The exchange operator formalism in polar coordinates, previously considered for the Calogero-Marchioro-Wolfes problem, is generalized to a recently introduced, infinite family of exactly solvable and integrable Hamiltonians $H_k$, $k=1$, 2, 3,..., on a plane. The elements of the dihedral group $D_{2k}$ are realized as operators on this plane and used to define some differential-difference operators $D_r$ and $D_{\varphi}$. The latter serve to construct $D_{2k}$-extended and invariant Hamiltonians $\chh_k$, from which the starting Hamiltonians $H_k$ can be retrieved by projection in the $D_{2k}$ identity representation space.
Deep Dive into Exchange operator formalism for an infinite family of solvable and integrable quantum systems on a plane.
The exchange operator formalism in polar coordinates, previously considered for the Calogero-Marchioro-Wolfes problem, is generalized to a recently introduced, infinite family of exactly solvable and integrable Hamiltonians $H_k$, $k=1$, 2, 3,…, on a plane. The elements of the dihedral group $D_{2k}$ are realized as operators on this plane and used to define some differential-difference operators $D_r$ and $D_{\varphi}$. The latter serve to construct $D_{2k}$-extended and invariant Hamiltonians $\chh_k$, from which the starting Hamiltonians $H_k$ can be retrieved by projection in the $D_{2k}$ identity representation space.
In a recent work, an infinite family of exactly solvable and integrable quantum Hamiltonians H k , k = 1, 2, 3, . . . , on a plane has been introduced [1]. Such a family includes all previously known Hamiltonians with the above properties, containing rational potentials and allowing separation of variables in polar coordinates. These correspond to the Smorodinsky-Winternitz (SW) system (k = 1) [2,3], the rational BC 2 model (k = 2) [4,5], and the Calogero-Marchioro-Wolfes (CMW) model (k = 3) [6,7] (reducing in a special case to the three-particle Calogero one [8]). Furthermore, it has been conjectured (and proved for the first few cases) that all members of the family are also superintegrable. In agreement with such a conjecture, all bounded classical trajectories have been shown to be closed and the classical motion to be periodic [9].
Since the pioneering work of Olshanetsky and Perelomov [4,5] on the integrability of Calogero-Sutherland type N-body models, i.e., the existence of N well-defined, commuting integrals of motion including the Hamiltonian, there have been several studies of such a problem using various approaches (see, e.g., Refs. [10,11] for some recent ones). One of the most interesting methods is based on the use of some differential-difference operators or covariant derivatives, known in the mathematical literature as Dunkl operators [12]. These operators were independently rediscovered by Polychronakos [13] and Brink et al. [14] in the context of the N-body Calogero model. Later on, they were generalized to the CMW model [15] and, in such a context, an interesting exchange operator formalism in polar coordinates was introduced [16].
Since the CMW Hamiltonian is one of the members of the infinite family of Hamiltonians H k , k = 1, 2, 3, . . . , considered in Ref. [1], it is worthwhile to extend the latter formalism to the whole family and to study some of its consequences. This is the purpose of this letter. To solve the problem, we shall have to distinguish between odd and even k values and to prove several nontrivial trigonometric identities.
Let us consider the subfamily of Hamiltonians
corresponding to k = 1, 3, 5, . . . . Here ฯ, a, b are three parameters such that ฯ > 0,
), and the configuration space is given by the sector 0 โค r < โ, 0 โค ฯ โค ฯ/(2k). In cartesian coordinates x = r cos ฯ, y = r sin ฯ, the Hamiltonian (2.1) can be rewritten as
and more and more complicated expressions as k is increasing. H 1 is known as the SW Hamiltonian [2,3], while H 3 is the relative motion Hamiltonian in the CMW problem, as shown below.
The three-particle Hamiltonian of the CMW problem is given by [7]
where x i , i = 1, 2, 3, denote the particle coordinates, x ij = x i -x j , i = j, and
The range of the particle coordinates is appropriately restricted as explained in Ref. [15]. In terms of the variables x =
x 12 / โ 2, y = y 12 / โ 6, and X = (x 1 + x 2 + x 3 )/ โ 3, the Hamiltonian can be separated into a centre-of-mass Hamiltonian H cm = -โ 2 X + ฯ 2 X 2 and a relative one H rel , coinciding with H 3 given in (2.3).
The CMW Hamiltonian (2.4) is known [4,5] to be related to the G 2 Lie algebra, whose Weyl group is the dihedral group D 6 . The 12 operators of the latter can be realized either in terms of the particle permutation operators K ij and the inversion operator I r in relative coordinate space [15] or in terms of the rotation operator R = exp 1 3 ฯโ ฯ through angle ฯ/3 in the plane (r, ฯ) and the operator I = exp(iฯฯโ ฯ ) changing ฯ into -ฯ [16]. These exchange operators can then be used to extend the partial derivatives
In terms of the former
the latter can be defined as
On using the correspondences
Eq. (2.6) can be rewritten as
Note that the small discrepancies existing between Eqs. (2.6) -(2.8) and the corresponding expressions ( 13) and ( 15) of Ref. [16] come from the fact that here we use the conventional definition of polar coordinates, while our former work was based on Wolfes’ definition [7].
We can now build on this exchange operator formalism in polar coordinates to go further and extend the Hamiltonian (2.4) itself. As a first step, we note that from the characteristic relations of D 6 ,
it is easy to prove that D r and D ฯ satisfy the equations
(2.12)
The next stage consists in expressing D 2 ฯ in terms of ฯ, โ ฯ , R, and I. This can be done using Eqs. (2.9) and (2.11), as well as some well-known trigonometric identities. The result reads
(2.13)
Finally, we may introduce some generalized CMW Hamiltonian, defined by
where
We now plan to show that the formalism developed for k = 3 in Sec. 2.1 can be extended to any other odd k value (including k = 1). For such a purpose, let us introduce the two operators R = exp 1 k ฯโ ฯ and I = exp(iฯฯโ ฯ ), satisfying the defining relations
of the dihedral group D 2k , whose elements may be realized as R i and R i I, i = 0, 1, . . . , 2k -1. It is then straightforward to show that the differential-difference operat
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