Abstract
We study the completeness property and the basis property of the root function system of the Sturm-Liouville operator defined on the segment [0, 1]. All possible types of two-point boundary conditions are considered.
Highlights
The spectral theory of two-point differential operators was begun by Birkhoff in his two papers 1, 2 of 1908 where he introduced regular boundary conditions for the first time
The present communication is a brief survey of results in the spectral theory of the Sturm-Liouville operator: Lu u − q x u, 1.1 with two-point boundary conditions
The set of all eigen- and associated functions or root functions corresponding to the same eigenvalue λ together with the function u x ≡ 0 forms a root linear manifold
Summary
The spectral theory of two-point differential operators was begun by Birkhoff in his two papers 1, 2 of 1908 where he introduced regular boundary conditions for the first time. It was continued by Tamarkin 3, 4 and Stone 5, 6. Afterwards their investigations were developed in many directions. The present communication is a brief survey of results in the spectral theory of the Sturm-Liouville operator: Lu u − q x u, 1.1 with two-point boundary conditions. We will study the completeness property and the basis property of the root function system of operator 1.1 , 1.2. The convergence of spectral expansions is investigated only in classical sense; that is, the question about the summability of divergent series by a generalized method is not considered
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