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We present an explicit two-parameter family of finite-band Jacobi ellipticpotentials for a non-self-adjoint Dirac operator which connects two previouslyknown limiting cases in which the elliptic parameter is zero or one. A fullcharacterization of the spectrum is obtained by relating the periodic andantiperiodic eigenvalue problems for the Dirac operator to correspondingeigenvalue problems for tridiagonal operators acting on Fourier coefficients ina weighted Hilbert space and to appropriate connection problems for Heun'sequation. In turn, these problems are related to four non-self-adjointunbounded tridiagonal operators, all of which nonetheless have only realeigenvalues. For certain parameter values, the corresponding ellipticpotentials generate finite-genus solutions for all the positive and negativeflows of the focusing nonlinear Schr\"odinger hierarchy.

Elliptic finite-band potentials of a non-self-adjoint Dirac operator