Sturm-Liouville difference equations and banded matrices. (English) Zbl 1072.39500

The author investigates the discrete higher order Sturm-Liouville eigenvalue problem of the form \[ L(y)_k := \sum ^n_{\mu =0} (-\Delta )^\mu \{r_\mu (k)\Delta ^\mu y_{k+1-\mu }\} = \lambda \rho (k) y_{k+1}, \] where \(N\) and \(n\) are integers, \(1 \leq n \leq N\), \(0\leq k\leq N-n\), with the assumptions that \(r_n(k) \neq 0,\, \rho (k)>0\) and with the zero boundary conditions \(y_{1-n}= \cdots = y_0 = y_{N+2-n}= \cdots = y_{N+1} = 0\). This problem corresponds to the eigenvalue problem for symmetric, banded matrix \({\mathcal A} \in \mathbb R^{(N+1-n)\times (N+1-n)}\) with band-width \(2n+1\). The main results of the paper can be characterized as follows: (i) a recursive formula for computing the characteristic polynomial of \({\mathcal A}\), (ii) an oscillation theorem, which generalizes Sturm’s well-known theorem on Sturmian chains, and (iii) an inversion formula, which shows that every symmetric, banded matrix corresponds uniquely to a Sturm-Liouville eigenvalue problem of the above form.


39A10 Additive difference equations
39A12 Discrete version of topics in analysis
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
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