Elliptic finite-band potentials of a non-self-adjoint Dirac operator
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| Main Authors: | , , , |
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| Format: | Preprint |
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2022
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| _version_ | 1866913574574620672 |
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| author | Biondini, Gino Luo, Xu-Dan Oregero, Jeffrey Tovbis, Alexander |
| author_facet | Biondini, Gino Luo, Xu-Dan Oregero, Jeffrey Tovbis, Alexander |
| contents | 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 Schrödinger hierarchy. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_07303 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Elliptic finite-band potentials of a non-self-adjoint Dirac operator Biondini, Gino Luo, Xu-Dan Oregero, Jeffrey Tovbis, Alexander Spectral Theory Analysis of PDEs Exactly Solvable and Integrable Systems 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 Schrödinger hierarchy. |
| title | Elliptic finite-band potentials of a non-self-adjoint Dirac operator |
| topic | Spectral Theory Analysis of PDEs Exactly Solvable and Integrable Systems |
| url | https://arxiv.org/abs/2210.07303 |