Semiclassical expansion for exactly solvable differential operators
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arXiv
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| Format: | Preprint |
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2024
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| author | Borrego-Morell, Jorge A. Shapiro, Boris |
| author_facet | Borrego-Morell, Jorge A. Shapiro, Boris |
| contents | Below we study a linear differential equation $\MM (v(z,η))=η^M{v(z,η)}$, where $η>0$ is a large spectral parameter and $\MM=\sum_{k=1}^{M}ρ_{k}(z)\frac{d^k}{dz^k},\; M\ge 2$ is a differential operator with polynomial coefficients such that the leading coefficient $ρ_M(z)$ is a monic complex-valued polynomial with $\dgr{ρ_M }=M$ and other $ρ_k(z)$'s are complex-valued polynomials with $\dgr{ρ_k }\leq k$. We prove the Borel summability of its WKB-solutions in the Stokes regions. For $M=3$ under the assumption that $ρ_M$ has simple zeros, we give the full description of the Stokes complex (i.e. the union of all Stokes curves) of this equation. Finally, we show that for the Euler-Cauchy equations, their WKB-solutions converge in the usual sense. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_19087 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Semiclassical expansion for exactly solvable differential operators Borrego-Morell, Jorge A. Shapiro, Boris Classical Analysis and ODEs Primary 34M60, Secondary 34E20, 34M40, 34M30, 34M35 Below we study a linear differential equation $\MM (v(z,η))=η^M{v(z,η)}$, where $η>0$ is a large spectral parameter and $\MM=\sum_{k=1}^{M}ρ_{k}(z)\frac{d^k}{dz^k},\; M\ge 2$ is a differential operator with polynomial coefficients such that the leading coefficient $ρ_M(z)$ is a monic complex-valued polynomial with $\dgr{ρ_M }=M$ and other $ρ_k(z)$'s are complex-valued polynomials with $\dgr{ρ_k }\leq k$. We prove the Borel summability of its WKB-solutions in the Stokes regions. For $M=3$ under the assumption that $ρ_M$ has simple zeros, we give the full description of the Stokes complex (i.e. the union of all Stokes curves) of this equation. Finally, we show that for the Euler-Cauchy equations, their WKB-solutions converge in the usual sense. |
| title | Semiclassical expansion for exactly solvable differential operators |
| topic | Classical Analysis and ODEs Primary 34M60, Secondary 34E20, 34M40, 34M30, 34M35 |
| url | https://arxiv.org/abs/2402.19087 |