Perturbed (2n-1)-dimensional Kepler problem and the nilpotent adjoint orbits of U(n,n)

We study the (2n-1)-Kepler problem and other Hamiltonian systems which are related to the nilpotent coadjoint orbits of U(n,n). The Kustaanheimo-Stiefel and Cayley regularization procedures are discussed and their equivalence is shown. Some integrable generalization (perturbation) of (2n-1)-Kepler problem is proposed.


Introduction
Kepler problem (not only by historical reasons) is one of the most fundamental subjects of celestial mechanics and quantum mechanics [9,10,11,12,18,24]. Such questions as Moser [6] and  regularization procedures as well as the relationship between of them [7] are well known for celestial mechanics specialists. Also the questions concerning the quantization of the Kepler system and the MIC-Kepler system, which is its natural generalization, are the subject of many publications, e.g. see [13,17,18,19,20,23]. There are 1 other interesting generalizations of Kepler and MIC-Kepler problems, for example see [2,14,15,16].
Initially the group U(2, 2), being a natural extension of the Poincare group, was recognized as the dynamical group [1,11] for the threedimensional Kepler problem. Consequently, the group U(n, n) plays the same role for higher dimensional case. Taking this fact into account, in the present paper we study various Hamiltonian systems which have U(n, n) as a dynamical group. They are related to the adjoint nilpotent orbits of U(n, n) and could be interpreted as some natural generalizations of Kepler problem.
In Section 3 we investigate the vector bundles over the Grassmannian Gr 0 (n, C 2n ) of isotropic (with respect to φ) n-dimensional subspaces of C 2n , which treated as a manifold is isomorphic with U(n), see Proposition 3.2, and show that T * U(n) has structure of U(n, n)-Hamiltonian space, see Proposition 3.2 and Proposition 3.3. In Proposition 3.3, we classify the orbits of U(n, n)-action on T * U(n) and specify the one-toone correspondence of these orbits with such nilpotent adjoint U(n, n)orbits whose elements X satisfy X 2 = 0.
In Section 4 we investigate the geometry of the orbit N 10 , which consists of the rank one nilpotent elements of u(n, n), see Proposition 4.1. We also discuss the equivalent realizations of the regularized (2n− 1)-dimensional Kepler problem, see Proposition 4.3.
In Section 5 we show the equivalence of Cayley and Kustaanheimo-Stiefel regularizations in the context of higher-dimensional Kepler problem, originating in this way a natural generalization of the Kustaanheimo-Stiefel transform for the arbitrary dimension.
Finally, in the last Section 6 we consider some integrable generalization of (2n − 1)-Kepler problem. For this generalized Kepler problem the Hamiltonian, see formula (6.2) for its definition, depends on the positions and momenta through the coordinates of angular momenta and Runge-Lenz vector. The integrability of this system is proved by the methods developed in [21].

