On the ``splitting'' of multiplicity in principal eigenvalues of multipoint boundary value problems
The author studies a boundary value problem for a differential operator with discontinuous coefficients of the form \[ \begin{aligned} - y_1''(x)+ q_1(x)y_1(x) &= \lambda y_1(x),\quad 0\leq x< x_1;\tag{1}\\ - y_2''(x)+ q_2(x)y_2(x) &= \lambda y_2(x),\quad x_1< x\leq \pi,\end{aligned} \] under the gluing conditions \(y^{(j)}_1(x_1- 0)= y^{(j)}_2(x_1+ 0)\), \(j= 0,1\), and the boundary conditions \(y(0)= 0\), \(y_2(\pi)= -5y_2(\pi/3)+ 4y_2(2\pi/3)\). The functions \(q_1\), \(q_2\) are supposed to be sufficiently smooth on \((0,x_1)\) and \((x_1,\pi)\). The author deduces an equation for the eigenvalues of the multipoint boundary value problem and finds asymptotic expansions for the zeros of the quasipolynomial on the left-hand side of the equation which sharpen those ones presented by \textit{R. Bellman} and \textit{K. L. Cooke} [Differential-difference equations. Academic Press (1963; Zbl 0105.06402)].
scientific article; zbMATH DE number 1175162