Superintegrability of Calogero model with oscillator and Coulomb potentials and their generalizations to (pseudo)spheres: Observation

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1 Superintegrability of Calogero model with oscillator and Coulomb potentials and their generalizations to (pseudo)spheres: Observation Armen Nersessian Yerevan State University Hannover 013

2 Whole non-triviality of Calogero model (and of other conformal mechanics) is encoded in spherical part. With the aid of that spherical part I will: Construct the analog of Calogero-oscillator and Calogero-Coulomb systems on spheres and hyperboloids, Demonstrate their classical superintegrability. As a bi-product, I ll show that the so-called Tremblay-Turbiner-Winternits and Post-Winternitz models ( Calogero-oscillator and Calogero-Coulomb systems related with positive roots of Coxeter system I (k)), are connected to each other by Bohlin transformation

3 Content Spherical part of conformal mechanics Spherical part of Calogero model Spherical Calogero in terms of action variables Superintegrable Calogero-oscillator and Calogero-Coulomb systems Tremblay-Turbiner-Winternitz and Post-Winternitz systems and their relation

4 Conformal mechanics {H, D} = H, {K, D} = K, {H, K} = D Most known example ω = dp dr, H = p + V (r), where r V (r) = V (r). D = p r, K = r The phase space of conformal mechanics is noncompact one: we can t formulate the system in action-angle variables.

5 Canonical basis and its spherical part SO(1.) Casimir : I = KH 1 D Introduce radial coordinate and momentum D p r r, K r : {P r, I} = {r, I} H = p r + I : {H, I} = 0 r One can choose the appropriate spherical coordinates, u A = (p α, φ α ) : {p r, u A } = {r, u A } = 0, {p β, φ α } = δ α β, separating the radial and spherical parts of the system. Thus, we can extract the compact ( spherical ) part of conformal mechanics: (I(u), dp α dφ α )

6 Importance of spherical part Splitting of noncompact motion from compact one Straightforward application of the textbook scattering theory N = 4 superconformal extentions (Super)Integrable spherical parts allows to construct new (super)integrable (deformations of known) systems

7 Superconformal extensions Let the spherical spherical part of conformal mechanics admits N = 4 supersymmetric extension {Θ aα, Θ bβ } = ıε ab ε αβ I SUSY. Then we can immediately construct N = 4 D(1.; α) superconformal extention of the whole system (T.Hakobyan, O.Lechtenfeld, S.Krivonos, A.N.) S aα = rη aα, Q aα = p r η aα + Θaα r +..., H SUSY = p r + I SUSY r ıθaα η aα r +...

8 Integrable deformations of N-dimensional Models IR N : H = p r + I { N 1(I i ) ω r + V(r), V(r) = r / γ/r S N : H = p χ r0 + I { N 1 r r0 + V (tan χ), V (r) = 0 ω tan χ/ sin χ (γ cot χ)/r 0 H N : H = p χ r0 + I { N 1 r r0 +V (tanh χ), V (r) = 0 ω tanh χ/ sinh χ (γ coth χ)/r 0

9 Quantum and classical spectra Quantum E(n r, l) E(n r, E n1,...,n N 1 ), where E n1,...,n N 1 ) is spectrum os spherical part. Classical H(I r, I 1 + I +... N 1 ) H(I r, I(I 1,..., I N 1 )), where I(I 1,..., I N 1 ) is spherical part written in action variables.

10 How to find hidden symmetries? Let we have Hamiltonian system formulated in action angle variables, with the Hamiltonian H = H(k 1 I 1 + k I k K I K, I K+1,..., I N ) where k i are integers. Then the system possesses hidden symmeties given by the functions I ij = cos(k j Φ i k i Φ j ), i, j = 1,..., K. Maximally superintegrable systems H = H(k 1 I 1 + k I k N I N ).

11 Classical spectra of deformed oscillator and Coulomb systems Oscillator ω(i r + I) for IR 1 H osc = (I χ + I + ω) ω for S 1 (I χ + I ω) + ω for H Coulomb γ /(I r + I) for IR H C = γ /(I χ + + (I χ + I) / for S ( γ / I χ + I) (Iχ + I) / for H

12 Superintegrable deformations of oscillator and Coulomb systems requires: I Sph = 1 ( i k i I i ), k i N Which systems on spheres have such Hamiltonian?

13 Simplest example: I MP = L N + gi xi, i=1 N xi = 1. (0.1) It defines nontrivial part of spherical mechanics of a particle in (n + 1)-dimensional extreme Myers-perry black hole i=1 Classical Spectrum : I MP = 1 N 1 ( i=1 I i + N g i ) i=1

14 Other example: Let us define the segment on (N 1)-dimensional sphere restricting the range of the angles by: φ i [0, π/k i ), k i Z. We can immediately find, that its classical spectrum is of required form. However, the configuration space of resulting system is N-dimensional cone!

15 Spherical part of Calogero model(s) The study of spherical part of CM at classical level has been done in our papers with T.Hakobyan, S.Krivonos, O.Lechtenfeld, A.Saghatelian, V.Yeghikyan. The spherical part of N-particle Calogero model defines multi-center Higgs oscillator on S N 1. For example, for A 3 -CM force centers are located at the vertexes of cuboctahedron. Spherical parts of rational CM s are maximally superintegrable systems. They allows immediately find hidden symmetries of rational CM Quantum properties of the spherical part of rational Calogero models were considered in the recent paper of M.Feigin, O.Lechtenfeld, A.Polychronakos.

16 Spherical part of conventional CM: from Quantum to Classical Quantum E sphcal = 1 q (q + g + (N )), where N(N 1) q g + k k N k N, with all k i take nonnegative integer values. Classical limit I sphcal = 1 ( ) N(N 1) + 1 g + I 1 + 3I NI N + N

17 Calogero-Coulomb and Calogero oscillator models on R n H CC = p + i g (x i x j ) + ω x Well-known system. H CC = p + i g (x i x j ) γ x Less known system suggested by A. Khare (1996). MAXIMAL SUPERINTEGRABILITY!!!

18 Generalization to spheres and hyperboloids V (p)s Coulomb = α + g α(α α)r0 (α x) γ x 0 r 0 x, x 0 ± x = r0, V (p)s osc = α + g α(α α)r0 (α x) + ω r0 x x0, x 0 ± x = r 0. The upper sign corresponds to the sphere, and lower sign to the hyperboloid. MAXIMAL SUPERINTEGRABILITY!!!

19 CO(scillator) - CC(oulomb) corredpondence. -particle case d =, 3, 5 Coulomb system can be with d =, 4, 8-oscillator via Hopf maps. Similar relation holds for respective Calogero-Coulomb and Calogero-Oscillator systems. H TTW = π π + f ((z/ z)k ) z z + ω z z, where z = x 1 + ıx = re φ, f (φ) = k α sin kφ + k β cos kφ. Performing canonical (Levi-Civita) transformation w = z, p = π z, we get H TTW = 4 w p p + f ( w/ w) w + ω w.

20 Fixing the level surface H TTW = E TTW, we get H PW = E PW, H PW p p + f ( w/ w) 4w w γ w : where γ = E TTW E PW = ω 4 4. Generalization of this transformation to the sphere and pseudosphere is straightforward (A.N., G.Pogosyan, 000). For the higher dimensions the relation between CO- and CC-systems is less trivial.

21 Thank you for your attention!

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