Spectral Properties of Hamiltonians of Charged Systems in a Homogeneous Magnetic Field: I. General Characteristic of the Spectrum

We study the spectrum of Hamiltonians of charged multiparticle systems in a homogeneous magnetic field with a fixed sum $P_{\Sigma }$ of the pseudomomentum components and without it. We prove that if $P_{\Sigma }$ is fixed, then the spectrum of Hamiltonians is independent of the value of $P_{\Sigma }$, while the spectrum without fixation of $P_{\Sigma }$ coincides with the spectrum with fixation and differs from the latter only by some additional infinite degeneration (this is a principal difference between problems with a homogeneous magnetic field and problems without any field in which the absence of any fixation of the total angular momentum results in “covering” the spectrum of the relative motion by a continuous spectrum). We find the continuous spectrum of the Hamiltonians and characterize the spectrum of Hamiltonians of two-cluster mutually noninteracting systems obtained by decomposing the original system in the state with a fixed value of $P_{\Sigma }$. The last result is necessary for the study of the purely point spectrum.