# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
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 ${u}_{zz}+\left(1/z-\lambda -{\left(z-1/\epsilon \right)}^{2}\right)u=0$ on the unbounded interval $z\in \left[-\infty ,\infty \right]$, where $\lambda$ is the eigenvalue and $u\left(z\right)\to 0$ 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 $1/\left(z-i\delta \right)$. The eigenfunction has a branch point of the form $zlog\left(z\right)$ 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 $\sqrt{\pi }exp\left(-1/{\epsilon }^{2}\right)\left\{1-2\epsilon log\epsilon +\epsilon log2+\gamma \epsilon \right\}$. Because $\text{Im}\left(\lambda \right)$ is transcendentally small in the small parameter $\epsilon$, it lies ‘beyond all orders’ in the usual Rayleigh-Schrödinger power series in $\epsilon$. Nonetheless, the authors develop special numerical algorithms that are effective in computing $\text{Im}\left(\lambda \right)$ for $\epsilon$ as small as $\frac{1}{100}$.
##### MSC:
 34L40 Particular ordinary differential operators 34B24 Sturm-Liouville theory 34E05 Asymptotic expansions (ODE) 76B65 Rossby waves