We consider a Dirichlet spectral problem for a second order differential operator, with piecewise constant coefficients, in a domain Omega(epsilon) in the plane R-2. Here Omega(epsilon) is Omega boolean OR omega(epsilon) boolean OR Gamma, where Omega is a fixed bounded domain with boundary Gamma, omega(epsilon) is a curvilinear band of width O(epsilon), and Gamma = (Omega) over bar boolean AND (omega) over bar (epsilon). The density and stiffness constants are of order epsilon(-m-t) and epsilon(-t) respectively in this band, while they are of order 1 in Omega; t >= 1, m > 2, and s is a small positive parameter. We address the asymptotic behavior, as epsilon -> 0, for the eigenvalues and the corresponding eigenfunctions. In particular, we show certain localization effects for eigenfunctions associated with low frequencies. This is deeply involved with the extrema of the curvature of Gamma. (C) 2020 Elsevier Inc. All rights reserved.