Duke Mathematical Journal

Dispersionless Toda and Toeplitz operators

A. Bloch,F. Golse,T. Paul, and A. Uribe

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Abstract

In this paper we present some results on the dispersionless limit of the Toda lattice equations viewed as the semiclassical limit of an equation involving certain Toeplitz operators. We consider both nonperiodic and periodic boundary conditions. For the nonperiodic case the phase space is the Riemann sphere, while in the periodic case it is the torus $\mathbb {C}/\mathbb {Z}\sp 2$. In both cases we prove precise estimates on the dispersionless limit. In addition, we show that the Toda equations, although they are nonlinear, propagate a Toeplitz operator into an operator arbitrarily close to a Toeplitz operator as long as the Toda partial differential equation (PDE) (dispersionless limit) admits smooth solutions.

Article information

Source
Duke Math. J. Volume 117, Number 1 (2003), 157-196.

Dates
First available: 26 May 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1085598341

Mathematical Reviews number (MathSciNet)
MR1962785

Digital Object Identifier
doi:10.1215/S0012-7094-03-11713-5

Zentralblatt MATH identifier
1024.37047

Subjects
Primary: 37K60: Lattice dynamics [See also 37L60]
Secondary: 35P20: Asymptotic distribution of eigenvalues and eigenfunctions 47B35: Toeplitz operators, Hankel operators, Wiener-Hopf operators [See also 45P05, 47G10 for other integral operators; see also 32A25, 32M15]

Citation

Bloch, A.; Golse, F.; Paul, T.; Uribe, A. Dispersionless Toda and Toeplitz operators. Duke Mathematical Journal 117 (2003), no. 1, 157--196. doi:10.1215/S0012-7094-03-11713-5. http://projecteuclid.org/euclid.dmj/1085598341.


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