Grassmannian Gr(n, C 2n ) and related vector bundles
In this section we will study some canonicaly defined bundles over the Grassmannian Gr(n, C 2n ) of n-dimensional complex vector subspaces 2 of C 2n . Let us recall that Gr(n, C 2n ) is a n 2 -dimensional compact complex analytic manifold homogenous with respect to the natural action of GL(2n, C). We begin with defining the following complex analytic bundles over Gr(n, C 2n ). Namely, we consider the bundle π N : N → Gr(n, C 2n ) whose fibres consist of nilpotent elements of gl(2n, C). The total space of this bundle is defined as and π N is the projection of N on the second component of the product gl(2n, C) × Gr(n, C 2n ). One easily sees that π N : N → Gr(n, C 2n ) is a complex vector bundle of rank n 2 . The subset pr 1 (N ) ⊂ gl(2n, C) consists of such elements Z ∈ gl(2n, C), which satisfy Z 2 = 0 and have rank k := dim C Im(Z), where 1 ≤ k ≤ n.
Next one is the bundle π P : P → Gr(n, C 2n ) of idempotents, i.e.
where π P is the projection of P on the second component of gl(2n, C)× Gr(n, C 2n ). We note that pr 1 (P) ⊂ gl(2n, C) consists of such idempotents that dim C (Im(p)) = n. In order to make the structure of π P : P → Gr(n, C 2n ) transparent, we formulate the following proposition.
Proposition 2.1. The bundle π P : P → Gr(n, C 2n ) is an affine bundle with π N : N → Gr(n, C 2n ) as the structural vector bundle, i.e. for any z ∈ Gr(n, C 2n ) the vector space N z := π −1 N (z) acts in a transitive and free way on the fibre P z := π −1 P (z). Proof. For p ∈ P z and Z ∈ N z we have This shows that p + Z ∈ P z . For p, p ′ ∈ P z we have and dim C Im(p ′ − p) ≤ n. Thus, p ′ − p =: Z ∈ N z . Due to the above facts one has free and transitive action of N z on P z .
We note that for p ′ , p ∈ P z the following equalities hold Subsequently, using the Cartan-Killing form we will identify the dual space gl(2n, C) * with the Lie algebra gl(2n, C). For any p ∈ pr 1 (P) one has the open subset of the Grassmannian. We define a chart φ p : Ω p → (1 − p)gl(2n, C)p ∼ = Mat n×n (C) in the following way. The decomposition z ⊕ (1 − p)C 2n = C 2n defines the projection q z of C 2n on subspace z ⊂ C 2n . For projections 1 − q z and 1 − p one has Im(1 − q z ) = Im(1 − p). So, according to Proposition 2.1 there exists Z ∈ (1 − p)gl(2n, C)p such that The equality (2.9) defines the chart Ω p ∋ z → φ p (z) = Z, mentioned above.
Hence we see that one can identify in a canonical way the tangent space T z Gr(n, C 2n ) with N * z . So, we have Proposition 2.2. The bundle of nilpotent elements π N : N → Gr(n, C 2n ) is isomorphic with the complex cotangent bundle T * Gr(n, C 2n ) of the Grassmannian.
Let us note that one has another canonical complex vector bundles and over Gr(n, C 2n ). The complex linear group GL(2n, C) acts on the above bundles in the following way
of the vector bundles.
(ii) The group GL(2n, C) acts on N , E ⊗ E ⊥ and T * Gr(n, C 2n ) by Σ g , T g ⊗T * g and T * σ g , respectively, preserving their vector bundle structures, and isomorphisms from (2.24) are GL(2n, C)-isomorphisms. (iii) The vector bundle N (and thus the vector bundles E ⊗ E ⊥ and T * Gr(n, C 2n )) splits into GL(2n, C) orbits:

