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On the collapse and restoration of condensates in n dimensions in the mean‐field approximation
Author(s) -
Cederbaum Lorenz S.,
Moiseyev Nimrod,
Cederbaum Lorenz S.,
Moiseyev Nimrod
Publication year - 2004
Publication title -
israel journal of chemistry
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.908
H-Index - 54
eISSN - 1869-5868
pISSN - 0021-2148
DOI - 10.1560/uvhd-erlq-3ygn-vpph
Subject(s) - ansatz , mean field theory , virial theorem , field (mathematics) , nonlinear system , function (biology) , chemistry , operator (biology) , range (aeronautics) , mathematical physics , quantum mechanics , physics , mathematics , pure mathematics , biochemistry , repressor , evolutionary biology , galaxy , gene , transcription factor , biology , materials science , composite material
The mean‐field equation of N bosons in an external potential interacting via a short‐range δ‐function potential is studied. This equation is formally equivalent to the nonlinear Schrödinger equation. Virial theorems are derived and the number of possible solutions is investigated for various dimensions n. The unboundedness from below of the underlying mean‐field operator for attractive nonlinearity and the related collapse of the wave function are studied in n dimensions. Within the symmetry preserving mean‐field approximation, an ansatz to avoid the collapse by modifying the spherically symmetric external potential is discussed and analyzed. Other suggestions to stabilize the mean‐field condensate are briefly mentioned. Illustrative numerical examples of the collapse and its restoration are given. Detailed conclusions are drawn.