# Norm estimates of discrete Schrödinger operators

Colloquium Mathematicae (1998)

• Volume: 76, Issue: 1, page 153-160
• ISSN: 0010-1354

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## Abstract

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Harper’s operator is defined on ${\ell }^{2}\left(Z\right)$ by ${H}_{\theta }\xi \left(n\right)=\xi \left(n+1\right)+\xi \left(n-1\right)+2cosn\theta \phantom{\rule{0.166667em}{0ex}}\xi \left(n\right),$ where $\theta \phantom{\rule{-0.166667em}{0ex}}\in \phantom{\rule{-0.166667em}{0ex}}\left[0,\pi \right]$. We show that the norm of $\parallel {H}_{\theta }\parallel$ is less than or equal to $2\sqrt{2}$ for $\pi /2\le \theta \le \pi$. This solves a conjecture stated in [1]. A general formula for estimating the norm of self-adjoint tridiagonal infinite matrices is also derived.

## How to cite

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Szwarc, Ryszard. "Norm estimates of discrete Schrödinger operators." Colloquium Mathematicae 76.1 (1998): 153-160. <http://eudml.org/doc/210547>.

@article{Szwarc1998,
abstract = {Harper’s operator is defined on $\ell ^2(\{Z\})$ by $H\_\theta \xi (n) = \xi (n+1) + \xi (n-1) + 2\cos n\theta \, \xi (n),$ where $\theta \! \in \![0,\pi ]$. We show that the norm of $\Vert H_\theta \Vert$ is less than or equal to $2\sqrt\{2\}$ for $\pi /2 \le \theta \le \pi$. This solves a conjecture stated in [1]. A general formula for estimating the norm of self-adjoint tridiagonal infinite matrices is also derived.},
author = {Szwarc, Ryszard},
journal = {Colloquium Mathematicae},
keywords = {norm estimate; Harper's operator; difference operator},
language = {eng},
number = {1},
pages = {153-160},
title = {Norm estimates of discrete Schrödinger operators},
url = {http://eudml.org/doc/210547},
volume = {76},
year = {1998},
}

TY - JOUR
AU - Szwarc, Ryszard
TI - Norm estimates of discrete Schrödinger operators
JO - Colloquium Mathematicae
PY - 1998
VL - 76
IS - 1
SP - 153
EP - 160
AB - Harper’s operator is defined on $\ell ^2({Z})$ by $H_\theta \xi (n) = \xi (n+1) + \xi (n-1) + 2\cos n\theta \, \xi (n),$ where $\theta \! \in \![0,\pi ]$. We show that the norm of $\Vert H_\theta \Vert$ is less than or equal to $2\sqrt{2}$ for $\pi /2 \le \theta \le \pi$. This solves a conjecture stated in [1]. A general formula for estimating the norm of self-adjoint tridiagonal infinite matrices is also derived.
LA - eng
KW - norm estimate; Harper's operator; difference operator
UR - http://eudml.org/doc/210547
ER -

## References

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1. [1] C. Béguin, A. Valette and A. Żuk, z On the spectrum of a random walk on the discrete Heisenberg group and the norm of Harper's operator, J. Geom. Phys. 21 (1997), 337-356. Zbl0871.60053
2. [2] T. Chihara, z An Introduction to Orthogonal Polynomials, Math. Appl. 13, Gordon and Breach, New York, 1978. Zbl0389.33008
3. [3] P. R. Halmos and V. S. Sunder, z Bounded Integral Operators on ${L}^{2}$ Spaces, Springer, Berlin, 1978. Zbl0389.47001
4. [4] P. Lancaster, z Theory of Matrices, Academic Press, New York, 1969. Zbl0186.05301

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