Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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On the eigentheory of operators which exhibit a slow variation


Authors: Lawrence Sirovich and B. W. Knight
Journal: Quart. Appl. Math. 38 (1981), 469-488
MSC: Primary 47A70; Secondary 35B40, 45C05, 92A09
DOI: https://doi.org/10.1090/qam/614554
MathSciNet review: 614554
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Abstract: A class of linear operators which exhibit slow variation is considered. If the kernel of the operator is $ K\left( {x - y,\frac{1}{2}\varepsilon \left( {x + y} \right)} \right)$, $ \varepsilon $ the parameter of slowness, then its Wigner transform is defined to be $ \tilde K\left( {p, q} \right) = \smallint K\left( {u, p} \right)\exp \left( { - iup} \right)du$. The eigenvalues of such operators are shown to follow an area rule: if the curve $ {\lambda _n} = \tilde K\left( {p, q} \right)$ contains the area $ A\left( \lambda \right) = \left( {2n + 1} \right)\pi \varepsilon $ then $ {\lambda _n}$ is an eigenvalue. Forms for the corresponding eigenfunctions are also obtained. Classical WKB theory is shown to be a special case and other examples are given.


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DOI: https://doi.org/10.1090/qam/614554
Article copyright: © Copyright 1981 American Mathematical Society


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