Geodesic
Norm estimates of discrete Schrödinger operators
Colloquium Mathematicum, Tome 76 (1998) no. 1, pp. 153-160
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Harper's operator is defined on $\ell^2({\sym Z})$ 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 $\|H_\theta\|$ 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.
DOI : 10.4064/cm-76-1-153-160
Keywords: norm estimate, Harper's operator, difference operator 
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Ryszard Szwarc. Norm estimates of discrete Schrödinger operators. Colloquium Mathematicum, Tome 76 (1998) no. 1, pp. 153-160. doi: 10.4064/cm-76-1-153-160

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