Asymptotics of eigenvalues and eigenfunctions of the Dirichlet problem on a thin spacial net with small knots.

We homogenize a thin net of quantum waveguides with small thickening nodes (the Dirchlet problem for the Laplace operator). In contrast to the boundary-value problem with the Neumann condition the lower but remote from the coordinate origin range of the spectrum of the Dirichlet problem is characterized by localization of the corresponding eigenfunctions near the angular junction zones and the ligaments themselves in dependence on distribution of eigenvalues of eigenvalues from the discrete spectra (necessarily nonempty) of the model problems about joints of semi-infinite cylindrical quantum waveguides. The behaviour of eigenvalues and eigenfunctions in the mid-frequency range of the spectrum of the net essentially depend on the threshold resonance phenomenon in the above-mentioned joints and the relation between the small parameters, the period of distribution of the nodes and their diameter which is of the same order but larger that diameter of the ligaments. We discuss concrete situations (rectangular and circular cross-sections) and formulate open questions.