Dependence of eigenvalues of Sturm-Liouville problems with eigenparameter-dependent boundary conditions and interface conditions
In this paper, the authors study continuous dependence and differential expressions of eigenvalues for the second-order Sturm-Liouville equation \[ -\left ( p(x) y'(x) \right )' + q(x)y =\lambda w(x)y, \quad x \in J = [a,c) \cup (c, b], \tag{1}\] with the boundary conditions \begin{align*} (\alpha_1' \lambda - \alpha_1) y(a) - (\alpha_2' \lambda - \alpha_2) p y'(a) =0, \\ (\beta_1' \lambda + \beta_1) y(b) - (\beta_2' \lambda + \beta_2) p y'(b) =0, \tag{2}\end{align*} and the interface conditions \begin{align*} y(c+) - \alpha_3 y(c-) - \beta_3 p y'(c-) = 0, \\ p y'(c+) - \alpha_4 y(c-) - \beta_4 p y'(c-) = 0. \tag{3}\end{align*} The authors prove that the eigenvalues for the equation depend not only continuously but also smoothly on the parameters of the equation: the coefficients, the boundary conditions, the interface conditions, as well as the endpoints. Finally, the authors list the derivative formulas of eigenvalues with respect to all the parameters.
eigenparameter-dependent boundary conditions
scientific article; zbMATH DE number 7493764
scientific article