### B P Rynne and F A Davidson

#### Abstract

Let $\T \subset \R$ be a bounded time-scale, with $a = \inf\T$, $b = \sup \T$. We consider the weighted, linear, eigenvalue problem \begin{gather*} -(p u^{\De})^{\De}(t) +q(t)u^\si(t) = \la w(t) u^\si(t) , \quad t \in \T^{\ka^2}, %\label{eval_prob.eq} \\ c_{00} u(a) + c_{01} u^\De(a) = 0, \quad c_{10} u(\rho(b)) + c_{11} u^\De(\rho(b)) = 0, %\label{bc.eq} \end{gather*} for suitable functions $p,\,q$ and $w$ and $\la \in \R$. Problems of this type on time-scales have normally been considered in a setting involving Banach spaces of continuous functions on $\T$. In this paper we formulate the problem in Sobolev-type spaces of functions with generalised $L^2$-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., $p \in H^1(\T^\ka)$ and $q,\, w \in L^2(\T^\ka)$ compared with the usual hypotheses that $p \in \Crd^1(\T^\ka)$, $q,\, w \in \Crd^0(\T^{\ka^2})$. Further to these conditions, we assume that $p \ge c >0$, on $\T^\ka$, $C \ge w \ge c > 0$ on $\T^{\ka^2}$, 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 operators associated with the left hand side of the problem is not in fact self-adjoint on general time-scales.