Sekiguchi-Debiard operators at infinity

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.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of symmetric functions and integrable many‑body systems: while the Sekiguchi‑Debiard (SD) operators provide a commuting family of differential operators whose eigenfunctions are Jack symmetric polynomials in a finite set of variables, no satisfactory infinite‑variable analogue existed. The authors construct such an infinite‑variable SD operators by first renormalising the finite‑N operators and then taking the limit (N\to\infty). The renormalisation removes the divergent leading terms that grow with (N) and yields operators that act purely on the power‑sum variables \