Some spectral properties of the family of a 3 × 3 operator matrices in statistical physics

This paper investigates the spectral properties of an operator matrix that describes a one-dimensional lattice system within statistical physics. The model under consideration is characterized by a variable number of three distinct types of particles, which are treated as bosons. System interactions that lead to the creation and annihilation of particles are described using corresponding operators. To analyze this complex initial system, an auxiliary "channel operator" is introduced. It is proven that the spectrum of this channel operator entirely determines the essential spectrum of the original operator matrix. During the study, a Faddeev-type matrix equation is derived and used to establish the validity of the Birman-Schwinger principle for this model. The results obtained are a crucial step toward determining whether the operator has a finite or infinite number of isolated eigenvalues, a question of fundamental importance for understanding its complete spectral structure.

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