Generic KAM Hamiltonians are not quantum ergodic. (English) Zbl 1528.37052
The author investigates a converse to quantum ergodicity, namely generic failure of quantum ergodicity for the quantization of a KAM perturbation of a completely integrable classical Hamiltonian. More precisely, for a KAM perturbation of a completely integrable Kolmogorov nondegenerate Gevrey smooth classical Hamiltonian (e.g., a completely integrable Schrödinger operator \(-\Delta +V\)), the main result establishes failure of quantum ergodicity for almost every perturbation size parameter.
The paper builds on results by G. Popov [Mat. Contemp. 26, 87–107 (2004; Zbl 1074.37031); Ergodic Theory Dyn. Syst. 24, No. 5, 1753–1786 (2004; Zbl 1088.37030); Ann. Henri Poincaré 1, No. 2, 249–279 (2000; Zbl 1002.37028); Ann. Henri Poincaré 1, No. 2, 223–248 (2000; Zbl 0970.37050)], who constructed a normal form for Gevrey smooth Hamiltonians (which is refined here) and who proved that perturbed elliptic operators in a closely related setting have quasimodes which are microlocalized in phase space near suitable KAM tori. The author extends the results from microlocalization of quasimodes to eigenfunctions by controlling the spectral concentration.
The paper builds on results by G. Popov [Mat. Contemp. 26, 87–107 (2004; Zbl 1074.37031); Ergodic Theory Dyn. Syst. 24, No. 5, 1753–1786 (2004; Zbl 1088.37030); Ann. Henri Poincaré 1, No. 2, 249–279 (2000; Zbl 1002.37028); Ann. Henri Poincaré 1, No. 2, 223–248 (2000; Zbl 0970.37050)], who constructed a normal form for Gevrey smooth Hamiltonians (which is refined here) and who proved that perturbed elliptic operators in a closely related setting have quasimodes which are microlocalized in phase space near suitable KAM tori. The author extends the results from microlocalization of quasimodes to eigenfunctions by controlling the spectral concentration.
Reviewer: Marius Lemm (Cambridge)
MSC:
| 37J40 | Perturbations of finite-dimensional Hamiltonian systems, normal forms, small divisors, KAM theory, Arnol’d diffusion |
| 37J35 | Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests |
| 81S08 | Canonical quantization |
| 81Q20 | Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory |
| 58J51 | Relations between spectral theory and ergodic theory, e.g., quantum unique ergodicity |
References:
| [1] | 10.1007/978-1-4612-1037-5 · doi:10.1007/978-1-4612-1037-5 |
| [2] | 10.1007/978-1-4757-2063-1 · doi:10.1007/978-1-4757-2063-1 |
| [3] | 10.1112/jlms/s2-22.3.495 · Zbl 0419.26010 · doi:10.1112/jlms/s2-22.3.495 |
| [4] | 10.1007/BF01390202 · Zbl 0449.53040 · doi:10.1007/BF01390202 |
| [5] | 10.1007/BF01209296 · Zbl 0592.58050 · doi:10.1007/BF01209296 |
| [6] | 10.1007/978-0-8176-8108-1 · Zbl 1214.35092 · doi:10.1007/978-0-8176-8108-1 |
| [7] | 10.24033/asens.2217 · Zbl 1297.58007 · doi:10.24033/asens.2217 |
| [8] | 10.1007/978-1-4899-6762-6_12 · doi:10.1007/978-1-4899-6762-6_12 |
| [9] | 10.1088/1361-6544/aa776f · Zbl 1396.37046 · doi:10.1088/1361-6544/aa776f |
| [10] | 10.1090/S0002-9939-09-09849-9 · Zbl 1171.81008 · doi:10.1090/S0002-9939-09-09849-9 |
| [11] | 10.4007/annals.2010.171.605 · Zbl 1196.58014 · doi:10.4007/annals.2010.171.605 |
| [12] | 10.1007/BF01215225 · Zbl 0624.58039 · doi:10.1007/BF01215225 |
| [13] | 10.24033/bsmf.2705 · Zbl 1375.37009 · doi:10.24033/bsmf.2705 |
| [14] | 10.1007/978-3-662-12678-3 · doi:10.1007/978-3-662-12678-3 |
| [15] | 10.1007/BF01390041 · Zbl 0431.53003 · doi:10.1007/BF01390041 |
| [16] | ; Kolmogorov, A. N., On conservation of conditionally periodic motions for a small change in Hamilton’s function, Dokl. Akad. Nauk SSSR (N.S.), 98, 527 (1954) · Zbl 0056.31502 |
| [17] | 10.1088/0951-7715/18/1/015 · Zbl 1109.81037 · doi:10.1088/0951-7715/18/1/015 |
| [18] | ; Moser, Jürgen, A rapidly convergent iteration method and non-linear partial differential equations, I, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 20, 2, 265 (1966) · Zbl 0144.18202 |
| [19] | 10.1007/BF01399536 · Zbl 0149.29903 · doi:10.1007/BF01399536 |
| [20] | 10.1007/BF01456977 · Zbl 0633.46033 · doi:10.1007/BF01456977 |
| [21] | 10.1007/PL00001004 · Zbl 0970.37050 · doi:10.1007/PL00001004 |
| [22] | 10.1007/PL00001005 · Zbl 1002.37028 · doi:10.1007/PL00001005 |
| [23] | ; Popov, Georgi, KAM theorem and quasimodes for Gevrey Hamiltonians, Mat. Contemp., 26, 87 (2004) · Zbl 1074.37031 |
| [24] | 10.1017/S0143385704000458 · Zbl 1088.37030 · doi:10.1017/S0143385704000458 |
| [25] | 10.1090/pspum/069/1858551 · Zbl 0999.37053 · doi:10.1090/pspum/069/1858551 |
| [26] | ; Shnirelman, A. I., Ergodic properties of eigenfunctions, Uspekhi Mat. Nauk, 29, 6(180), 181 (1974) · Zbl 0324.58020 |
| [27] | 10.1090/conm/320/05622 · Zbl 1044.58038 · doi:10.1090/conm/320/05622 |
| [28] | 10.1215/S0012-7094-87-05546-3 · Zbl 0643.58029 · doi:10.1215/S0012-7094-87-05546-3 |
| [29] | 10.1090/S0002-9939-03-07298-8 · Zbl 1055.58016 · doi:10.1090/S0002-9939-03-07298-8 |
| [30] | 10.1090/gsm/138 · doi:10.1090/gsm/138 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
