Spectrum density of the first zone of linear coupled oscillators Part I
The aim of this work is to estimate the density of the electron spectrum of objects with a dispersion bond, such as liquid helium, for example. The interest in its electronic properties is caused by the recently discovered dynamic polarization in the second sound wave, which made the study of its zone structure relevant. In this paper, the objects under study are modeled after born by a linear chain of coupled oscillators. The dispersion bond is considered on the basis of the quasi-classical London approximation, which takes into account the correlations of the oscillating dipole moments of atoms. The first part of the work is preparatory. A brief review of the new experimental data for helium II is given, and the problem statement is formulated. On simple examples (lines of two and three coupled oscillators) the research methodology is worked out. This section is of independent importance for rapid estimates of the density of the electron spectrum in dispersion coupling, and identifies trends in the change of spectrum parameters with an increase in the number of structural units. It is obtained, in particular, that the band width increases and the specific binding energy decreases slightly. In the second part, a long chain of coupled oscillators is investigated on the basis of the developed techniques. The density of the electron spectrum is calculated both for specific chains and in the thermodynamic limit. The results obtained in this work are planned to be used in the future for the development of dynamic polarization models for helium II