Eigenvalues of regular Sturm-Liouville problems
The Sturm-Liouville differential equation \[ -(py')' + qy= \lambda wy \text{ on } (a',b'),\;-\infty \leq a' \leq b' \leq\infty, \;\lambda \in\mathbb{R} \tag{1} \] is considered, where \(p,q,w: (a',b') \to \mathbb{R}\), \(1/p,q,w \in L^1_{\text{loc}} (a',b')\), \(w>0\) a.e. on \((a',b')\). For \(I=[a,b]\), \(a'<a <b <b'\), separated or coupled selfadjoint two-point boundary conditions are taken. For example, the separated boundary conditions can be written in the form \[ \cos \alpha y(a) - \sin \alpha (py')(a) = 0,\;\cos \beta y(b) - \sin \beta (py') (b)=0, \tag{2} \] with parameters \(0\leq \alpha < \pi\), \(0< \beta \leq\pi\). The eigenvalues \(\lambda\) and (normalized) eigenfunctions \(u\) of this boundary eigenvalue problem depend on \(a,b, 1/p,q,w\), and the parameters in the boundary conditions. Therefore, considering \(\lambda\) and \(u\) as functions of these parameters, the question of continuous and differentiable dependence arises. The continuity is shown in Theorem 3.1. This result is somehow mathematical folklore, but no explicit proofs have been known to the authors or the reviewer. Although some results about differentiable dependence can be found in earlier publications, comprehensive results are given for the first time in this paper. Differentiability is proved and explicit formulas for the derivatives with respect to each of the parameters are given. Let us state some of these formulas from Theorems 4.1 and 4.2. We shall assume that \(\lambda\) is a simple eigenvalue and \(u\) a normalized eigenfunction of \(\lambda\) for the boundary eigenvalue problem. Then the derivatives with respect to the indicated parameters are \[ \lambda' (a) = {1\over p(a)} |pu' |^2 (a) - |u|^2 (a) \bigl[q(a) - \lambda (a)w(a) \bigr] \text{ a.e. in } (a',b), \] \[ d\lambda_{(1/p)} (h)= -\int^b_a |pu' |^2h,\;h\in L^1 (a, b). \] If the boundary conditions are separable and written in the form (2), then \(\lambda' (\alpha) = - |u |^2 (a)- |pu' |^2(a)-|pu'|^2(a)\).
differentiable dependence of eigenvalues
coupled self-adjoint boundary conditions
scientific article; zbMATH DE number 946922