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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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[1] 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
[2] Adami, R., Golse, F., Teta, A.: Rigorous derivation of the cubic NLS in dimension one. Preprint: mp-arc 05-211 · Zbl 1118.81021
[3] 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
[4] Bourgain, J.: Global Solutions of Nonlinear Schrödinger Equations. American Mathematical Society Colloquium Publications, vol. 46. American Mathematical Society, Providence, RI (1999) · Zbl 0933.35178
[5] Cazenave, T.: Semilinear Schrödinger Equations. Courant Lect. Notes Math., vol. 10. American Mathematical Society, Providence, RI (2003) · Zbl 1055.35003
[6] Davies, E.B.: The functional calculus. J. Lond. Math. Soc. (2) 52(1), 166–176 (1995) · Zbl 0858.47012
[7] 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
[8] 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
[9] 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
[10] 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
[11] 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
[12] 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)
[13] Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin, New York (1977) · Zbl 0361.35003
[14] 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
[15] Hepp, K.: The classical limit for quantum mechanical correlation functions. Commun. Math. Phys. 35, 265–277 (1974)
[16] Lieb, E.H., Seiringer, R.: Proof of Bose–Einstein condensation for dilute trapped gases. Phys. Rev. Lett. 88, 170409 (2002) · Zbl 1041.81107
[17] 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
[18] 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
[19] Reed, M., Simon, B.: Methods of Mathematical Physics. Vol. I. Academic Press, New-York, London (1975) · Zbl 0308.47002
[20] Rudin, W.: Functional analysis. McGraw-Hill Series in Higher Mathematics, McGraw-Hill Book Co., New York (1973) · Zbl 0253.46001
[21] Salmhofer, M.: Renormalization. An Introduction. Text and Monograph in Physics. Springer, Berlin (1999) · Zbl 0913.00014
[22] Spohn, H.: Kinetic equations from hamiltonian dynamics. Rev. Mod. Phys. 52(3), 569–615 (1980) · Zbl 0465.76069
[23] Spruch, L., Rosenberg, L.: Upper bounds on scattering lengths for static potentials. Phys. Rev. 116(4), 1034–1040 (1959) · Zbl 0089.21304
[24] 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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