  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.
