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Through exact numerical solutions we show Bose-Einstein condensation (BEC)for the one-dimensional (1D) bosons with repulsive short-range interactions atzero temperature by taking a particular large size limit. Following thePenrose-Onsager criterion of BEC, we define condensate fraction by the fractionof the largest eigenvalue of the one-particle reduced density matrix. % We showthe finite-size scaling behavior such that condensate fraction is given by ascaling function of one-variable: interaction parameter multiplied by a powerof particle number. Condensate fraction is nonzero and constant for any largevalue of particle number or system size, if the interaction parameter isproportional to the negative power of particle number. %Here the interactionparameter is defined by the coupling constant of the delta-function potentialsdevided by the density. %With the scaling behavior we derive variousthemodynamic limits where condensate fraction is constant for any large systemsize; for instance, it is the case even in the system of a finite particlenumber.

Finite-size scaling behavior of Bose-Einstein condensation in the 1D Bose gas