Semiclassical WKB Problem for the non-self-adjoint Dirac operator
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| Format: | Preprint |
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2026
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| _version_ | 1866910047317000192 |
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| author | Fujiié, Setsuro Hatzizisis, Nicholas Kamvissis, Spyridon |
| author_facet | Fujiié, Setsuro Hatzizisis, Nicholas Kamvissis, Spyridon |
| contents | We review some recent rigorous results on the semiclassical behavior ($ε\downarrow0$) of the scattering data of a non-self-adjoint Dirac operator with potential $A\exp\{iS/ε\}$ where both $A$ and $S$ are differentiable functions tending to constants as $x \to \pm \infty$. We have either employed the so-called exact WKB method, or the older WKB theory of Olver. Our analysis is motivated by the need to understand the semiclassical behaviour of the focusing cubic NLS equation with initial data $A\exp\{iS/ε\}$, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables us to obtain the solution of the NLS equation via inverse scattering theory. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_09204 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Semiclassical WKB Problem for the non-self-adjoint Dirac operator Fujiié, Setsuro Hatzizisis, Nicholas Kamvissis, Spyridon Spectral Theory Analysis of PDEs We review some recent rigorous results on the semiclassical behavior ($ε\downarrow0$) of the scattering data of a non-self-adjoint Dirac operator with potential $A\exp\{iS/ε\}$ where both $A$ and $S$ are differentiable functions tending to constants as $x \to \pm \infty$. We have either employed the so-called exact WKB method, or the older WKB theory of Olver. Our analysis is motivated by the need to understand the semiclassical behaviour of the focusing cubic NLS equation with initial data $A\exp\{iS/ε\}$, in view of the well-known fact discovered by Zakharov and Shabat that the spectral analysis of the Dirac operator enables us to obtain the solution of the NLS equation via inverse scattering theory. |
| title | Semiclassical WKB Problem for the non-self-adjoint Dirac operator |
| topic | Spectral Theory Analysis of PDEs |
| url | https://arxiv.org/abs/2603.09204 |