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

We present an explicit two-parameter family of finite-band Jacobi elliptic potentials given by q≡Adn(x;m), where m∈(0,1) and A can be taken to positive without loss generality, for a non-self-adjoint Dirac operator L, which connects two well-known limiting cases the plane wave (m=0) sech potential (m=1). show that, if A∈N, then spectrum consists R plus 2A Schwarz symmetric segments (bands) on iR. This characterization is obtained relating periodic antiperiodic eigenvalue problems corresponding tridiagonal operators acting Fourier coefficients in weighted Hilbert space, appropriate connection Heun's equation. Conversely, A∉N, L infinitely many bands C. When generate finite-genus solutions all negative flows associated with focusing nonlinear Schrödinger hierarchy, including modified Korteweg-deVries equation sine-Gordon