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Eigenfunction expansions in \(L^{2}\) spaces for boundary value problems on time-scales. (English) Zbl 1125.34069

Summary: Let \(\mathbb{T}\subset\mathbb{R}\) be a bounded time-scale, with \(a=\inf \mathbb{T}\), \(b=\sup \mathbb{T}\). We consider the weighted, linear, eigenvalue problem \[ -(pu^\Delta)^\Delta(t)+q(t)u^\sigma(t)=\lambda w(t) u^\sigma(t),\;t\in \mathbb{T}^{\kappa^2}. \]
\[ c_{00}u(a)+c_{01}u^\Delta (a)=0,\quad c_{10}u(\rho(b))+ c_{11}u^\Delta(\rho(b))=0, \] for suitable functions \(p,q\) and \(w\) and \(\lambda \in\mathbb{R}\). Problems of this type on time-scales have normally been considered in a setting involving Banach spaces of continuous functions on \(\mathbb{T}\). In this paper we formulate the problem in Sobolev-type spaces of functions with generalized \(L^2\)-type derivatives. This approach allows us to use the functional analytic theory of Hilbert spaces rather than Banach spaces. Moreover, it allows us to use more general coefficient functions \(p,q\), and weight function \(w\), than usual, viz., \(p\in H^1(\mathbb{T}^\kappa)\) and \(q\), \(w\in L^2(\mathbb{T}^\kappa)\) compared with the usual hypotheses that \(p\in C^1_{\text{rd}} (\mathbb{T}^\kappa)\), \(q\), \(w\in C^0_{\text{rd}} (\mathbb{T}^{\kappa^2})\). Further to these conditions, we assume that \(p\geq c>0\) on \(\mathbb{T}^\kappa\), \(C\geq w\geq c>0\) on \(\mathbb{T}^{\kappa^2}\), for some constants \(C>c>0\). These conditions are similar to the usual assumptions imposed on Sturm-Liouville, ordinary differential equation problems. We obtain a min-max characterization of the eigenvalues of the above problem, and various eigenfunction expansions for functions in suitable function spaces. These results extend certain aspects of the standard theory of self-adjoint operators with compact resolvent to the above problem, even though the linear operator associated with the left-hand side of the problem is not in fact self-adjoint on general time-scales.

MSC:

34L10 Eigenfunctions, eigenfunction expansions, completeness of eigenfunctions of ordinary differential operators
34B24 Sturm-Liouville theory
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