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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References:

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