Inequalities among eigenvalues of Sturm-Liouville problems
The Sturm-Liouville problem is considered consisting of the equation \[ -(py')'+qy=\lambda wy,\quad x\in J=(a,b), \] and the coupled selfadjoint boundary conditions \[ Y(B)=e^{i\theta}KY(a), \] with \(\theta\in(-\pi,\pi]\), the vector-column \(Y=(y,py')\), \(-\infty<a<b<\infty\), \[ K\in SL(2,R)=:\left\{K=\left(\begin{matrix} k_{11} &k_{12}\\ k_{21} &k_{22}\end{matrix}\right),\;k_{ij}\in R,\;\det K=1\right\}, \] \(p^{-1}\), \(q\), \(w\in L^{1}(J,\mathbb{R})\), \(p>0\), \(w>0\) a.e. on \(J\). A well-known result [see \textit{M. P. S. Eastham}, The spectral theory of periodic differential equations, Scottish Academic Press, Edinburgh-London (1973; Zbl 0287.34016)] gives the following inequalities for \(K=I\), \(I\) is the identity matrix: \[ \begin{gathered}\lambda_{0}^{N}\leq\lambda_0(I)< \lambda_0(e^{i\theta}I)< \lambda_0(-I) \leq\{\lambda_0^{D},\lambda_1^{N}\}\leq\lambda_1(-I)<\lambda_1(e^{i\theta}I) <\lambda_1(I)\leq\{\lambda_1^{D},\lambda_2^{N}\}\\ \leq\lambda_{2}(I)<\lambda_2(e^{i\theta}I)<\lambda_2(-I) \leq\{\lambda_2^{D},\lambda_3^{N}\}\leq\lambda_3(-I)<\lambda_3(e^{i\theta}I) <\lambda_3(I)\leq\{\lambda_3^{D},\lambda_4^{N}\}\leq\dots{}\end{gathered} \] with \(\theta\in (-\pi,0)\cup (0,\pi)\), \(\lambda_n^D\) and \(\lambda_n^N\) denote \(n\)th Dirichlet and Neumann eigenvalues, respectively, \(\lambda_n(K)\) is the \(n\)th eigenvalue of the problem above and the notation \(\{\lambda_0^D,\lambda_1^N\}\) means either of \(\lambda_0^D\) or \(\lambda_1^n\) (all eigenvalues are listed in nondecreasing order). The authors present an analogue of the mentioned inequalities for an arbitrary \(K\in SL(2,\mathbb{R})\).
Dirichlet and Neumann boundary conditions
periodic and semiperiodic boundary conditions
geometric and algebraic multiplicities of eigenvalues
scientific article; zbMATH DE number 1300149
scientific article