# Norm estimates of discrete Schrödinger operators

Colloquium Mathematicae (1998)

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

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topSzwarc, 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

top- [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] T. Chihara, z An Introduction to Orthogonal Polynomials, Math. Appl. 13, Gordon and Breach, New York, 1978. Zbl0389.33008
- [3] P. R. Halmos and V. S. Sunder, z Bounded Integral Operators on ${L}^{2}$ Spaces, Springer, Berlin, 1978. Zbl0389.47001
- [4] P. Lancaster, z Theory of Matrices, Academic Press, New York, 1969. Zbl0186.05301

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