# zbMATH — the first resource for mathematics

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
Full Text:
##### 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
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.