknowledge gratefully partial support from the followinggrants: grant No. 201/05/0857 of the Grant Agency of Czech Republic (P. ˇS.) andgrant No. LC06002 of the Ministry of Education of Czech Republic (M. T.).
V. GeylerDepartment of MathematicsMordovian State UniversitySaransk, RussiaP. ˇSˇtov´ıˇcekDepartment of MathematicsFaculty of Nuclear SciencesCzech Technical UniversityPrague, Czech Republice-mail: [email protected]
Quantum Dot with Impurity in the Lobachevsky Plane 15