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We discuss the Hermitian treatment of the Dyson-type boson expansion theory. We show that the basic assumption of the conventional treatment does not hold in general and that the method is only approximately valid. The exception is the case in which the multi-phonon states are mutually orthogonal, which is not expected in realistic nuclear systems. We also show that the approximation is of the same order as that of the truncation of the expansion usually done in the Hermitian-type boson expansion theory. §1. IntroductionThe boson expansion theory (BET) 1) -3) has been widely used as a many-body technique to analyze anharmonic effects in nuclear collective motion beyond the random phase approximation. This theory is a systematic method to transform the eigenvalue problem in fermion space into that in boson space. With this method, an arbitrary fermion pair operator is expressed as an expansion of the boson operators, which enables us to calculate the wave functions more easily.However, the BET is not a complete theory and contains some problems to be solved. First, one must apply the theory to the limited subspace of the original fermion space, which we call the truncated fermion subspace. Otherwise, the boson space on which the transcribed boson operators act contains an unphysical part having no one-to-one correspondence to the original fermion space. One cannot give the exact prescription how the limited fermion subspace should be selected in general. Second, for reasons closely related to the above problem, one must adopt the so-called closed algebra approximation. 1) The validity of this approximation is not clear, although it is indispensable for the BET. Third, the convergence of the expansion for the non-collective mode is not assured. * )Concerning the last point, the Dyson-type (D-type) BET 6) -11) has a distinctive feature compared with the others such as the Holstein-Primakoff-type (HP-type) BET. 12) -17) In the D-type BET, the fermion space is mapped to the physical boson subspace by the special biunitary transformation so that a transformed pair operator * ) To solve these problems we proposed an alternative mapping theory in which only the collective modes were mapped into the boson modes. However, the purpose of the present paper is not to discuss this issue. The interested reader may refer to Refs. 4) and 5).

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