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

Abstract

By means of the extended Ito integral and the shooting method, this paper concerns the random problem containing white noise B ( t , ω ) B\left ( {t, \omega } \right ) : \[ [ − p ( t ) u ′ ( t ) ] ′ + [ q ( t ) + ε B ( t , ω ) ] u = λ u , u ( 0 ) = 0 , u ( 1 ) = 0. {\left [ { - p\left ( t \right )u’\left ( t \right )} \right ]’} + \left [ {q\left ( t \right ) + \varepsilon B\left ( {t, \omega } \right )} \right ] u = \lambda u, \qquad u\left ( 0 \right ) = 0, \qquad u\left ( 1 \right ) = 0. \] When ε \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.