Abstract: We consider spectral problems for the Laplace operator in 3D rod structures with a small cross section of diameter (Formula presented.), (Formula presented.) being a positive parameter. The boundary conditions are Dirichlet (Neumann, respectively) on the bases of this structure, and Neumann on the lateral boundary. As (Formula presented.), we show the convergence of the spectrum with conservation of the multiplicity toward that of a 1D spectral model with Dirichlet (Neumann, respectively) boundary conditions. This 1D model may arise in diffusion or vibration models of nonhomogeneous media with different physical characteristics and it takes into account the geometry of the 3D domain. We deal with the low frequencies and the approach to eigenfunctions in the suitable Sobolev spaces is also outlined.

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