Elliptic finite-band potentials of a non-self-adjoint Dirac operator
Elliptic finite-band potentials of a non-self-adjoint Dirac operator
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DOI：
10.1016/j.aim.2023.109188
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发表时间：
2022-10
影响因子：
1.7
通讯作者：
G. Biondini;Xu‐Dan Luo;Jeffrey Oregero;A. Tovbis
中科院分区：
文献类型：
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作者：
G. Biondini;Xu‐Dan Luo;Jeffrey Oregero;A. Tovbis
对于非自伴Dirac算子L，我们给出了一个显式的两参数有限带Jacobi椭圆势族q ∈ Adn（x; m），其中m∈（0，1）且A可以取正而不失一般性，它连接了平面波（m= 0）和sech势（m= 1）的两个著名极限情形.我们证明了，如果A∈ N，则谱由R加上2A个在i R上的施瓦茨对称段（带）组成.这种表征的频谱是通过有关的周期和反周期的狄拉克算子的特征值问题，相应的特征值问题的三对角算子作用于傅立叶系数在加权希尔伯特空间，并适当的连接问题Heun方程。相反，如果A <$N，则L的谱由C中的无限多个带组成。当A∈ N时，相应的势函数为所有与聚焦非线性薛定谔方程族相关的正流和负流（包括修正的Korteweg-deVries方程和sine-Gordon方程）生成有限亏格解.
We present an explicit two-parameter family of finite-band Jacobi elliptic potentials given by q≡ A dn (x; m), where m∈(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∈ 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∉ N, then the spectrum of L consists of infinitely many bands in C. When A∈ N, the corresponding potentials generate finite-genus solutions for all the positive and negative flows associated with the focusing nonlinear Schrödinger hierarchy, including the modified Korteweg-deVries equation and the sine-Gordon equation.