Spectral Properties of Unbounded Jacobi Matrices with Almost Monotonic Weights

We present a unified framework to identify spectra of Jacobi matrices. We give applications of the long-standing problem of Chihara (Mt J Math 21(1):121-137, 1991, J Comput Appl Math 153(1-2):535-536, 2003) concerning one-quarter class of orthogonal polynomials, to the conjecture posed by Roehner and Valent (SIAM J Appl Math 42(5):1020-1046, 1982) concerning continuous spectra of generators of birth and death processes, and to spectral properties of operators studied by Janas and MoszyA"ki (Integral Equ Oper Theory 43(4):397-416, 2002) and Pedersen (Proc Am Math Soc 130(8):2369-2376, 2002).