Eigenvalues of discontinuous Sturm-Liouville problems with symmetric potentials
The first 15 eigenvalues of two discontinuous Sturm-Liouville systems, \(- u''=\lambda u\) and \(-u''+(2 \cos 2x)u=\lambda u\) respectively, with boundary conditions \(u'(0)=u'(\pi)=0\) and symmetric jump conditions \(u(d_ 1+)=au(d_ 1-),\) \(u'(d_ 1+)=a^{-1}u'(d_ 1-)+bu(d_ 1-),\) \(u(d_ 2-)=au(d_ 2+),\quad u'(d_ 2-)=a^{-1}u'(d_ 2+)-bu(d_ 2- )\) were found by the classical fourth order Runge-Kutta method. The eigenvalues are plotted. The eigenvalues are used to reconstruct the potential function \(q=2 \cos 2x\) of the second Sturm-Liouville system by means of the author's algorithm [ibid. 18, No.4, 349-356 (1989; reviewed above)]. Then as the third example of the Sturm-Liouville system a symmetric potential function with symmetrically located discontinuities \(d_ 1\), \(d_ 2\) satisfying symmetric jump conditions is reconstructed by the same algorithm.
scientific article; zbMATH DE number 4117779