Iakovenko A. Asymptotic analysis of spectral problems on small-periodic networks.

The asymptotic expansions for the low-frequency spectrum of the spectral problem on the small-periodic network are constructed . An estimate, which depends on s-th eigenvalue of the homogenized problem, being justification of the obtained asymptotics for s-th eigenvalue and s-th eigenfunction of problem on the network, is found. Exact complex solutions of the problems on the small-periodic network, the spectrum of which is dense (in square scale for large N, where N denotes the number of network cells), are obtained. These solutions coincide with the asymptotics of the low-frequency spectrum. The influence of arithmetic properties of network size on the asymptotic behavior of the spectral problem on the small-periodic network solutions is researched.