A Note on a Discrete Version of Borg's Theorem via Toeplitz-Laurent Operators with Matrix-Valued Symbols

Golinskii, L. ; Kumar, K. ; Namboodiri, M. N. N. ; Serra-Capizzano, S.

Bollettino della Unione matematica italiana, Série 9, Tome 6 (2013) no. 1, pp. 205-218

Résumé

Consider a one dimensional Schrödinger operator $\tilde{A} = -\ddot{u} + V \cdot u$ with a periodic potential $V(\cdot)$, defined on a suitable subspace of $L^{2}(\mathbb{R})$. Its spectrum is the union of closed intervals, and in general these intervals are separated by open intervals (spectral gaps). The Borg theorem states that we have no gaps if and only if the periodic potential $V(\cdot)$ is constant almost everywhere. In this paper we consider families of Finite Difference approximations of the operator $\tilde{A}$, which depend upon two parameters $n$, i.e., the number of periodicity intervals possibly infinite, and $p$, i.e., the precision of the approximation in each interval. We show that the approach, with fixed $p$, leads to families of sequences $\{A_{n}(p)\}$, where every matrix $A_{n}(p)$ can be interpreted as a block Toeplitz matrix generated by a $p \times p$ matrix-valued symbol $f$. In other words, every $A_{n}(p)$ with finite $n$ is a finite section of the double infinite Toeplitz-Laurent operator $A_{\infty}(p) = L(f)$. The specific feature of the symbol $f$ , which is a trigonometric polynomial of 1st degree, allows to identify the distribution of the collective spectra of the matrix-sequence $\{A_{n}(p)\}$, and, in particular, provide a simple way for proving a discrete version of Borg's theorem: the discrete operator $L(f)$ has no gaps if and only if the corresponding ``potential'' is constant. The result partly overlaps with known results by Flaschka from the operator theory. The main novelty here is the purely linear algebra approach.