We present an explicit two-parameter family of finite-band Jacobi elliptic potentials given by $q\equiv A\dn(x;m)$, where $m\in(0,1)$ and $A$ can be taken to be positive without loss of generality, for a non-self-adjoint Dirac operator $L$, which connects two well-known limiting cases of the plane wave ($m=0$) and of the $\sech$ potential ($m=1$). We show that, if $A\in\N$, then the spectrum consists of $\R$ plus $2A$ Schwarz symmetric segments (bands) on $\i\R$. This 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. Conversely, if $A\not\in\N$, then the spectrum of $L$ consists of infinitely many bands in $\C$. When $A\in\N$, the corresponding potentials generate finite-genus solutions for all the positive and negative flows associated with the focusing nonlinear Schr\"odinger hierarchy,  including the modified Korteweg-deVries equation and the sine-Gordon equation. 

　　此论文在审稿结束后直接被录用，无需任何修改。  

　　Publication:  

　　Advances in Mathematics, Volume 429, 15 September 2023, 109188  

　　http://dx.doi.org/10.1016/j.aim.2023.109188  

　　Author:  

　　Gino Biondini  

　　Department of Mathematics, State University of New York at Buffalo, Buffalo, NY 14260, United States of America  

　　Xu-Dan Luo  

　　Key Laboratory of Mathematics Mechanization, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China  

　　Email: lxd@amss.ac.cn  

　　Jeffrey Oregero  

　　Department of Mathematics, University of Central Florida, Orlando, FL 32816, United States of America  

　　Alexander Tovbis  

　　Department of Mathematics, University of Central Florida, Orlando, FL 32816, United States of America