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Beyond Bogoliubov dynamics. (English) Zbl 1496.35332

The starting point of the investigations of the authors is the model describing bosons in the mean-field or Hartree regime. This is characterized by weak, long-range interactions. The authors construct a norm approximation such that all corrections to correlation functions and expectation values of bounded operators are given in terms of the two-point correlation functions of a quasifree state. The advantage of their construction that it reduces the complexity of the \(N\)-body problem and makes it possible to numerical calculate these quantities to arbitrary precision. Hence the computation of the higher-order corrections reduces to solve first the well-studied Hartree equation and second the Bogoliubov equation. The second is equivalent to solving a \(2 \times 2\) matrix differential equation. Moreover, the \(N\)-independent corrections fulfill a generalized form of Wick’s theorem.

MSC:

35Q40 PDEs in connection with quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
81Q05 Closed and approximate solutions to the Schrödinger, Dirac, Klein-Gordon and other equations of quantum mechanics
82C10 Quantum dynamics and nonequilibrium statistical mechanics (general)
81V70 Many-body theory; quantum Hall effect
35C20 Asymptotic expansions of solutions to PDEs
35B20 Perturbations in context of PDEs
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