On the eigentheory of operators which exhibit a slow variation

Sirovich, Lawrence; Knight, B. W.

doi:10.1090/qam/614554

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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