## Sekiguchi-Debiard operators at infinity

Research output: Contribution to journalArticle

## Department/unit(s)

### Publication details

Journal Communications in Mathematical Physics E-pub ahead of print - 18 Oct 2013 Published (current) - Oct 2013 n/a Early Online 20 n/a 18/10/13 English

### Abstract

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.

## Discover related content

Find related publications, people, projects, datasets and more using interactive charts.