Derivation of the cubic nonlinear Schrödinger equation from quantum dynamics of many-body systems. (English) Zbl 1123.35066

Erdős, László; Schlein, Benjamin; Yau, Horng-Tzer

Invent. Math. 167, No. 3, 515-614 (2007).

A system of \(N\) interacting bosons in \(d=3\) dimensions is considered. The state space of the \(N\)-boson systems is \(L_{s}^2(\mathbb{R}^{3N},d\mathbf{x})\), its subspace \(L^2(\mathbb{R}^{3N},d\mathbf{x})\) contains all functions symmetric with respect to permutations of \(N\) particles. For a smooth compactly supported, non-negative symmetric function \(V(x)=V(-x)\) the rescaling of \(V\) is defined by the equality \(V_N(x):=N^{3\beta}V(N^\beta x)\), \(\beta\geq 0\). The Hamiltonian with pair interaction \(\frac{1}{N}V_N(x_i-x_j)\) is given by the non-negative selfadjoint operator \(H_N=-\sum_{j=1}^N\Delta_h+\frac{1}{N}\sum_{i<j}V_N(x_i-x_j)\), acting on \(L_s^2(\mathbb{R}^{3N},d\mathbf{x})\).
The wave function \(\psi_{N,t}\) at time \(t\) satisfies the Schrödinger equation \(i\partial_t\psi_{N,t}=H_N\psi_{N,t}\) with initial condition \(\psi_{N,0}\), which conserves the energy, the \(L^2\) norm and the permutation symmetry of the wave function. Instead of the wave function \(\psi_{N,t}\) the corresponding density matrix \(\gamma_{N,t}\) is introduced, which is defined as the orthogonal projection onto \(\psi_{N,t}\) in the space \(L^2(\mathbb{R}^{3N},d\mathbf{x})\), i.e. \(\gamma_{N,t}=\pi\psi_{N,t}\). By definition, density matrices are non-negative trace class operators \(\gamma^k\geq 0\), acting on \(L^2(\mathbb{R}^{3k})\) with permutational symmetry. The two-body potential of this model is \(U=N^{-1}V_N\). By scaling the scattering length \(a_U=O(N^{-1})\) the range of interaction \(r_U\) is of order \(O(N^{-\beta})\).
The main result of the article is presented in Theorem 1.1 for the case \(0<\beta<1\), which states that the time evolution of the one-particle density matrix is given by a cubic nonlinear Schrödinger equation, provided \(0<\beta<\frac{1}{2}\). The same result is expected for all \(0<\beta<1\), however the regime \(\beta\geq\frac{1}{2}\) is an open problem.

Reviewer: Boris V. Loginov (Ul’yanovsk)

MSC:

35Q55	NLS equations (nonlinear Schrödinger equations)
81Q15	Perturbation theories for operators and differential equations in quantum theory
81V70	Many-body theory; quantum Hall effect
81T18	Feynman diagrams

Keywords:

systems of interacting bosons; Hamiltonian with pair interaction; scaling limit; wave function; density matrix; wave function; density matrix

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