  We present an explicit two-parameter family of finite-band Jacobi elliptic
potentials for a non-self-adjoint Dirac operator which connects two previously
known limiting cases in which the elliptic parameter is zero or one. A full
characterization of the spectrum is obtained by relating the periodic and
antiperiodic eigenvalue problems for the Dirac operator to corresponding
eigenvalue problems for tridiagonal operators acting on Fourier coefficients in
a weighted Hilbert space and to appropriate connection problems for Heun's
equation. In turn, these problems are related to four non-self-adjoint
unbounded tridiagonal operators, all of which nonetheless have only real
eigenvalues. For certain parameter values, the corresponding elliptic
potentials generate finite-genus solutions for all the positive and negative
flows of the focusing nonlinear Schr\"odinger hierarchy.
