Monodromy of generalized Lame equations with Darboux-Treibich-Verdier potentials: A universal law
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866910396131049472 |
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| author | Chen, Zhijie Lin, Chang-Shou |
| author_facet | Chen, Zhijie Lin, Chang-Shou |
| contents | The Darboux-Treibich-Verdier (DTV) potential $\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2};τ)$ is well-known as doubly-periodic solutions of the stationary KdV hierarchy (Treibich-Verdier, Duke Math. J. {\bf 68} (1992), 217-236). In this paper, we study the generalized Lamé equation with the DTV potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2};τ)+B\bigg] y(z),\quad n_{k}\in \mathbb{N} \end{equation*} from the monodromy aspect. We prove that the map from $(τ, B)$ to the monodromy data $(r,s)$ satisfies a surprising universal law $dτ\wedge dB\equiv8π^2 dr\wedge ds.$ Our proof applies Panlevé VI equation and modular forms. We also give applications to the algebraic multiplicity of (anti)periodic eigenvalues for the associated Hill operator. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01879 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Monodromy of generalized Lame equations with Darboux-Treibich-Verdier potentials: A universal law Chen, Zhijie Lin, Chang-Shou Classical Analysis and ODEs Mathematical Physics The Darboux-Treibich-Verdier (DTV) potential $\sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2};τ)$ is well-known as doubly-periodic solutions of the stationary KdV hierarchy (Treibich-Verdier, Duke Math. J. {\bf 68} (1992), 217-236). In this paper, we study the generalized Lamé equation with the DTV potential \begin{equation*} y^{\prime \prime }(z)=\bigg[ \sum_{k=0}^{3}n_{k}(n_{k}+1)\wp(z+\tfrac{ ω_{k}}{2};τ)+B\bigg] y(z),\quad n_{k}\in \mathbb{N} \end{equation*} from the monodromy aspect. We prove that the map from $(τ, B)$ to the monodromy data $(r,s)$ satisfies a surprising universal law $dτ\wedge dB\equiv8π^2 dr\wedge ds.$ Our proof applies Panlevé VI equation and modular forms. We also give applications to the algebraic multiplicity of (anti)periodic eigenvalues for the associated Hill operator. |
| title | Monodromy of generalized Lame equations with Darboux-Treibich-Verdier potentials: A universal law |
| topic | Classical Analysis and ODEs Mathematical Physics |
| url | https://arxiv.org/abs/2404.01879 |