Abstract
It is long known that the rational Calogero model describing n identical particles on a line with inverse-square mutual interaction potential is quantum superintegrable. We review the (nonlinear) algebra of the conserved quantum charges and the intertwiners which relate the Liouville charges at couplings g and g±1. For integer values of g, these intertwiners give rise to additional conserved charges commuting with all Liouville charges and known since the 1990s. We give a direct construction of such a charge, the unique one being totally antisymmetric under particle permutations. It is of order \( \frac{1}{2} \) n(n−1)(2g−1) in the momenta and squares to a polynomial in the Liouville charges. With a natural \( \mathbb{Z} \) 2 grading, this charge extends the algebra of conserved charges to a nonlinear supersymmetric one. We provide explicit expressions for intertwiners, charges and their algebra in the cases of two, three and four particles.
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References
F. Calogero, Solution of the one-dimensional N-body problems with quadratic and/or inversely quadratic pair potentials, J. Math. Phys. 12 (1971) 419 [Erratum ibid. 37 (1996) 3646] [INSPIRE].
M.A. Olshanetsky and A.M. Perelomov, Classical integrable finite dimensional systems related to Lie algebras, Phys. Rept. 71 (1981) 313 [INSPIRE].
M.A. Olshanetsky and A.M. Perelomov, Quantum integrable systems related to Lie algebras, Phys. Rept. 94 (1983) 313 [INSPIRE].
A.P. Polychronakos, Exchange operator formalism for integrable systems of particles, Phys. Rev. Lett. 69 (1992) 703 [hep-th/9202057] [INSPIRE].
A.P. Polychronakos, Physics and mathematics of Calogero particles, J. Phys. A 39 (2006) 12793 [hep-th/0607033] [INSPIRE].
S. Wojciechowski, Superintegrability of the Calogero-Moser system, Phys. Lett. A 95 (1983) 279.
V.B. Kuznetsov, Hidden symmetry of the quantum Calogero-Moser system, Phys. Lett. A 218 (1996) 212 [solv-int/9509001] [INSPIRE].
G. Barucchi and T. Regge, Conformal properties of a class of exactly solvable n body problems in space dimension one, J. Math. Phys. 18 (1977) 1149 [INSPIRE].
I.M. Krichever, Methods of algebraic geometry in the theory of nonlinear equations, Russ. Math. Surv. 32 (1977) 180.
J.L. Burchnall and T.W. Chaundy, Commutative ordinary differential operators, Proc. London Math. Soc. s2-21 (1923) 420.
Proc. Royal Soc. London A 118 (1928) 557.
H.F. Baker, Note on the foregoing paper, Proc. Royal Soc. London A 118 (1928) 584.
E.L. Ince, Ordinary differential equations, Dover (1956).
I.M. Krichever, Commutative rings of ordinary linear differential operators, Funct. Anal. Appl. 12 (1978) 175.
Yu.V. Brezhnev, Spectral/quadrature duality: Picard-Vessiot theory and finite-gap potentials, Contemporary Mathematics 563 (2012) 1 [arXiv:1011.1642].
O.A. Chalykh and A.P. Veselov, Commutative rings of partial differential operators and Lie algebras, Commun. Math. Phys. 126 (1990) 597.
O.A. Chalykh, K.L. Styrkas and A.P. Veselov, Algebraic integrability for the Schrödinger equation and finite reflection groups, Theor. Math. Phys. 94 (1993) 182.
O.A. Chalykh and A.P. Veselov, Integrability in the theory of Schrödinger operator and harmonic analysis, Commun. Math. Phys. 152 (1993) 29.
O.A. Chalykh, Additional integrals of the generalized quantum Calogero-Moser system, Theor. Math. Phys. 109 (1996) 1269.
Yu. Berest, Huygens’ principle and the bispectral problem, CRM Proceedings and Lecture Notes 14 (1998) 11.
O.A. Chalykh, M.V. Feigin and A.P. Veselov, Multidimensional Baker-Akhiezer functions and Huygens’ principle, Commun. Math. Phys. 206 (1999) 533 [math-ph/9903019].
M.V. Feigin and A.P. Veselov, Quasiinvariants of Coxeter groups and m-harmonic polynomials, Intern. Math. Res. Notices 10 (2002) 521 [math-ph/0105014].
P. Etingof and V. Ginzburg, On m-quasiinvariants of Coxeter groups, Mosc. Math. J. 2 (2002) 555 [math/0106175].
O.A. Chalykh, Algebro-geometric Schrödinger operators in many dimensions, Phil. Trans. R. Soc. A 366 (2008) 947.
C.F. Dunkl and Y. Xu, Orthogonal polynomials of several variables, Cambridge University Press (2001).
