Existence of deep-water limits for higher quantum ILW Hamiltonians

Establish whether the higher Hamiltonians of the quantum Intermediate Long Wave hierarchy admit well-defined limits under the deep-water specialization \(\gamma=\sqrt{\mu}\) as \(\mu\to+\infty\), and determine how those limits relate to the higher Hamiltonians of the Nazarov–Sklyanin quantum Benjamin–Ono hierarchy.

Background

The paper identifies the deep-water limit of the first quantum ILW Hamiltonian with the first nontrivial Hamiltonian of the Nazarov–Sklyanin quantum Benjamin–Ono hierarchy, after the relevant parameter identifications. It then raises the unresolved issue of whether analogous limits exist for all higher quantum ILW Hamiltonians and, if they do, how the resulting operators correspond to the quantum Benjamin–Ono Hamiltonians.

References

It is natural to study whether the limits of the higher Hamiltonians $G_d$ of the quantum ILW hierarchy exist, and, if yes, how to relate them to the higher Hamiltonians of the quantum BO hierarchy of Nazarov and Sklyanin.

— Quantum hierarchies of the ILW and BO equations and the DR cycles on the moduli spaces of curves  (2610.01841 - Buryak et al., 1 Oct 2026) in Section 1, Introduction

However, for $d\ge 3$ it can happen that the radius of convergence of~$h_{d,j,\lambda,\rho}(\mu)$ is zero, and in this case we don't know a reasonable way to take the limit when $\mu\to+\infty$.

— Quantum hierarchies of the ILW and BO equations and the DR cycles on the moduli spaces of curves  (2610.01841 - Buryak et al., 1 Oct 2026) in Section 2, subsection “Conjecture”

We see that for $d=1$ the formal power series $h_{1,j,\lambda,\rho}(\mu)$ is the power series expansion at $0$ of a function from $(\mu,e{\mu})$, and moreover this function has a limit when $\mu\inR$ and $\mu\to+\infty$. However, for $d\ge 3$ it can happen that the radius of convergence of~$h_{d,j,\lambda,\rho}(\mu)$ is zero, and in this case we don't know a reasonable way to take the limit when $\mu\to+\infty$.

— Quantum hierarchies of the ILW and BO equations and the DR cycles on the moduli spaces of curves  (2610.01841 - Buryak et al., 1 Oct 2026) in Section 2, subsection “Conjecture”