The bispectral problem, the Darboux process, monodromy and the Hermite operator
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| Formato: | Preprint |
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2025
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| _version_ | 1866908857773588480 |
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| author | Castro, M. M. Grünbaum, F. A. |
| author_facet | Castro, M. M. Grünbaum, F. A. |
| contents | The complete solution of the bispectral problem for the Schrödinger operator $L=-\tfrac{d^2}{dx^2}+V(x)$ in [DG] (J. J. Duistermaat and F. A. Grünbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177-240) is obtained by the application of the Darboux process to the cases of $V=0$ and $V(x)=-\tfrac{1}{4x^2}$. Both of these cases are trivially bispectral and after repeated applications of the Darboux process one gets either a pair of rank one bundles of bispectral situations (when starting from $V=0$) or a rank two bispectral bundle (when starting from $V(x)=-\tfrac{1}{4x^2}$). In the first case all operators have ''trivial monodromy'' as defined in [DG]. In the second case the monodromy group of all operators is given by the integers. In this paper we start from $V(x)=x^2$, use the Darboux process and explore the connection between the rank of certain non-polynomial bispectral families and trivial monodromy by means of examples. The main conclusion is that the results in [DG] do not apply verbatim in this case. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_04158 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The bispectral problem, the Darboux process, monodromy and the Hermite operator Castro, M. M. Grünbaum, F. A. Classical Analysis and ODEs 33C45, 22E45, 33C47 The complete solution of the bispectral problem for the Schrödinger operator $L=-\tfrac{d^2}{dx^2}+V(x)$ in [DG] (J. J. Duistermaat and F. A. Grünbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103 (1986), 177-240) is obtained by the application of the Darboux process to the cases of $V=0$ and $V(x)=-\tfrac{1}{4x^2}$. Both of these cases are trivially bispectral and after repeated applications of the Darboux process one gets either a pair of rank one bundles of bispectral situations (when starting from $V=0$) or a rank two bispectral bundle (when starting from $V(x)=-\tfrac{1}{4x^2}$). In the first case all operators have ''trivial monodromy'' as defined in [DG]. In the second case the monodromy group of all operators is given by the integers. In this paper we start from $V(x)=x^2$, use the Darboux process and explore the connection between the rank of certain non-polynomial bispectral families and trivial monodromy by means of examples. The main conclusion is that the results in [DG] do not apply verbatim in this case. |
| title | The bispectral problem, the Darboux process, monodromy and the Hermite operator |
| topic | Classical Analysis and ODEs 33C45, 22E45, 33C47 |
| url | https://arxiv.org/abs/2509.04158 |