We consider a family X^{(n)}, n \in \bbN_+, of continuous-time
nearest-neighbor random walks on the one dimensional lattice Z. We reduce the
spectral analysis of the Markov generator of X^{(n)} with Dirichlet conditions
outside (0,n) to the analogous problem for a suitable generalized second order
differential operator -D_{m_n} D_x, with Dirichlet conditions outside a given
interval. If the measures dm_n weakly converge to some measure dm_*, we prove a
limit theorem for the eigenvalues and eigenfunctions of -D_{m_n}D_x to the
corresponding spectral quantities of -D_{m_*} D_x. As second result, we prove
the Dirichlet-Neumann bracketing for the operators -D_m D_x and, as a
consequence, we establish lower and upper bounds for the asymptotic annealed
eigenvalue counting functions in the case that m is a self--similar stochastic
process. Finally, we apply the above results to investigate the spectral
structure of some classes of subdiffusive random trap and barrier models coming
from one-dimensional physics.Comment: Published on EJP. the Dirichlet-Neumann bracketing has been
corrected. shorter, improved version. 35 page
