This paper is a natural continuation of the previous paper J.Phys. A: Math.Theor. 44 (2011) 425204, arXiv 0907.1736 [quant-ph] where oscillator representations for nonnegative Calogero Hamiltonians with coupling constant $α\geq-1/4$ were constructed. Here, we present generalized oscillator representations for all Calogero Hamiltonians with $α\geq-1/4$.These representations are generally highly nonunique, but there exists an optimum representation for each Hamiltonian.
We construct the Hamiltonians and symmetry generators of Calogero-oscillator and Calogero-Coulomb models on the N-dimensional sphere within the matrix-model reduction approach. Our method also produces the integrable Calogero-Coulomb-Stark model on the sphere and proves the integrability of the spin extensions of all these systems.
We consider the quantum mechanics of Calogero models in an oscillator or Coulomb potential on the N-dimensional sphere. Their Hamiltonians are obtained by an appropriate Dunkl deformation of the oscillator/Coulomb system on the sphere and its restriction to (Coxeter reflection) symmetric wave functions. By the same method we also find the symmetry generators and compute their algebras.
We propose a new ${\cal N}$-extended supersymmetric $su(n)$ spin-Calogero model. Employing a generalized Hamiltonian reduction adopted to the supersymmetric case, we explicitly construct a novel rational $n$-particle Calogero model with an arbitrary even number of supersymmetries. It features ${\cal N}n^2$ rather than ${\cal N}n$ fermionic coordinates and increasingly high fermionic powers in the supercharges and the Hamiltonian.
The generalized Mironov-Morozov-Natanson (MMN) equation includes a set of commuting operators, which can be considered as Hamiltonians for the quantum Calogero-Sutherland problem with a special value of the coupling constant (free fermion point). These Hamiltonians can be considered as the center of the enveloping algebra of the group $GL_N(C)$. Another commuting series of Hamiltonians is presented, parametrized by an arbitrary matrix $A\in GL_N$. These Hamiltonians are related to the Hurwitz numbers in the same way as in the case of the MMN equation and generate a generalized variant of the C…
Starting from the Hamiltonian formulation of supersymmetric Calogero models associated with the classical $A_n$, $B_n$, $C_n$ and $D_n$ series we construct the ${\cal N}{=}\,2$ and ${\cal N}{=}\,4$ supersymmetric extensions of the their hyperbolic/trigonometric Calogero-Sutherland cousins. The bosonic core of these models are the standard Calogero-Sutherland hyperbolic/trigonometric systems.
We consider generalised Calogero-Moser-Sutherland quantum Hamiltonian $H$ associated with a configuration of vectors $AG_2$ on the plane which is a union of $A_2$ and $G_2$ root systems. The Hamiltonian $H$ depends on one parameter. We find an intertwining operator between $H$ and the Calogero-Moser-Sutherland Hamiltonian for the root system $G_2$. This gives a quantum integral for $H$ of order 6 in an explicit form thus establishing integrability of $H$.
We consider trigonometric solutions of the KP hierarchy. It is known that their poles move as particles of the Calogero-Moser model with trigonometric potential. We show that this correspondence can be extended to the level of hierarchies: the evolution of the poles with respect to the $k$-th hierarchical time of the KP hierarchy is governed by a Hamiltonian which is a linear combination of the first $k$ higher Hamiltonians of the trigonometric Calogero-Moser hierarchy.
This thesis presents our results on Liouville integrable systems of Calogero-Ruijsenaars type: 1. We prove an explicit formula providing canonical spectral coordinates for the rational Calogero-Moser system. 2. We explore action-angle duality for the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 3. We derive a Poisson-Lie deformation of the trigonometric BC(n) Sutherland system using Hamiltonian reduction. 4. We construct a Lax pair for the hyperbolic Ruijsenaars-Schneider system with two couplings. 5. We present an explicit construction of compactified trigonometric and e…
We consider solutions of the matrix KP hierarchy that are elliptic functions of the first hierarchical time $t_1=x$. It is known that poles $x_i$ and matrix residues at the poles $ρ_i^{αβ}=a_i^αb_i^β$ of such solutions as functions of the time $t_2$ move as particles of spin generalization of the elliptic Calogero-Moser model (elliptic Gibbons-Hermsen model). In this paper we establish the correspondence with the spin elliptic Calogero-Moser model for the whole matrix KP hierarchy. Namely, we show that the dynamics of poles and matrix residues of the solutions with respect to the $k$-th hierar…
In the two preceding parts of this series of papers, we introduced and studied a recursion scheme for constructing joint eigenfunctions $J_N(a_+, a_-,b;x,y)$ of the Hamiltonians arising in the integrable $N$-particle systems of hyperbolic relativistic Calogero-Moser type. We focused on the first steps of the scheme in Part I, and on the cases $N=2$ and $N=3$ in Part II. In this paper, we determine the dominant asymptotics of a similarity transformed function $\rE_N(b;x,y)$ for $y_j-y_{j+1}\to\infty$, $j=1,\ldots, N-1$, and thereby confirm the long standing conjecture that the particles in the …
