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Eigenfunction expansions for a Sturm-Liouville problem on time scales. (English) Zbl 1145.39005
The author considers the second order Sturm-Liouville eigenvalue problem $$ -y^{\Delta\nabla}(t)=\lambda\,y(t), \quad t\in(a,b), \quad y(a)=y(b)=0, $$ over a time scale interval $[a,b]$. Here a time scale is any nonempty closed subset of ${\mathbb R}$, and $y^\Delta$ and $y^\nabla$ are the time scale delta and nabla derivatives, respectively. The above given time scale problem generalizes and unifies the corresponding differential and difference eigenvalue problems. The main goal of this paper is to establish the eigenfunction expansions in the time scale $L^2$ and uniform norms. The main tools are the Hilbert-Schmidt theorem for completely continuous symmetric linear operators in a Hilbert space (applied to the inverse of the operator associated with the above problem), and certain new integration by parts formulas relating the delta and nabla integrals.

MSC:
39A12Discrete version of topics in analysis
34L10Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions (ODE)
34B24Sturm-Liouville theory