×

A priori estimates for many-body Hamiltonian evolution of interacting Boson system. (English) Zbl 1160.35357

Summary: We study the evolution of a many-particle system whose wave function obeys the \(N\)-body Schrödinger equation under Bose symmetry. The system Hamiltonian describes pairwise particle interactions in the absence of an external potential. We derive a priori dispersive estimates that express the overall repulsive nature of the particle interactions. These estimates hold for a wide class of two-body interaction potentials which are independent of the particle number, \(N\). We discuss applications of these estimates to the BBGKY hierarchy for reduced density matrices analyzed by Elgart, Erdős, Schlein and Yau.

MSC:

35B45 A priori estimates in context of PDEs
35Q40 PDEs in connection with quantum mechanics
76Y05 Quantum hydrodynamics and relativistic hydrodynamics
81V70 Many-body theory; quantum Hall effect
82C22 Interacting particle systems in time-dependent statistical mechanics
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] Berezin F. A., Method of Second Quantization (1966) · Zbl 0151.44001
[2] J. M. Blatt and V. F. Weisskopf, Theoretical Nuclear Physics (Wiley, 1952) pp. 74–76. · Zbl 0049.14005
[3] DOI: 10.1103/PhysRev.106.20 · Zbl 0077.23503 · doi:10.1103/PhysRev.106.20
[4] DOI: 10.1007/s00205-005-0388-z · Zbl 1086.81035 · doi:10.1007/s00205-005-0388-z
[5] DOI: 10.1002/cpa.20123 · Zbl 1122.82018 · doi:10.1002/cpa.20123
[6] DOI: 10.1007/s00222-006-0022-1 · Zbl 1123.35066 · doi:10.1007/s00222-006-0022-1
[7] DOI: 10.1103/PhysRevLett.98.040404 · doi:10.1103/PhysRevLett.98.040404
[8] DOI: 10.1007/BF02731494 · Zbl 0100.42403 · doi:10.1007/BF02731494
[9] DOI: 10.1063/1.1703944 · doi:10.1063/1.1703944
[10] DOI: 10.1103/PhysRev.105.767 · Zbl 0077.20905 · doi:10.1103/PhysRev.105.767
[11] DOI: 10.1103/PhysRev.105.776 · Zbl 0077.21001 · doi:10.1103/PhysRev.105.776
[12] DOI: 10.1007/s00220-008-0426-4 · Zbl 1147.37034 · doi:10.1007/s00220-008-0426-4
[13] DOI: 10.1103/PhysRev.106.1135 · Zbl 0077.45003 · doi:10.1103/PhysRev.106.1135
[14] DOI: 10.1103/PhysRev.105.1119 · doi:10.1103/PhysRev.105.1119
[15] DOI: 10.1007/s00220-006-1524-9 · Zbl 1233.82004 · doi:10.1007/s00220-006-1524-9
[16] Lieb E. H., The Mathematics of the Bose Gas and Its Condensation (2005) · Zbl 1104.82012
[17] DOI: 10.1103/PhysRevA.61.043602 · doi:10.1103/PhysRevA.61.043602
[18] DOI: 10.1007/s002200100533 · Zbl 0996.82010 · doi:10.1007/s002200100533
[19] DOI: 10.1016/0022-1236(78)90073-3 · Zbl 0395.35070 · doi:10.1016/0022-1236(78)90073-3
[20] Pitaevskii L. P., Soviet Phys. JETP 13 pp 451–
[21] DOI: 10.1103/RevModPhys.52.569 · doi:10.1103/RevModPhys.52.569
[22] DOI: 10.1007/978-3-642-84371-6 · doi:10.1007/978-3-642-84371-6
[23] DOI: 10.1090/cbms/106 · doi:10.1090/cbms/106
[24] DOI: 10.1103/PhysRev.115.1390 · Zbl 0096.23701 · doi:10.1103/PhysRev.115.1390
[25] DOI: 10.1063/1.1724205 · Zbl 0151.44907 · doi:10.1063/1.1724205
[26] DOI: 10.1103/PhysRevA.58.1465 · doi:10.1103/PhysRevA.58.1465
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.