  We study spectral properties of the quantum Korteweg-de Vries hierarchy
defined by Buryak and Rossi. We prove that eigenvalues to first order in the
dispersion parameter are given by shifted symmetric functions. The proof is
based on the boson-fermion correspondence and an analysis of quartic
expressions in fermions. As an application, we obtain a new closed evaluation
of certain double Hodge integrals on the moduli spaces of curves. Finally, we
provide an explicit formula for the eigenvectors to first order in the
dispersion parameter. In particular, we show that its Schur expansion is
supported on partitions for which the Hamming distance is minimal.
