Norm estimates of discrete Schrödinger operators

• Autorzy: Ryszard Szwarc
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

norm estimate; Harper's operator; difference operator

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Colloquium Mathematicum
• Rocznik: 1998
• Tom: 76
• Numer: 1
• Strony: 153-160

Daty

wydano

1998

otrzymano

1997-09-04

poprawiono

1997-11-17

Twórcy

autor

Ryszard Szwarc

• Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

Bibliografia

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