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Quantum Ergodicity and Orthonormal Basis of States

Didier Robert
Département de Mathématiques, Laboratoire Jean Leray, CNRS-UMR 6629 Université de Nantes
Wednesday, April 11, 2018 - 16:30
Our aim in this talk is to give an introduction to a domain of Mathematical Physics often named "Quantum Chaos”. 
A natural physical definition of this concept is always problematic (what means instability in quantum mechanics?).
In mathematics the central problem is to understand the specific properties of states of a quantum system when the corresponding classical system is chaotic; typically Ergodic in L. Boltzman sense [1871].The main mathematical result connecting the classical world and the quantum world is the famous Shnirelman Theorem [1974]. The result was extended later in different settings including billiards (P. Gérard-E. Leichtnam [1993]). 

It was noticed by S. Zelditch [1992] that some classical integrable systems may have random quantum states with a chaotic behaviour.
Finally we shall mention a beautiful result by G. Rivière [2013] proving a quantum analogous of the general Birkhoff Ergodic Theorem. 


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