# Systemes Integrables en Mecanique Classique et Quantique

Abstract : Our main motivation here is to develop the methods of study of the classical integrable systems which can be directly generalized to the quantum case. For this, we start by using the algebro-geometric tools and the ideas of the method of separation of variables to explicitly construct a matrix model of the Jacobian variety of a spectral curve of any order $N$, thus generalizing a construction previously known only for hyperelliptic case ($N=2$). Using this model we investigate the structure of the singular cohomologies of the affine Jacobian and we find a new formula for its Euler characteristic. By analyzing its behaviour we discover that the cohomologies are much more complicated in general case than in the hyperelliptic one. From the integrable systems point of view our main achievement is the discovery that the entire algebra of observables is generated by the action of some Hamiltonian vector fields from a finite number of the coefficients of the cohomologies of the highest degree. The importance of this observation is especially clear in the quantum case, to which all our results can be generalized in agreement with our initially stated motivation. Indeed, in this case such structure of the algebra implies that calculation of correlation functions of any observable is reduced to computing the correlation functions of only a finite number of the (deformed) highest degree cohomology coefficients. Finally, using the known results of the hyperelliptic case as well as the semi-classical limit considerations, we conjecture a simple formula for the scalar product on the Hilbert space in which the algebra of the quantum observables acts.
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https://tel.archives-ouvertes.fr/tel-00001770
Submitted on : Friday, October 4, 2002 - 4:26:11 PM
Last modification on : Wednesday, December 9, 2020 - 3:09:21 PM
Long-term archiving on: : Tuesday, September 11, 2012 - 6:00:30 PM

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• HAL Id : tel-00001770, version 1

### Citation

Vadim Zeitlin. Systemes Integrables en Mecanique Classique et Quantique. Mathématiques [math]. Université Paris-Diderot - Paris VII, 2002. Français. ⟨tel-00001770⟩

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