From Proposition 2.3 we conclude:
Remark 2.4. (i) The orbit N 0 is the zero section of N → Gr(n, C 2n ) so, one can identify it with Gr(n, C 2n ).
(ii) The orbit N n is an open-dense subset of N .
We mention here that N k is the total space of the following GL(2n, C)homogeneous bundles: where the bundle projections are defined by π im (Z, z) = Im(Z), π(Z, z) = z, (2.27) π ker (Z, z) = Ker(Z).
3. T * U(n) as a Hamiltonian U(n, n)-space Now we will describe some real versions of the structures described in the previous section and their relation to the structure of the cotangent bundle T * U(n) as a U(n, n)-Hamiltonian space. For this reason we fix a scalar product v, w := v + φw (3.1) of v, w ∈ C 2n , defined by a hermitian matrix φ = φ + ∈ Mat 2n×2n (C) which has signature (+ . . . + n − . . . − n ) and satisfies φ 2 = ½ 2n . Hence we define the group U(n, n) and Lie algebra u(n, n) of U(n, n) by and by 3) respectively, where by definition g ∈ U(n, n) and X ∈ u(n, n). Since for n = 2 the vector space C 2n provided with scalar product (3.1) is known as twistor space [22], in the subsequent we will use the same terminology for an arbitrary dimension.
Using scalar product (3.1) we also define on gl(2n, C), Gr(n, C 2n ) and N , respectively, the following involutions where z ⊥ ⊂ C 2n is the orthogonal complement of z ∈ Gr(n, C) with respect to (3.1) and Z ∈ gl(2n, C). Let us note that (3.4) is an antilinear map of gl(2n, C) and (3.6) is a fibre-wise anti-linear map of the bundle π N : N → Gr(n, C 2n ). Hence, taking into account the equivalent equalities we obtain the anti-holomorphic bundle isomorphisms which are equivariant with respect to the actions of U(n, n) ⊂ GL(2n, C) defined in (2.20) and (2.23). By π N 0 : N 0 → Gr 0 (n, C 2n ) we denote the vector bundle over the Grassmannian Gr 0 (n, C 2n ) of complex n-dimensional isotropic with respect to (3.1) subspaces of C 2n . By definition z ∈ Gr 0 (n, C 2n ) if and only if z = z ⊥ . The total space N 0 of this bundle is defined as the subset N 0 ⊂ N of fixed points of the involutionĨ : N → N defined in (3.6). Let us note here that dim R Gr 0 (n, C 2n ) = n 2 .
Let us define the map of the vector bundle N 0 into the Lie algebra u(n, n) by pr 1 (X, z) := X. (3.10) The set of values of this map is determined in the following way.
Proposition 3.1. An element X ∈ u(n, n) belongs to pr 1 (N 0 ) if and only if X 2 = 0.
Proof. If a X ∈ u(n, n) satisfies X 2 = 0 then because of I(X) = X and (3.7) we find that From the above and nonsingularity of the scalar product (3.1) we obtain So, 0 ≤ k ≤ n and thus, there exists z ∈ Gr 0 (n, C 2n ) such that i.e. X ∈ pr 1 (N 0 ). Note that Im(X) as an isotropic subspace of (C 2n , ·, · ) could be extended to maximal isotropic subspace z, which has dimension n and is contained in (Im(X)) ⊥ . By definition of N 0 any element X ∈ pr 1 (N 0 ) satisfies (3.11), so, one has X 2 = 0.
Next, in this section, taking the decomposition C 2n = C n ⊕ C n , we will choose the hermitian matrix from the definition (3.1) in the following diagonal block form where E and 0 are unit and zero n × n-matrices. Hence, we obtain of linearly independent vectors which span z ∈ Gr 0 (n, C 2n ). Since v k , v l = 0 it follows that for k, l = 1, . . . , n. From (3.14) we see that there exists such Z ∈ U(n) that η k = Zξ k for k = 1, . . . , n. So, vectors ξ 1 , . . . , ξ n form a basis in C n . The above considerations shows that there is a natural diffeomorphism U(n) ∼ = Gr 0 (n, C 2n ) between the unitary group U(n) and the Grassmannian Gr 0 (n, C 2n ) of n-dimensional isotropic subspaces of (C 2n , ·, · ) defined in the following way One easily see that for φ d the block matrix elements A, B, C, D ∈ Mat n×n (C) of g = A B C D ∈ U(n, n) satisfy From (3.15) one find that U(n, n) acts on U(n) as follows Subsequently we will need the explicit description of the stabilizer U(n, n) E := {g ∈ U(n, n) : σ g (E) = E} of the group unit E ∈ U(n).

