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Nonlinear corrections in the quantization of a weakly nonideal Bose gas at zero temperature

In the present paper, quantization of a weakly nonideal Bose gas at zero temperature along the lines of the well-known Bogolyubov approach is performed. The analysis presented in this paper is based, in addition to the steps of the original Bogolyubov approach, on the use of nonoscillation modes (which are also solutions of the linearized Heisenberg equation) for recovering the canonical commutation relations in the linear approximation, as well as on the calculation of the first nonlinear correction to the solution of the linearized Heisenberg equation which satisfies the canonical commutation relations at the next order. It is shown that, at least in the case of free quasi-particles, consideration of the nonlinear correction automatically solves the problem of nonconserved particle number, which is inherent to the original approach.