Spectral Theory of the Difference Equation f(n + 1) + f(n − 1) = [E − φ(n)]f(n)

In this work, the spectrum of the second‐order difference equation $$\text{f(n+1)+f(n−1)=[E−φ(n)]f(n)}$$ in the l² Hilbert space is studied for the case in which the limit of the sequence ${\mathrm{\{\phi (n)\}}}_{\text{n=1}}^{\infty }$ exists. By means of a simple representation the problem is transferred to one about the spectrum of an abstract operator in a separable Hilbert space. This operator T has a form analogous to the Schrödinger operator, namely T = T₀ + A, where T₀ is self‐adjoint with a purely continuous spectrum but bounded, while A depends on the sequence {φ(n)}. In fact, A is of Hilbert‐Schmidt type for any {φ(n)} in l², and of trace class if the series ${\sum }_{\text{n=1}}^{\infty }\mathrm{|\phi (n)|}$ converges. Sufficient conditions for the existence of a discrete spectrum and more generally, of proper values, are found. Using the theory of the wave operators ${\Omega }_{\pm }\mathrm{=s-}\lim {\displaystyle {lim}_{\mathrm{t\to \pm \infty }}}\exp \mathrm{\hspace{0.17em}(iTt)}$ range exp (−iT₀t), results on the existence of a mixed spectrum are obtained.