Research output: Contribution to journal › Article

Journal | Communications in Mathematical Physics |
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Date | E-pub ahead of print - 18 Oct 2013 |

Date | Published (current) - Oct 2013 |

Issue number | n/a |

Volume | Early Online |

Number of pages | 20 |

Pages (from-to) | n/a |

Early online date | 18/10/13 |

Original language | English |

We construct a family of pairwise commuting operators such that the Jack symmetric functions of infinitely many variables $x_1,x_2,...$ are their eigenfunctions. These operators are defined as limits at $N\to\infty$ of renormalised Sekiguchi-Debiard operators acting on symmetric polynomials in the variables $x_1,...,x_N$. They are differential operators in terms of the power sum variables $p_n=x_1^n+x_2^n+...$ and we compute their symbols by using the Jack reproducing kernel. Our result yields a hierarchy of commuting Hamiltonians for the quantum Calogero-Sutherland model with infinite number of bosonic particles in terms of the collective variables of the model. Our result also yields explicit shift operators for the Jack symmetric functions.

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