Eigenparameter dependent Sturm-Liouville problems in boundary conditions with transmission conditions
Let \(\Omega_1 := (0, c)\), \(\Omega_2 := (c, \infty)\), \(\Omega = \Omega_1 \cup \Omega_2\), \[ l(y):= \frac{1}{\omega(x)} \bigl [-y^{\prime \prime}+\frac{\nu^2-1/4}{x^2} y + q(x)y],\;x \in \Omega, \] where \(0 \leq \nu < 1\), \(\omega, q\) are real-valued Lebesgue measurable functions on \(\Omega\), \(\omega (x) > 0\) a.e., and \(\omega, q\) are in \(L_{\mathrm{loc}}^1\) on \(\Omega_1\) and \(\Omega_2\), respectively. \(c\) is called regular if \(\omega, q \in L^1[c-\epsilon, c+\epsilon]\) for some \(\epsilon > 0\). The BVTP (boundary value transmission problem) considered is \[ l(y) = \lambda y,\;y \in D,\;x \in \Omega \] under the boundary conditions \[ \begin{aligned} &\alpha_1 [y,u]_0 - \alpha_2 [y, v]_0 = \lambda (\alpha_1'[y,u]_0-\alpha_2'[y,v]_0);\\ & [y,u]_\infty -h[y,v]_\infty = 0,\;\mathfrak{I} h > 0;\\ &\gamma_1 y(c-) = \delta_1y(c+);\;\gamma_2y'(c-) = \delta_2y'(c+),\end{aligned} \] where \(\lambda\) is a complex spectral parameter, \(\alpha_1, \alpha_1', \alpha_2, \alpha_2'\) are real, \(\rho := \alpha_1' \alpha_2 - \alpha_2' \alpha_1 > 0\), \(\gamma_1 \gamma_2 > 0\) and \(\delta_1 \delta_2 > 0\); \(D\) is the subset of \(L_\omega^2(\Omega)\) with \(y'\) being locally absolutely continuous on \(\Omega_1\) and \(\Omega_2\) and \(l(y) \in L_\omega\) for \(y,z \in D\), \([y,z]_x :=(y \bar z' - y' \bar z)(x)\), \([y,z]_0 := \lim_{x \rightarrow 0^+}[y,z]_x\), \([y,z]_{c \pm} := \lim_{x \rightarrow c{\pm}}[y,z]_x\), \([y,z]_\infty := \lim_{x \rightarrow \infty}[y,z]_x\). Define the maximum operator \(L\) on \(D\) by \(Ly = l(y)\), let \(D_0\) be the linear set of all function \(y \in D\) such that \([y, z]_0 = y(c-) = y'(c-) = y(c+) = y'(c+)= [y,z ]_\infty = 0\) for arbitrary \(z \in D\). Denote by \(L_0\) the restriction of \(L\) to \(D_0\). The authors conclude that, if \(c\) is regular and Weyl's limit-circle case holds at \(0\) and \(\infty\) for \(l\) (i.e., if \(L_0\) has the deficiency indices \((4,4)\)), then the spectrum of the above BVTP is purely discrete and belongs to the open upper half-plane. Furthermore, for all \(h\) with \(\mathfrak{I} h > 0\), except possibly for a single value \(h = h_0\), this BVTP \((h \not = h_0)\) has a countable number of isolated eigenvalues with finite multiplicity and limit points at infinity. The system of eigenfunctions and associated functions of this problem \((h \not = h_0)\) is complete in \(L_\omega^2 (\Omega)\). A similar results holds for Weyl's limit-circle case holds at \(0, c\) and \(\infty\) for \(l\).
scientific article; zbMATH DE number 6143679