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Homogeneous one-dimensional Bose-Einstein Condensate in the Bogoliubov's Regime

We analyze the corrections caused by finite size effects upon the ground state properties of a homogeneous one-dimensional Bose-Einstein condensate. We assume from the very beginning that the Bogoliubov's formalism is valid and consequently we show that in order to obtain a well defined ground state properties, finite size effects of the system must be taken into account. Indeed, the formalism described in the present work allows to recover the usual properties related to the ground state of a homogeneous one-dimensional Bose-Einstein condensate but corrected by finite size effects of the system. Finally, this scenario allows us to analyze the sensitivity of the system when the Bogoliubov's regime is valid and when finite size effects are present. These facts open the possibility to apply these ideas to more realistic scenarios, e.g., low-dimensional trapped Bose-Einstein condensates.