The random eigenvalue problem for a differential equation containing small white noise

Xia, Ning Mao

doi:10.1090/qam/973379

Author: Ning Mao Xia
Journal: Quart. Appl. Math. 46 (1988), 611-630
MSC: Primary 60H10; Secondary 34B25, 34F05
DOI: https://doi.org/10.1090/qam/973379
MathSciNet review: 973379
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Abstract: By means of the extended Ito integral and the shooting method, this paper concerns the random problem containing white noise $B\left ( {t, \omega } \right )$: \[ {\left [ { - p\left ( t \right )u’\left ( t \right )} \right ]’} + \left [ {q\left ( t \right ) + \varepsilon B\left ( {t, \omega } \right )} \right ] u = \lambda u, \qquad u\left ( 0 \right ) = 0, \qquad u\left ( 1 \right ) = 0.\] When $\varepsilon$ is small, the existence and asymptotic expansions for the solutions can be obtained, and the normal properties for the first correction terms can be proved. Formulas for evaluation are derived and one example of the Schrödinger equation is given to illustrate the whole procedure.

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