We present the deformed (for the presence of Calogero potential terms) one-dimensional quantum oscillator with the exceptional Lie superalgebra $F(4)$ as spectrum-generating superconformal algebra. The Hilbert space is given by a $16$-ple of square-integrable functions. The energy levels are $\frac{2}{3}+n$, with $n=0,1,2,\ldots$. The ground state is $7$ times degenerate. The excited states are $8$ times degenerate. The $(7,8,8,8,\ldots )$ semi-infinite tower of states is recovered from the $(7;8;1)$ supermultiplet of the ${\cal N}=8$ worldline supersymmetry. The model is unique, up to similar…
Starting from the Hamiltonian formulation of ${\cal N}{=}\,2$ supersymmetric Calogero models associated with the classical $A_n, B_n, C_n$ and $D_n$ series and their hyperbolic/trigonometric cousins, we provide their superspace description. The key ingredients include $n$ bosonic and $2n(n{-}1)$ fermionic ${\cal N}{=}\,2$ superfields, the latter being subject to a nonlinear chirality constraint. This constraint has a universal form valid for all Calogero models. With its help we find more general supercharges (and a superspace Lagrangian), which provide the ${\cal N}{=}\,2$ supersymmetrization…
Classical mechanics on GL(n, R) group and Euler-Calogero-Sutherland model
在哈密顿约化的框架下，讨论了GL^+（n，R）群流形上的自由运动与自旋自由度为1/sinh^2 x "势“（Euler-Calogero-Sutherland模型）相互作用的n粒子系统的动力学之间的关系.两种减少的自由度被认为是：由于连续不变性和由于离散对称性。它表明，投影后相应的不变流形的哈密顿系统表示Euler-Calogero-Sutherland模型在这两种情况下。
We consider solutions of the KP hierarchy which are elliptic functions of $x=t_1$. It is known that their poles as functions of $t_2$ move as particles of the elliptic Calogero-Moser model. We extend this correspondence to the level of hierarchies and find the Hamiltonian $H_k$ of the elliptic Calogero-Moser model which governs the dynamics of poles with respect to the $k$-th hierarchical time. The Hamiltonians $H_k$ are obtained as coefficients of the expansion of the spectral curve near the marked point in which the Baker-Akhiezer function has essential singularity.
We use Hamiltonian reduction to simplify Falqui and Mencattini's recent proof of Sklyanin's expression providing spectral Darboux coordinates of the rational Calogero-Moser system. This viewpoint enables us to verify a conjecture of Falqui and Mencattini, and to obtain Sklyanin's formula as a corollary.
We use generalized kernel functions to construct explicit solutions by integrals of the non-stationary Schrödinger equation for the Hamiltonian of the elliptic Calogero-Sutherland model (also known as elliptic Knizhnik-Zamolodchikov-Bernard equation). Our solutions provide integral represenations of elliptic generalizations of the Jack polyomials.
In a previous paper we introduced and developed a recursive construction of joint eigenfunctions $J_N(a_+,a_-,b;x,y)$ for the Hamiltonians of the hyperbolic relativistic Calogero-Moser system with arbitrary particle number $N$. In this paper we focus on the cases $N=2$ and $N=3$, and establish a number of conjectured features of the corresponding joint eigenfunctions. More specifically, choosing $a_+,a_-$ positive, we prove that $J_2(b;x,y)$ and $J_3(b;x,y)$ extend to globally meromorphic functions that satisfy various invariance properties as well as a duality relation. We also obtain detaile…
We consider the isomonodromic formulation of the Calogero-Painlevé multi-particle systems and proceed to their canonical quantization. We then proceed to the quantum Hamiltonian reduction on a special representation to radial variables, in analogy with the classical case and also with the theory of quantum Calogero equations. This quantized version is compared to the generalization of a result of Nagoya on integral representations of certain solutions of the quantum Painlevé equations. We also provide multi-particle generalizations of these integral representations.
Parent Hamiltonian construction of generalized Calogero-Sutherland models
Indian Institute of Technology Madras（印度理工学院马德拉斯分校）；Würzburg–Dresden Cluster of Excellence ctd.qmat（维尔茨堡-德累斯顿卓越集群 ctd.qmat）；Julius-Maximilians-Universität Würzburg（维尔茨堡朱利叶斯-马克西米利安大学）
The 8-parameter elliptic Sakai difference Painlevé equation admits a Lax formulation. We show that a suitable specialization of the Lax equation gives rise to the time-independent Schrödinger equation for the $BC_1$ 8-parameter 'relativistic' Calogero-Moser Hamiltonian due to van Diejen. This amounts to a generalization of previous results concerning the Painlevé-Calogero correspondence to the highest level in the two hierarchies.
We construct ${\mathcal N}=4 \,$ $\, D(2,1;α)$ superconformal quantum mechanical system for any configuration of vectors forming a V-system. In the case of a Coxeter root system the bosonic potential of the supersymmetric Hamiltonian is the corresponding generalised Calogero-Moser potential. We also construct supersymmetric generalised trigonometric Calogero-Moser-Sutherland Hamiltonians for some root systems including $BC_N$.
