Sturm-Liouville difference equations and banded matrices

Werner Kratz

Address. Abteilung Angewandte Analysis, Universitat Ulm, D - 89069 Ulm, Germany


Abstract. In this paper we consider {\it discrete} Sturm-Liouville eigenvalue problems of the form $$ L(y)_k := \sum^n_{\mu =0} (-\Delta)^\mu \{r_\mu(k)\Delta^\mu y_{k+1-\mu}\} = \lambda \rho(k) y_{k+1} $$ $$ for \;\; 0\le k \le N-n \;\; with \;\; y_{1-n}= \cdots = y_0 = y_{N+2-n}= \cdots = y_{N+1} = 0, $$ where $N$ and $n$ are integers with $ 1 \le n \le N$ and with the assumptions that $r_n(k) \not=0,\, \rho(k)>0$ for all $k.$ These problems correspond to eigenvalue problems for symmetric, banded matrices ${\cal A} \in \IR^{(N+1-n)\times(N+1-n)}$ with band-width $2n+1.$ We present the following results: - a formula for the chracteristic polynomial of ${\cal A},$ which yields a {\it recursion} for its calculation - an {\it oscillation theorem}, which generalizes Sturm's well-known theorem on Sturmian chains, and - an inversion formula, which shows that {\it every} symmetric, banded matrix corresponds uniquely to a Sturm-Liouville eigenvalue problem of the above form.

AMSclassification. 39A10, 39A12, 65F15, 15A18

Keywords. Sturm-Liouville equations, banded matrices, eigenvalue problems; Sturmian chains.