Simple considerations shows that
where F ∈ GL(n, C) and H ∈ Mat n×n (C) satisfy we denote the vector tangent to Z(t) at Z and by τ : where by H(n) we denote the real vector space of n × n-hermitian matrices. Using the above notation, from (3.17) and (3.16) we obtain (3.20) Since ofŻ = Zτ , we have the isomorphism of vector bundles T U(n) ∼ = U(n) × iH(n). It follows from (3.20) that the covector ρ ∈ T * E U(n) ∼ = iH(n) is transformed in the following way (3.21) 9 The elements of Lie algebra u(n, n) in the diagonal realization (3.13) of φ are given by matrices where β ∈ Mat n×n (C) and α, δ ∈ iH(n).
is a U(n, n)-equivariant isomorphism of the vector bundles. The action Σ g : N 0 → N 0 , g ∈ U(n, n), is a restriction to U(n, n) and N 0 ⊂ N of the action defined in (2.20). The action Λ g : U(n) × iH(n) → U(n) × iH(n) is defined by where g = A B C D .
(ii) The canonical one-form γ 0 on T * U(n) ∼ = U(n) × iH(n) written in the coordinates (Z, δ) ∈ U(n) × iH(n) assumes the form and it is invariant with respect to the action (3.25).
(iii) The map J 0 : T * U(n) → u(n, n) defined by is the momentum map for symplectic form dγ 0 , i.e. it is a U(n, n)equivariant  for f, g ∈ C ∞ (u(n, n), R).
(ii) One obtains (3.26) directly from the definition of canonical form γ 0 on T * U(n) and from the isomorphism T * U(n) ∼ = U(n) × iH(n).
(ii) From Proposition 3.1 and point (i) of Proposition 3.2 it follows that any Ad(U(n, n))-orbit in pr 1 (N 0 ) has form J 0 (O kl ). Since for g ∈ U(n, n) E we have the momentum map J 0 : T * U(n) → pr 1 (N 0 ) maps O kl on the one Ad(U(n, n))-orbit N kl ⊂ pr 1 (N 0 ) only.
As it follows from general theory, the Ad(U(n, n))-orbit N kl is a homogenous symplectic manifold with the symplectic form ω kl , obtained in a canonical way by Kirillov construction, see [5]. From point (ii) of Proposition 3.3 we have J −1 0 (N kl ) = O kl . Hence, one can obtain (N kl , ω kl ) reducing standard symplectic form dγ 0 on T * U(n) to the orbit O kl . Let us note here that fibres J −1 0 (X), X ∈ N kl , are degeneracy submanifolds for the 2-form dγ 0 | O kl , so, N kl = O kl / ∼ and ω kl = dγ 0 | O kl / ∼ , where "∼" is equivalence relation on O kl defined by the submersion J 0 : O kl → N kl .
Ending this section, we mention that in the case when k + l = n one has N kl ∼ = O kl and the orbits O kl are open subsets of the cotangent bundle T * U(n). For symplectic forms ω kl we have ω kl = dγ 0 .

Regularized (2n − 1)-dimensional Kepler problem
In this section we will describe in details the various Hamiltonian systems having U(n, n) as their dynamical group. As we will show in the next section, these systems give the equivalent description of the regularized (2n − 1)-dimensional Kepler system.
(ii) One can also consider N 10 as the total space of the fibre bundlė The total space of the tangent bundle T CP(n − 1) → CP(n − 1) has the form

) and its complementary subbundle
is isomorphic to the trivial bundle CP(n − 1) × C.
Summing the above facts we conclude from the point (ii) of Proposition 4.1 that one can identify N 10 ∼ = S 2n−1 ×C n U (1) → CP(n − 1) with the vector bundle S 2n−1 ×C n U (1) → CP(n − 1) with removed null section. To explain the role of U(n, n) as the dynamical group for (2n − 1)dimensional regularized Kepler problem we discuss now other description of N 10 corresponding to the choice of anti-diagonal realization of twistor form (3.1). Subsequently we will denote the realizations (C 2n , φ d ) and (C 2n , φ a ) of twistor space by T andT , respectively. The same convention will be assumed for their groups of symmetry, i.e. g = A B C D ∈ U(n, n) iff g + φ d g = φ d and g = ÃB CD ∈ U(n, n) iffg + φ ag = φ a . Hence, forg ∈ U(n, n) andX = u(n, n) one hasÃ respectively, whereβ + =β andγ + =γ. The canonical one-form (4.1) and the momentum map (4.3) forT are given byγ

