Spectra of Jacobi matrices, differential equations on the circle, and the su (1,1) Lie algebra

A family of differential operators on the circle is shown to be isospectral to a certain family of bilaterally infinite Jacobi matrices. The spectral properties of the differential operators are then used to explain a previously noted isospectral deformation of the Jacobi matrices. Differential operators on the circle are used to provide realizations of principle and complementary series representations of $su(1,1)$.