Combescure M., Robert D., Distribution of matrix elements and level spacings for classically chaotic systems, Ann. Inst. H. Poincar 61,4 (1994), 443-483. Home/Search Document Details and Download
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Quantum Chaos: A Brief First Visit - De Bièvre (Correct)
....the Schnirelman theorem is clearly only a rst step in the semi classical analysis of spectral problems associated to classically chaotic systems. Mixing does play a role when investigating the behaviour of o diagonal matrix elements of the form (k 6= j) k i H as explained in [Z2] Z3][CR][Bo] The role of exponential mixing is explored in [BonDB] An interesting question raised by the Schnirelman theorem is whether there are any exceptional sequences of eigenfunctions j N , for which the limit i H = f) exists but does not equal T 2 f(x)dx. Exercise 14.6. ....
Combescure M., Robert D., Distribution of matrix elements and level spacings for classically chaotic systems, Ann. Inst. H. Poincar 61,4 (1994), 443-483.
Egorov Theorems and equidistribution of eigenfunctions.. - De Bièvre.. (1996) (Correct)
....quantized observables between eigenstates and is commonly referred to as the Schnirelman Theorem. It has been proven in many cases [Z1, CdV, HMR, GL, Z3, Z4, DIG, Sa, BD, ZZ] If the system is in addition mixing, more can be inferred: in that case (most) off diagonal matrix elements tend to zero [Z2, CR]. We will be interested here in the classical limit of quantized, discontinuous, ergodic or mixing symplectic transformations of the two torus. Combining ideas of [Z4] and [ZZ] with the approach of [BD] we will show (Theorem 2) that, in this case, the equipartition result will follow provided one ....
M. Combescure, D. Robert, Distribution of matrix elements and level spacings for classical chaotic systems, Ann. Inst. H. Poincar'e 61, 4 (1994), 443-483.
Equipartition of the Eigenfunctions of Quantized.. - Bouzouina, De.. (1997) (1 citation) (Correct)
....hand, it permits to obtain additional results on the limit N 1 of off diagonal matrix elements OE N j(N) Op AW (f)OE N k(N) N) j(N) 6= k(N) which are related to the mixing properties of the classical map. Similar results should be accessible with our approach along the lines of [CR]. All the examples we considered so far are obtained by periodization of affine transformations on the plane. On the plane they are obtained by integrating the flow of quadratic Hamiltonians, and as such they are exactly solvable both classically and quantum mechanically. Although the situation is ....
M.Combescure, D.Robert, Distribution of matrix elements and level spacings for classically chaotic systems, Ann. Inst. H. Poincar'e 61,4 (1994), 443-483.
Semi Classical Sum Rules And Generalized Coherent States - Combescure, Robert (1994) (2 citations) Self-citation (Combescure) (Correct)
....with the stationary phase theorems, allow a simple derivation of various sum rules in the semi classical limit, with a precise estimate of the errors terms, and they can be applied to situations where the corresponding classical motion is chaotic. It is an alternative of our preceeding work [4], where similar results were obtained through the usual pseudo differential calculus. This approach, has several advantages and drawbacks connected with the use of the coherent states, that we shall stress in the course of the paper. 2 An elementary sum rule for diagonal, matrix elements. We ....
....E Gamma E j h ) g(0) 2 h) Gamman 1 f Z Sigma E doe E (z)a E (z) O( h)g (2.6) 8 h 2 [0; h T ] where Sigma E is the energy surface p 2 V (q) E, with oe E (z) the Liouville measure on it, and where aE (z) is the restriction of the symbol a(q; p) to Sigma E . Proof: As in [4], we introduce a C 1 0 (I cl ) function , and denote by A ( h) the self adjoint operator : A ( h) P ( h) A (P ( h) 2.7) and we have S(E) X E j 2I cl j ; A ( h) j g( E Gamma E j h ) O( h 1 ) 2.8) We then write X E j 2I cl j ; A ( h) j g( E Gamma E ....
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Combescure M., Robert D., Distribution of Matrix Elements and level spacings for classically chaotic systems, Nantes preprint (1994).
Chaos, Quantization And The Classical Limit On The Torus - De Bièvre (Correct)
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M. Combescure, D. Robert, Distribution of matrix elements and level spacings for classical chaotic systems, Ann. Inst. H. Poincar'e 61 4: 443-483 (1994)
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