Summary: It is proved that the first eigenfunction of the mixed boundary value problem for Laplacian in a thin domain \(\Omega _h\) is localized either at the whole lateral surface \(\Gamma _h\) of the domain, or at a point of \(\Gamma _h\), while the eigenfunction decays exponentially inside \(\Omega _h\). Other effects, attributed to the high-frequency range of the spectrum, are discussed for eigenfunctions of the mixed boundary value and Neumann problems, too.