Spectral properties of the Klein--Gordon \(s\)-wave equation with spectral parameter-dependent boundary condition
Let \(L_{\lambda}\) denote the operator generated in \(L^2(\mathbb{R}_+)\) by the Klein--Gordon s-wave equation \[ y''(x) + (\lambda - q(x))^2 y(x) =0 \] and the boundary condition \[ y'(0) - (a\lambda +b) y(0) =0, \] where \(a,b \in {\mathbb{C}}, a \neq \pm i, 0,\) and \(q\) is a complex-valued function which is absolutely continuous on each finite subinterval of \( \mathbb{R}_+\) and satisfies the condition \[ \sup_{x\in \mathbb{R}_+} \{ \exp(\varepsilon \sqrt{x}) | q'(x)| \} < \infty, \quad \varepsilon >0. \] The eigenvalues of \(L_{\lambda}\) are poles of its Green function in the upper and lower open half-planes \( \mathbb{C}_{\pm}\) and the spectral singularities are poles on \( {\mathbb{R}}\setminus \{0\}.\) It is proved that \(L_{\lambda}\) has a finite number of eigenvalues and spectral singularities, and these are of finite multiplicity. To be specific, the poles of the Green function are the zeros of functions \(g^{\pm}\) which are analytic in \( \mathbb{C}_{\pm}\), and it is shown that \(g^+, g^-\) can be continued analytically into the half-planes \( \text{Im} \lambda > -\varepsilon/2\), \(\text{Im} \lambda < \varepsilon/2,\) respectively.
scientific article; zbMATH DE number 2168695