E.M. Opdam, Root systems and hypergeometric functions III, Comp. Math. 67 (1988) 21.
E.M. Opdam, Root systems and hypergeometric functions IV, Comp. Math. 67 (1988) 191.
G.J. Heckman, A remark on the Dunkl differential-difference operators, in Harmonic analysis on reductive groups, W. Barker and P. Sally eds., Progr. Math. 101 (1991) 181, Birkhäuser.
M. Feigin, O. Lechtenfeld and A. Polychronakos, The quantum angular Calogero-Moser model, JHEP 07 (2013) 162 [arXiv:1305.5841] [INSPIRE].
T. Hakobyan, A. Nersessian and V. Yeghikyan, Cuboctahedric Higgs oscillator from the Calogero model, J. Phys. A 42 (2009) 205206 [arXiv:0808.0430] [INSPIRE].
A. Fring, A note on the integrability of non-Hermitian extensions of Calogero-Moser-Sutherland models, Mod. Phys. Lett. A 21 (2006) 691 [hep-th/0511097] [INSPIRE].
A. Fring, PT-symmetric deformations of integrable models, Phil. Trans. Roy. Soc. Lond. A 371 (2013) 20120046 [arXiv:1204.2291] [INSPIRE].
F. Correa and M.S. Plyushchay, Spectral singularities in PT-symmetric periodic finite-gap systems, Phys. Rev. D 86 (2012) 085028 [arXiv:1208.4448] [INSPIRE].
C. Leiva and M.S. Plyushchay, Superconformal mechanics and nonlinear supersymmetry, JHEP 10 (2003) 069 [hep-th/0304257] [INSPIRE].
A. Anabalon and M.S. Plyushchay, Interaction via reduction and nonlinear superconformal symmetry, Phys. Lett. B 572 (2003) 202 [hep-th/0306210] [INSPIRE].
F. Correa, M.A. del Olmo and M.S. Plyushchay, On hidden broken nonlinear superconformal symmetry of conformal mechanics and nature of double nonlinear superconformal symmetry, Phys. Lett. B 628 (2005) 157 [hep-th/0508223] [INSPIRE].
F. Correa, V. Jakubsky and M.S. Plyushchay, Aharonov-Bohm effect on AdS 2 and nonlinear supersymmetry of reflectionless Pöschl-Teller system, Annals Phys. 324 (2009) 1078 [arXiv:0809.2854] [INSPIRE].
M.S. Plyushchay and L.-M. Nieto, Self-isospectrality, mirror symmetry and exotic nonlinear supersymmetry, Phys. Rev. D 82 (2010) 065022 [arXiv:1007.1962] [INSPIRE].
A. Arancibia, J.M. Guilarte and M.S. Plyushchay, Effect of scalings and translations on the supersymmetric quantum mechanical structure of soliton systems, Phys. Rev. D 87 (2013) 045009 [arXiv:1210.3666] [INSPIRE].
A.A. Andrianov and M.V. Ioffe, Nonlinear supersymmetric quantum mechanics: concepts and realizations, J. Phys. A 45 (2012) 503001 [arXiv:1207.6799] [INSPIRE].
F. Calogero, Solution of a three-body problem in one-dimension, J. Math. Phys. 10 (1969) 2191 [INSPIRE].
F. Calogero and C. Marchioro, Exact solution of a one-dimensional three-body scattering problem with two-body and/or three-body inverse-square potentials, J. Math. Phys. 15 (1974) 1425 [INSPIRE].
C.F. Dunkl, Some orthogonal polynomials in four variables, SIGMA 4 (2008) 82 [arXiv:0812.0063].
S. Krivonos, O. Lechtenfeld and K. Polovnikov, N = 4 superconformal n-particle mechanics via superspace, Nucl. Phys. B 817 (2009) 265 [arXiv:0812.5062] [INSPIRE].
S. Krivonos and O. Lechtenfeld, Many-particle mechanics with D(2, 1; α) superconformal symmetry, JHEP 02 (2011) 042 [arXiv:1012.4639] [INSPIRE].
F. Correa, V. Jakubský, L.-M. Nieto and M.S. Plyushchay, Self-isospectrality, special supersymmetry and their effect on the band structure, Phys. Rev. Lett. 101 (2008) 030403 [arXiv:0801.1671] [INSPIRE].
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Correa, F., Lechtenfeld, O. & Plyushchay, M. Nonlinear supersymmetry in the quantum Calogero model. J. High Energ. Phys. 2014, 151 (2014). https://doi.org/10.1007/JHEP04(2014)151
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DOI: https://doi.org/10.1007/JHEP04(2014)151