  An amended proof of Pitaevskii-Stringari's theorem is given in mathematically
coherent manner without resorting to the Bogoliubov approximation. This
approach is based on the orthodox quantum field theory which rigorously
maintains the canonical commutation relations. Moreover, we make a sound
argument by taking the thermodynamic limit which the authors of the original
papers did not refer to. We conclude that there is no Bose Einstein
Condensation (BEC) as a phase transition based on the Spontaneous Symmetry
Breaking (SSB) of the global $U(1)$ gauge symmetry in flat one dimensional
geometry even absolute zero temperature.
