This paper is a part of the project suggested by A. M. Vershik and the author and aimed to combine known results on the representation theory of the finite and infinite symmetric groups and a circle of results related to the quantum inverse scattering method and Bethe ansatz. In this first part, we consider the simplest spectral properties of a distinguished operator in the group algebra of the symmetric group, which we call the periodic Coxeter Laplacian. Namely, we study this operator in the two-row representations of symmetric groups and in the "ferromagnetic" asymptotic mode. Bibliography: 11 titles.