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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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##### References:
  Adami, R., Bardos, C., Golse, F., Teta, A.: Towards a rigorous derivation of the cubic nonlinear Schrödinger equation in dimension one. Asymptot. Anal. 40(2), 93–108 (2004) · Zbl 1069.35082  Adami, R., Golse, F., Teta, A.: Rigorous derivation of the cubic NLS in dimension one. Preprint: mp-arc 05-211 · Zbl 1118.81021  Bardos, C., Golse, F., Mauser, N.: Weak coupling limit of the N-particle Schrödinger equation. Methods Appl. Anal. 7, 275–293 (2000) · Zbl 1003.81027  Bourgain, J.: Global Solutions of Nonlinear Schrödinger Equations. American Mathematical Society Colloquium Publications, vol. 46. American Mathematical Society, Providence, RI (1999) · Zbl 0933.35178  Cazenave, T.: Semilinear Schrödinger Equations. Courant Lect. Notes Math., vol. 10. American Mathematical Society, Providence, RI (2003) · Zbl 1055.35003  Davies, E.B.: The functional calculus. J. Lond. Math. Soc. (2) 52(1), 166–176 (1995) · Zbl 0858.47012  Elgart, A., Erdos, L., Schlein, B., Yau, H.-T.: Gross–Pitaevskii equation as the mean filed limit of weakly coupled bosons. Arch. Rat. Mech. Anal. 179(2), 265–283 (2006) · Zbl 1086.81035  Elgart, A., Schlein, B.: Mean field dynamics of boson stars. To appear in Commun. Pure Appl. Math. Preprint arXiv:math-ph/0504051 · Zbl 1113.81032  Erdos, L., Schlein, B., Yau, H.-T.: Derivation of the Gross–Pitaevskii equation for the dynamics of Bose–Einstein condensate. Preprint arXiv:math-ph/0410005  Erdos, L., Yau, H.-T.: Linear Boltzmann equation as the weak coupling limit of the random Schrödinger equation. Commun. Pure Appl. Math. 53, 667–735 (2000) · Zbl 1028.82010  Erdos, L., Yau, H.-T.: Derivation of the nonlinear Schrödinger equation from a many body Coulomb system. Adv. Theor. Math. Phys. 5(6), 1169–1205 (2001) · Zbl 1014.81063  Fröhlich, J., Lenzmann, E.: Mean-Field Limit of Quantum Bose Gases and Nonlinear Hartree Equation. Séminaire Équations aux Dérivées Partielles. 2003–2004, Exp. No. XIX, École Polytech., Palaiseau (2004)  Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, New York (1977) · Zbl 0361.35003  Ginibre, J., Velo, G.: The classical field limit of scattering theory for non-relativistic many-boson systems, I and II. Commun. Math. Phys. 66, 37–76 (1979) and 68, 45–68 (1979) · Zbl 0443.35067  Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974)  Lieb, E.H., Seiringer, R.: Proof of Bose–Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88, 170409 (2002) · Zbl 1041.81107  Lieb, E.H., Seiringer, R., Solovej, J.-P., Yngvason, J.: The mathematics of the Bose gas and its condensation. Oberwolfach Seminars, Birkhäuser, Basel (2005) · Zbl 1104.82012  Lieb, E.H., Yngvason, J.: The ground state energy of a dilute Bose gas. Differential Equations and Mathematical Physics, University of Alabama, Birmingham (1999), Weikard, R., Weinstein, G. (eds.), pp. 295–306. Am. Math. Soc./Internat. Press, Providence, RI (2000) · Zbl 1056.82001  Reed, M., Simon, B.: Methods of Mathematical Physics. Vol. I. Academic Press, New-York, London (1975) · Zbl 0308.47002  Rudin, W.: Functional analysis. McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York (1973) · Zbl 0253.46001  Salmhofer, M.: Renormalization. An Introduction. Text and Monograph in Physics. Springer, Berlin (1999) · Zbl 0913.00014  Spohn, H.: Kinetic equations from hamiltonian dynamics. Rev. Mod. Phys. 52(3), 569–615 (1980) · Zbl 0465.76069  Spruch, L., Rosenberg, L.: Upper bounds on scattering lengths for static potentials. Phys. Rev. 116(4), 1034–1040 (1959) · Zbl 0089.21304  Tao, T.: Local and global analysis of nonlinear dispersive and wave equations. http://www.math.ucla.edu/ao/preprints/books.html · Zbl 1106.35001
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