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Geometric aspects of Sturm-Liouville problems. I: Structures on spaces of boundary conditions. (English) Zbl 0957.34024

A regular Sturm-Liouville problem is considered consisting of the equation \[ -(py')'+qy=\lambda wy\quad \text{ on} \;I=(a,b), \tag{1} \] and the complex boundary conditions \[ (A|B)\begin{pmatrix} y(a)\\ (py')(a)\\ y(b) \\ (py')(b)\end{pmatrix}=0, \tag{2} \] with \(-\infty\leq a<b\leq+\infty\) and \(p\), \(q\) and \(w\) are real-valued functions on \(I\) satisfying \(w\not=0 \;\text{ a.e. \;on } \;I, \;{1\over p}, q, w\in L^1(I)\), \((A|B)\in M^*_{2\times 4}(C)\). While general results of perturbation theory [T. Kato, Perturbation theory for linear operators. Berlin etc.: Springer-Verlag (1966; Zbl 0148.12601); V. M. Eni, Dokl. Akad. Nauk SSSR 173, 1251-1254 (1967); translation from Sov. Math., Dokl. 8, 542-545 (1967; Zbl 0157.45302); and I. C. Gohberg and E. I. Sigal, Math. USSR, Sob. 13(1971), 603-625 (1972); translation from Mat. Sb., n. Ser. 84(126), 607-629 (1971; Zbl 0254.47046)] can be successfully applied in the case when \(p\), \(q\) or \(w\) are perturbed, those results cannot be used if the boundary conditions are under perturbation. Therefore the authors establish (in a series of publications) a perturbation theory for this case. They prove in particular that if \(\Omega\in C\) is a normal domain (open bounded set whose boundary does not contain any eigenvalue of (1),(2)) then the number (counting multiplicities) of eigenvalues inside \(\Omega\) is stable if the perturbation is in certain class and it is small in some sense.
The authors characterize subsets of complex boundary conditions that have a given complex number as an eigenvalue as well as those that have this given number as an eigenvalue of multiplicity 2. The case of real eigenvalue and real boundary conditions is also considered. The authors clarify a geometric structure on the space of complex boundary conditions and on the space of real boundary conditions.

MSC:

34B24 Sturm-Liouville theory
34B08 Parameter dependent boundary value problems for ordinary differential equations
34B09 Boundary eigenvalue problems for ordinary differential equations
34L15 Eigenvalues, estimation of eigenvalues, upper and lower bounds of ordinary differential operators
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