Two-body problem on spaces of constant curvature A.V. Shchepetilov∗, I.E. Stepanova† Abstract 6 0 The two-body problem with a central interaction on simply connected constant 0 curvaturespacesofanarbitrary dimensionisconsidered. Theexplicit expressionfor 2 thequantumtwo-bodyHamiltonian viaaradialdifferentialoperator andgenerators n of the isometry group is found. We construct a self-adjoint extension of this Hamil- a tonian. Someitsexactspectralseriesarecalculatedforseveralpotentialinthespace J S3. 7 Wedescribealsothereducedclassicalmechanicalsystemonahomogeneousspace 1 ofaLiegroupintermsofthecoadjointactionofthisgroup. Usingthisapproachthe description of the reduced classical two-body problem on constant curvature spaces 4 is given. v 5 1 PACS numbers: 03.65.Fd, 02.40.Vh, 02.40.Ky, 02.40.Yy. 0 Mathematical Subject Classification: 70F05, 43A85, 22E70, 57S25, 70G65. 1 0 5 1 Introduction 0 / h The simply connected constant curvature spaces Sn and Hn posses isometry groups as p wide as the isometry group for the Euclidean space En and have no selected points or - h directions [1]. The one-body classical and quantum problems in the central field on these t spaces were studied in many papers, among which we indicate basic ones [2]-[13]. a m IncontrasttotheEuclideancase,theconfigurationspacesSn Sn,Hn Hnofthetwo- × × body problem on spaces Sn and Hn are not spaces of a constant curvature. Only space : v isometries that preserve an interaction potential enter in the symmetry group of such i X problem a priori. This group does not suffice to ensure the integrability of the two-body problem. Atthesametime,no”hidden”symmetriesorotherintegrabilitytoolsareknown r a fornontrivialpotential. Numericalexperimentsforthereducedclassicaltwo-bodyproblem in spaces Sn and Hn, n = 2,3 [14] show the soft chaos in this system for some natural interactive potentials. Numerical experiments ([15]) and analytical results ([16]-[18]) for the restricted classical two-body problem on S2 and H2 also prove its nonintegrability. Theclassicalmechanicaltwo-bodyprobleminconstantcurvaturespaceswasfirstcon- sideredin[19],wherethemethodoftheHamiltonianreductionofsystemswithsymmetries [20] was used to exclude the motion of a system as a whole. The description of reduced mechanical systems, their classification, and conditions for existence of a global dynamic were obtained using explicit analytic coordinate calculations on a computer. In [22], an analogousquantummechanicalsystemwasconsideredinthe two-dimensionalcase,i.e.on the spaces S2 H2. There, the quantum mechanical two-body Hamiltonian was expressed through isometry group generators and a radial differential operator. The structure of this expression is similar to the structure of reduced Hamilton function. The idea arises ∗Department of Physics, Moscow State University, 119992 Moscow, Russia, e-mail address: alexey@quant.phys.msu.su †Institute forPhysicsofEarth,RAS,bol.Gruzinskaja10,123995Moscow,Russia 1 2 to seek a general procedure for simultaneous simplification both classical and quantum problems without performing cumbersome calculations. We present such a procedure in the present paper. The obtained expression for the quantum two-body Hamiltonian is useful for solving at least three problems. First it enables us to prove that the two-body Hamiltonian with the proper domain is self-adjoint. Secondly, using this expression, one can reduce the spectral problem for the two-body Hamiltonian to a sequence of systems of ordinary differential equations enumeratedbyirreduciblerepresentationsoftheisometrygroup. Inthecaseofthesphere S3 we found all separate differential equations for spectral values, which can be solved in an explicit form for some interaction potentials. Thus, although the two-body problem on spaces Sn and Hn seems to be non-integrable in any sense, some its energy levels can be explicitly found. Such a situation appears in so called quasi exactly solvable models [23]-[25]. There aretwoessentialdifferences however. First,usually quasiexactly solvable models are artificially constructed. Second exact energy levels are obtained, as a rule, from one differential equation with the specially selected potential, but this equation has also other unknown spectral values. Conversely, in the problem