Decay bounds on eigenfunctions and the singular spectrum of unbounded Jacobi matrices

Bounds on the exponential decay of generalized eigenfunctions of bounded and unbounded selfadjoint Jacobi matrices in l2(N) are established. Two cases are considered separately and lead to different results: (i) the case in which the spectral parameter lies in a general gap of the spectrum of the Jacobi matrix and (ii) the case of a lower semibounded Jacobi matrix with values of the spectral parameter below the spectrum. It is demonstrated by examples that both results are sharp. We apply these results to obtain a “many barriers-type” criterion for the existence of square-summable generalized eigenfunctions of an unbounded Jacobi matrix at almost every value of the spectral parameter in suitable open sets. In particular, this leads to examples of unbounded Jacobi matrices with a spectral mobility edge, i.e. a transition from purely absolutely continuous spectrum to dense pure point spectrum.