  We characterize the low-lying excitations of a quantum vortex in a
quasi-two-dimensional Bose-Einstein condensate (BEC) using the standard
definition of the density of states (DOS) and a modified version that is
sensitive to complementary aspects of the excitation's spectrum. The latter
proves to be particularly relevant to studying the polaronic state realized
when an impurity is embedded in a quantum vortex. We establish that the
impurity becomes sensitive to the transversal fluctuations of the vortex, via
its remnant kelvon mode, and to the phase fluctuations of the BEC
Nambu-Goldstone mode. The presence of the vortex yields an anomalous excitation
spectrum with a finite energy gap and non-linear DOS at low energies. We find
that the high sensitivity of the kelvon mode to external potentials provides a
channel of quantum-level control over impurities trapped in a vortex. This
extra control channel may be of practical use for the proposal of using
vortex-trapped impurities as qubit units for quantum information processing.