under consideration we select a separate differential equation from systems of ordinary differential equations and find all its spectral values. Finally, from the obtained expression for the two-body Hamiltonian we derive the Hamilton function of the reduced two-body classical mechanical system, using the description of a reduced classical mechanical system on a homogeneous space in terms of coadjoint orbits of the corresponding Lie group, founded in section 6. This paper is an essential revision of papers [26] and [27], made by the first author. Some results in those papers were not properly grounded. This was later done in [28]. Proofs in the present paper are extended and in some technically difficult cases they contain references to [28]. More serious revision concerns the theorem 5 of the present paper. Because of a confusion between left and right shifts on a group, the statement in [27] corresponding to this theorem contains some additional erroneous eigenvectors. 2 Notations Consider the sphere Sn as the space Rn with the metric ∪{∞} n n 2 g = 4R2 dx2 1+ x2 , (1) s i i !, ! i=1 i=1 X X where x , i = 1,...,n are Cartesian coordinates in Rn and R is the curvature radius. i Let ρs(, ) denotes the distance between two points in Sn. The identity component of a · · whole isometry group for Sn, acting from the left, is SO(n+1). The set ∂ ∂ Xs =x x , 16i2 this is impossible r sincetheSO(n+1)-action on F is notfreeandtheprojection ofleft-invariant vectorfields r from SO(n+1) onto SO(n+1)/SO(n 1) is not well defined. By lifting the Hamiltonian − onto the symmetry group one can express it through group generators. 3.2 Two-particle Hamiltonian on the space Hn Hn × Theformalsubstitutionx ix ,r ir,R iR,j =1,...,n,(hereiistheimaginary j j →− →− → unit) transforms objects on the sphere Sn into objects on the hyperbolic space Hn (see also [19]). Therefore from results of the previous section one can obtain the two-particle Hamiltonian on the space Hn Hn: × (1 r2)n ∂ rn−1 ∂ 1 1 1 Hh = − D¯2 A D¯ C D¯ B D¯ , (17) 0 −8mR2rn−1∂r (1 r2)n−2∂r − a 0− 2 h 2− 2 h 1− h 3 (cid:18) − (cid:19) wherbe 0 k∇ k ∇ ∇ k∇ k f − f f2 − f2 (cid:13) (cid:13) =(cid:13)(cid:13)(cid:13)ψ2fg˜2srr(f′)2− 2fψ2(cid:13)(cid:13)(cid:13)g˜srrf′∂∂r (fψ) Equation(15)impliesthatg˜rr =(1+r2)2/(8R2m). IntegratingoverQ withthemeasure s 3Allfunctionalspacesareassumedtocomprisecomplex-valuedfunctions. 10 dµ , one gets s 1 (f′)2ψ2 2ψf′ ∂ ψ 2dµ > (fψ) (1+r2)2dµ k∇ k s 8R2m f2 − f2 ∂r s Z Z (cid:20) (cid:21) Qs Qs (19) ∞ 1 (f′)2ψ2 1 ψf′rn−1 ∂ = (1+r2)2dµ (fψ)drdµ . 8R2m f2 s− 4R2m f2(1+r2)n−2∂r f Z Z Z Qs Frs 0 We now want to find a function f such that for every smooth function ψ, the function ψf′rn−1 ∂ (fψ) f2(1+r2)n−2∂r has the form ∂ φ(r)ψ2 . Then the last integral in (19) can vanish identically. Solving ∂r the system of equations (cid:0) (cid:1) (f′)2rn−1 f′rn−1 =φ′, =2φ, f2(1+r2)n−2 f(1+r2)n−2 one gets (1+r2)n−2 −1 (n 2)rn−2/4, n>3 φ(r)= 4 dr , φ(r) − , r 0, − rn−1 ∼ (4 lnr)−1, n=2 → (cid:20) Z (cid:21) (cid:26) | | (n 2)r2−n/4, n>3 φ(r) − − , r , (20) ∼ (4 lnr)−1, n=2 →∞ (cid:26) − | | f′ 2 (n 2)2/(4r2), n>3 f (r)∼ 4−r2ln2r −1, n=2 , r →0, r →∞. (cid:18) (cid:19) (cid:26) (cid:0) (cid:1) It is easy to verify that for any choice of the integration constant the function φ(r) is discontinues at some point, which does not occur in the Euclidean case for n > 3. Let ω := x Q r(x)<δ and ω′ := x Q r(x)>δ−1 . Choose δ > 0 in such a way δ { ∈ s| } δ ∈ s| that the function φ(r) is continues in the domain (ω ω′ ) (Fs Fs ). On the space (cid:8) 2δ∪(cid:9)2δ \ 0 ∪ ∞ Q (Fs Fs ) the continuity domains of the functions f′/f and φ(r) coincide; moreover s\ 0 ∪ ∞ both functions are nonzero for r =0, . We set 6 ∞ r−2+r2, n>3 u (r)= , n r−2+r2 /ln2r, n=2 (cid:26) and choose the constant κ>0 such t(cid:0)hat the in(cid:1)equality (1+r2)2(f′)2 κu (r)6 n 8R2mf2 holds. Then due to (19) and (20) one gets the following inequality κ u (r)ψ 2dµ 6 ψ 2dµ (21) n s s | | k∇ k Z Z Qs Qs forthefunctionψ C∞(Q )withsupp ψ ω ω′. Writinginequality(21)separatelyfor ∈ s ⊂ δ∪ δ therealandimaginarypartsofacomplex-valuedfunction,weobtain(21)foranarbitrary function ψ C∞(Q ) with suppψ ω ω′. ∈ s ⊂ δ∪ δ

Two-body problem on spaces of constant curvature

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