.16) and byJ
where υ ζ ∈T . The null twistors space is defined asT 0 (4.18) The Hamiltonian flow on C 2n generated byĨ +− is given bỹ Both realizations T andT of the twistor space are related by the following unitary transformation of C 2n : Let us note here that the above action is not defined globally, i.e. the formula (4.23) is valid only if det(CY +D) = 0.
The momentum mapJ 0 : H(n) × H(n) → u(n, n) corresponding to dγ 0 andσg has the form and it satisfies the equivariance propertỹ The following diagram  Proof. By straightforward verification.
is a smooth one-to-one map of H(n) into U(n), which is known as Cayley transform, see e.g. [3]. Hence, the unitary group U(n) could be considered as a compactification of H(n), Namely, in order to obtain the full group U(n) one adds to Cayles image of H(n) such unitary matrices Z, which satisfy the condition det(iZ + E) = 0. One sees this observing that the inverse Cayley map is defined by Taking into account the properties of Poisson maps presented in the diagram (4.26) as well as (4.33) and (4.32) one obtains the following morphisms of U(n, n)-Hamiltonian spaces which are symplectic isomorphisms, except of In particular cases when X ++ = i Rewriting the above formula in the anti-diagonal realization, wherẽ The functions I ++ , I 0 , andĨ ++ ,Ĩ 0 are invariants of the Hamiltonian flows presented in (4.6) and (4.19), respectively. Note that these flows are generated by i E 0 0 −E ∈ u(n, n). So, the above functions could be considered as Hamiltonians I ++ / ∼ , I 0 / ∼ andĨ ++ / ∼ ,Ĩ 0 / ∼ on the reduced symplectic manifolds T 0 +− / ∼ , O 10 / ∼ andT 0 +− / ∼ ,Ȯ 10 / ∼ , respectively. Taking into account the symplectic manifolds morphisms mentioned in the diagram (4.34) we conclude Since the Hamiltonian L X +− and, thus Hamiltonians I ++ / ∼ , I 0 / ∼ , I ++ / ∼ andĨ 0 / ∼ are defined by the element X +− of the Lie algebra u(n, n) one can consider U(n, n) as a dynamical group for all systems mentioned in (i) of Proposition 4.3. As a matter of fact we can treat all of them as various realizations of the same Hamiltonian system. 18 The easiest way to find the symmetry groups of these systems, and thus, their integrals of motion, is to consider the case (T 0 +− / ∼ , I ++ / ∼ ). In this case the symmetry group is the subgroup of U(n, n), which preserve the canonical form γ +− , defined in (4.1), and the Hamiltonian I ++ , i.e. it is U(n, n) ∩ U(2n) ∼ = U(n) × U(n). So, the corresponding integrals of motion one obtains restricting the matrix functions Reducing them to (Ȯ 10 / ∼ ,Ĩ 0 / ∼ ) we obtain their coresspondence to the integrals of motion I + and I − : (4.47) where K reg :Ȯ 10 / ∼ →T 0 +− / ∼ is defined by The Hamilton equations defined byĨ 0 are i.e. they could be classified as a matrix Riccati type equations. In order to obtain their solution we note that after passing to (T 0 +− / ∼ , I ++ / ∼ ) they asssume the form of a linear equations which are solved by i.e. the Hamiltonian flow σ t ++ is one-parameter subgroup of U(n, n) generated by X +− ∈ u(n, n). Therefore, going through the symplectic manifold isomorphisms presented in (4.34), we obtain the solution 19 of (4.49) by specifying the transformation formula (4.23) to the one- C of the group U(n, n).
Ending this section let us mention the papers [17,18,19,20,23], where Kepler and MIC-Kepler problems were considered on the classical and quantum levels. Let us also mention some interesting generalizations of these problems [2,13,14,15,16] based on the theory of Jordan algebras.

Cayley and Kustaanheimo-Stiefel transformations
In this section we discuss two regularizations of the Hamiltonian system (Ȯ 10 / ∼ ,Ĩ 0 ) which were mentioned in the point (ii) of Proposition 4.3. At first we will show that the regularization K reg :Ȯ 10 / ∼ → T 0 +− / ∼ could be interpreted as a generalization for arbitrary dimension of Kustaanheimo-Stiefel regularization, which was introduced in [8] for the case n = 2. Then we will discuss shortly the regularization We will also show the equivalence of the both considered regularizations.