Differential-parametric method for calculating the eigenvalues of the third boundary problem for the Sturm-Liouville operator (Q584482)

The author presents a method for calculating the eigenvalues of a non- self-adjoint, Sturm-Liouville problem \[ (1)\quad \{L[\psi]=\{\frac{d}{dx}p(x)\frac{d}{dx}+p(x)\}\psi (x)=-\lambda^ 2f(x)\psi (x),\quad a<x<b;\quad | \psi (a)| <\infty,\quad \alpha \psi '(b)+\beta \psi (b)=0 \] where \(p(a)=0\), Re p\(>0\) and \(\alpha\),\(\beta\),p,q,f are all complex. It reduces the problem to a Cauchy problem for a first-order equation with an algebraic right-hand side and initial conditions determined by solutions of \(L\psi =-\mu^ 2f(x)\psi (x)\) with either Dirichlet or Neumann conditions at b.