On the spectral problem of the quantum KdV hierarchy

TL;DR

This paper investigates the spectral properties of the quantum KdV hierarchy, establishing key symmetries, eigenvectors, and connections to symmetric group characters, advancing understanding of quantum integrable systems.

Contribution

It proves symmetry and non-degeneracy of quantum KdV Hamiltonians, constructs eigenvectors, and links spectral problems to symmetric group characters.

Findings

Proved symmetry and non-degeneracy of quantum KdV Hamiltonians

Constructed a complete set of common eigenvectors

Established vanishing identities for symmetric group characters

Abstract

The spectral problem for the quantum dispersionless Korteweg-de Vries (KdV) hierarchy, aka the quantum Hopf hierarchy, is solved by Dubrovin. In this article, following Dubrovin, we study Buryak-Rossi's quantum KdV hierarchy. In particular, we prove a symmetry property and a non-degeneracy property for the quantum KdV Hamiltonians. On the basis of this we construct a complete set of common eigenvectors. The analysis underlying this spectral problem implies certain vanishing identities for combinations of characters of the symmetric group. We also comment on the geometry of the spectral curves of the quantum KdV hierarchy and we give a representation of the quantum dispersionless KdV Hamiltonians in terms of multiplication operators in the class algebra of the symmetric group.