Asymptotic approximations of eigen-functions for regular Sturm-Liouville problems with eigenvalue parameter in the boundary condition for integrable potential (Q2857598)

Consider the boundary eigenvalue problem NEWLINE\[CARRIAGE_RETURNNEWLINE \begin{gathered} -y''+qy=\lambda y ,\tag{1}\\ a_1y(a)-a_2y'(a)=\lambda (a_3y(a)-a_4y'(a)),\tag{2}\\ y(b)\cos \beta +y'(b)\sin\beta =0, \tag{3} \end{gathered}CARRIAGE_RETURNNEWLINE\]NEWLINE where \(q\) is integrable on \([a,b]\), \(a_1,a_2,a_3,a_4\in \mathbb{R}\) such that \(a_1a_4\neq a_2a_3\), and \(\beta \in [0,\pi)\). The authors then consider solutions \(\Psi\) and \(\Phi\) of the initial value problems (1), (2), and (1), (3), respectively, although the initial values for \(\Psi \) and \(\Psi '\) are interchanged, which appears to be a misprint. Making use of a relation between solutions of (1) and the Riccati equation NEWLINE\[CARRIAGE_RETURNNEWLINEv'=-\lambda +q-v^2CARRIAGE_RETURNNEWLINE\]NEWLINE which has been developed in [\textit{H. Coskun} and \textit{B. J. Harris}, Proc. R. Soc. Edinb., Sect. A, Math. 130, No. 5, 991--998 (2000; Zbl 0977.34077)], asymptotic formulas for \(\Psi \) and \(\Phi \) are obtained. The asymptotic approximation of the eigenfunctions follows now due to the eigenvalue asymptotics found in [\textit{H. Coskun} and \textit{E. Baskaya}, Math. Scand. 107, No. 2, 209--223 (2010; Zbl 1216.34091)].