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

##### 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
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