Spectral properties of Hamiltonians with magnetic field under fixation of pseudomomentum. I

It is established that the energy operator of an $n$-particle neutral system in a homogeneous magnetic field with a fixed pseudomomentum can be written as some operator in the space of relative motion. For this operator, the HVZ-theorem for the localization of the essential spectrum is proved, accounting for the permutational symmetry for any $n\geq 2$. For $n=2$, the conditions of finiteness and infinity of the discrete spectrum and spectral asymptotic behavior are found. The result can be applied, in particular, to the Hamiltonian of the hydrogen atom in the homogeneous magnetic field.