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A Sturm-Liouville eigenproblem of the fourth kind: a critical latitude with equatorial trapping. (English) Zbl 1020.34080
Summary: Through both analytical and numerical methods, the authors solve the eigenproblem $u_{zz}+(1/z-\lambda-(z-1/\varepsilon)^2)u=0$ on the unbounded interval $z\in[-\infty,\infty]$, where $\lambda$ is the eigenvalue and $u(z)\to 0$ as $|z|\to\infty$. This models an equatorially trapped Rossby wave in a shear flow in the ocean or atmosphere. It is the usual parabolic cylinder equation with Hermite functions as the eigenfunctions except for the addition of an extra term, which is a simple pole. The pole, which is on the interior of the interval, is interpreted as the limit $\delta\to 0$ of $1/(z-i\delta)$. The eigenfunction has a branch point of the form $z\log(z)$ at $z=0$, where the branch cut is on the upper imaginary axis. The eigenvalue is complex valued with an imaginary part, which the authors show, through matched asymptotics, to be approximately $\sqrt\pi\exp(-1/\varepsilon^2)\{1-2\varepsilon\log\varepsilon+\varepsilon\log 2+\gamma\varepsilon\}$. Because $\text{Im}(\lambda)$ is transcendentally small in the small parameter $\varepsilon$, it lies `beyond all orders’ in the usual Rayleigh-Schrödinger power series in $\varepsilon$. Nonetheless, the authors develop special numerical algorithms that are effective in computing $\text{Im}(\lambda)$ for $\varepsilon$ as small as $\frac{1}{100}$.

34L40Particular ordinary differential operators
34B24Sturm-Liouville theory
34E05Asymptotic expansions (ODE)
76B65Rossby waves
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