Spectral analysis of 1D nearest-neighbor random walks and applications to subdiffusive trap and barrier models

Abstract

We consider a   sequence  $X^{(n)}$, $n \geq 1 $,   of continuous-time nearest-neighbor random walks on the one dimensional lattice $\mathbb{Z}$.  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 giveninterval. If  the measures $dm_n$ weakly converge to some measure $dm_\infty$,  we prove a limit theorem for the eigenvalues and eigenfunctions of $-D_{m_n}D_x$ to the corresponding spectral quantities of $-D_{m_\infty}  D_x$.  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.