Tigran Hakobyan

Hidden symmetries of integrable conformal mechanical systems

Abstract We split the generic conformal mechanical system into a “radial” and an “angular” part, where the latter is defined as the Hamiltonian system on the orbit of the conformal group, with the Casimir function in the role of the Hamiltonian. We reduce the analysis of the constants of motion of the full system to the study of certain differential equations on this orbit. For integrable mechanical systems, the conformal invariance renders them superintegrable, yielding an additional series of conserved quantities originally found by Wojciechowski in the rational Calogero model. Finally, we show that, starting from any N = 4 supersymmetric “angular” Hamiltonian system one may construct a new system with full N = 4 superconformal D ( 1 , 2 ; α ) symmetry.

The cuboctahedric Higgs oscillator from the rational Calogero model

We exclude the center of mass of the N-particle rational Calogero model and consider the angular part of the resulting Hamiltonian. We show that it describes the motion of the particle on (N-2)-dimensional sphere interacting with N(N-1)/2 force centers with Higgs oscillator potential. In the case of four-particle system these force centers define the vertexes of an Archimedean solid called cuboctahedron.

Invariants of the spherical sector in conformal mechanics

A direct relation is established between the constants of motion for conformal mechanics and those for its spherical part. In this way, we find the complete set of functionally independent constants of motion for the so-called cuboctahedric Higgs oscillator, which is just the spherical part of the rational A3 Calogero model (describing four Calogero particles after decoupling their center of mass).

Integrable generalizations of oscillator and Coulomb systems via action–angle variables

Abstract Oscillator and Coulomb systems on N-dimensional spaces of constant curvature can be generalized by replacing their angular degrees of freedom with a compact integrable ( N − 1 ) -dimensional system. We present the action–angle formulation of such models in terms of the radial degree of freedom and the action–angle variables of the angular subsystem. As an example, we construct the spherical and pseudospherical generalization of the two-dimensional superintegrable models introduced by Tremblay, Turbiner and Winternitz and by Post and Winternitz. We demonstrate the superintegrability of these systems and give their hidden constant of motion.

The spherical sector of the Calogero model as a reduced matrix model

Abstract We investigate the matrix-model origin of the spherical sector of the rational Calogero model and its constants of motion. We develop a diagrammatic technique which allows us to find explicit expressions of the constants of motion and calculate their Poisson brackets. In this way we obtain all functionally independent constants of motion to any given order in the momenta. Our technique is related to the valence-bond basis for singlet states.

Phase diagram of the frustrated two-leg ladder model

12 chains one next to the other. Many weakly coupled ladder systems have now been synthesized. Among them, the family Srn21Cun11O2n consists in weakly coupled 1 2 (n11)-legged ladders which are obtained from the CuO2 planes of the parent compound SrCuO2. After the suggestion that mechanisms similar to those occurring in the CuO2 planes of cuprate ceramics may also lead to

On Dunkl angular momenta algebra

A bstractWe consider the quantum angular momentum generators, deformed by means of the Dunkl operators. Together with the reflection operators they generate a subalgebra in the rational Cherednik algebra associated with a finite real reflection group. We find all the defining relations of the algebra, which appear to be quadratic, and we show that the algebra is of Poincaré-Birkhoff-Witt (PBW) type. We show that this algebra contains the angular part of the Calogero-Moser Hamiltonian and that together with constants it generates the centre of the algebra. We also consider the gl(N ) version of the subalge-bra of the rational Cherednik algebra and show that it is a non-homogeneous quadratic algebra of PBW type as well. In this case the central generator can be identified with the usual Calogero-Moser Hamiltonian associated with the Coxeter group in the harmonic confinement.

Lobachevsky geometry of (super)conformal mechanics

Abstract We give a simple geometric explanation for the similarity transformation mapping one-dimensional conformal mechanics to free-particle system. Namely, we show that this transformation corresponds to the inversion of the Klein model of Lobachevsky space (non-compact complex projective plane) C P ˜ 1 . We also extend this picture to the N = 2 k superconformal mechanics described in terms of Lobachevsky superspace C P ˜ 1 | k .

Delocalization of states in two-component superlattices with correlated disorder

Electron and phonon states in two different models of intentionally disordered superlattices are studied analytically as well as numerically. The localization length is calculated exactly and we found that it diverges for particular energies or frequencies, suggesting the existence of delocalized states for both electrons and phonons. Numerical calculations for the transmission coefficient support the existence of these delocalized states.

Spin chain Hamiltonians with affine Uqg symmetry

We construct the family of spin chain Hamiltonians, which have affine