Eigenvalues and eigenfunctions of discontinuous Sturm-Liouville problems with eigenparameter in the boundary conditions. (Q1396271)

The authors derive asymptotic expressions for the eigenvalues and eigenfunctions of the Sturm-Liouville systems \(-u''+qu=\lambda u\) on \([-1,0)\) and \((0,1]\), coupled by the ``transmission'' conditions \(u(-0)=\delta u(+0)\), \(u'(-0)=\delta u'(+0)\), with boundary conditions \(\alpha _1 u(-1)+\alpha _2 u'(-1)=0= (\beta _{1}'\lambda +\beta _1 )u(1)-(\beta _{2}'\lambda + \beta _2 )u'(1)\), where \(\delta ,\alpha _i ,\beta _{i}', \beta _i\) are real numbers, \(i=1,2\).