  We argue that quantum fluctuations of the phase of the order parameter may
strongly affect the electron density of states (DOS) in ultrathin
superconducting wires. We demonstrate that the effect of such fluctuations is
equivalent to that of a quantum dissipative environment formed by sound-like
plasma modes propagating along the wire. We derive a non-perturbative
expression for the local electron DOS in superconducting nanowires which fully
accounts for quantum phase fluctuations. At any non-zero temperature these
fluctuations smear out the square-root singularity in DOS near the
superconducting gap and generate quasiparticle states at subgap energies.
Furthermore, at sufficiently large values of the wire impedance this
singularity is suppressed down to $T=0$ in which case DOS tends to zero at
subgap energies and exhibits the power-law behavior above the gap. Our
predictions can be directly tested in tunneling experiments with
superconducting nanowires.
