Norm estimates of discrete Schrödinger operators

Ryszard Szwarc

Colloquium Mathematicae (1998)

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

Abstract

top
Harper’s operator is defined on ℓ 2 ( Z ) by H θ ξ ( n ) = ξ ( n + 1 ) + ξ ( n - 1 ) + 2 cos n θ ξ ( n ) , where θ ∈ [ 0 , π ] . We show that the norm of ∥ H θ ∥ is less than or equal to 2 2 for π / 2 ≤ θ ≤ π . 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

top

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

top
  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

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.