Sekiguchi-Debiard operators at infinity

TY - JOUR

T1 - Sekiguchi-Debiard operators at infinity

AU - Nazarov, Maxim L

AU - Sklyanin FRS, Evgeny

PY - 2013/10

Y1 - 2013/10

N2 - 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.

AB - 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.

U2 - 10.1007/s00220-013-1821-z

DO - 10.1007/s00220-013-1821-z

M3 - Article

SN - 0010-3616

VL - Early Online

SP - n/a

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - n/a

ER -