  We calculate the breathing mode frequency $\omega$ in a one-dimensional Bose
gas confined to a harmonic trap of frequency $\omega_z$. We predict Exciting
temporal oscillations of the density distribution is a high-precision method
for probing ultracold trapped atomic gases. Interaction effects in their
many-body dynamics are particularly puzzling and counter-intuitive in one
spatial dimension (1D) due to enhanced quantum correlations. We consider 1D
quantum Bose gas in a parabolic trap at zero temperature and explain,
analytically and numerically, how oscillation frequency depends on the number
of particles, their repulsion and the trap strength. We identify the frequency
with the energy difference between the ground state and a particular excited
state. This way we avoided resolving the dynamical evolution of the system,
simplifying the problem immensely. We find an excellent quantitative agreement
of our results with the data from the Innsbruck experiment [Science 325, 1224
(2009)].
