On a class of random Sturm-Liouville problems (Q2761894)

The author investigates a class of random Sturm-Liouville problems \(-y^{\prime\prime }+q(x)y=\lambda y\), \(0<x<1\), with separated boundary conditions \((a_{11}+\xi_{11})y(0,\lambda)-(a_{12}+\xi_{12})y'(0,\lambda)=0\) and \((a_{21}+\xi_{21})y(1,\lambda)-(a_{22}+\xi_{22})y'(1,\lambda)=0\), where \((\xi_{11},\xi_{12})\), \((\xi_{21},\xi_{22})\) are two-dimensional independent random variables with zero mean and known probability density functions \(p_{\xi_{11},\xi_{12}}\) and \(p_{\xi_{21},\xi_{22}}\) and \(a_{11}^2+a_{12}^2=1\), \(a_{21}^2+a_{22}^2=1\), \(q\in L_{\text{loc}}^1(0,1)\). The probability density functions of the random eigenvalues \(\lambda \) are derived by the transmutation operator of Gelfand and Levitan. The author makes use of the Whitaker-Shannon-Kotelnikov sampling theorem to recover certain boundary functions associated with the considered problem. The main result is that the above stated problem has a sequence of random eigenvalues \(\lambda_k\), \(k=0,1,2,\dots \), with probability density function \(p_{\lambda_k}\) given in explicit form.