Weizhu Bao Department of Mathematics & Center of Computational Science and Engineering

1. Analysis and Efficient Computation for Nonlinear Eigenvalue Problems in Quantum Physics and Chemistry Weizhu Bao Department of Mathematics & Center of Computational Science and Engineering National University of Singapore Email: bao@math.nus.edu.sg URL: http://www.math.nus.edu.sg/~bao Collaborators: Fong Ying Lim (IHPC, Singapore), Yanzhi Zhang (FSU) Ming-Huang Chai (NUSHS); Yongyong Cai (NUS)

2. Outline • Motivation • Singularly perturbed nonlineareigenvalue problems • Existence, uniqueness & nonexistence • Asymptotic approximations • Numerical methods & results • Extension to systems • Conclusions

3. Motivation: NLS • The nonlinear Schrodinger (NLS) equation • t : time & : spatial coordinate (d=1,2,3) • : complex-valued wave function • : real-valued external potential • : interaction constant • =0: linear; >0: repulsive interaction • <0: attractive interaction

4. Motivation • In quantum physics & nonlinear optics: • Interaction between particles with quantum effect • Bose-Einstein condensation (BEC): bosons at low temperature • Superfluids: liquid Helium, • Propagation of laser beams, ……. • In plasma physics; quantum chemistry; particle physics; biology; materials science (DFT, KS theory,…); …. • Conservation laws

5. Motivation • Stationary states (ground & excited states) • Nonlinear eigenvalue problems: Find • Time-independent NLS or Gross-Pitaevskii equation (GPE): • Eigenfunctions are • Orthogonal in linear case & Superposition is valid for dynamics!! • Not orthogonal in nonlinear case !!!! No superposition for dynamics!!!

6. Motivation • The eigenvalue is also called as chemical potential • With energy • Special solutions • Soliton in 1D with attractive interaction • Vortex states in 2D

7. Motivation • Ground state: Non-convex minimization problem • Euler-Lagrange equation  Nonlinear eigenvalue problem • Theorem(Lieb, etc, PRA, 02’) • Existence d-dimensions (d=1,2,3): • Positive minimizer is unique in d-dimensions (d=1,2,3)!! • No minimizer in 3D (and 2D) when • Existence in 1D for both repulsive & attractive • Nonuniquness in attractive interaction – quantum phase transition!!!!

8. Symmetry breaking in ground state • Attractive interaction with double-well potential

9. Motivation • Excited states: • Open question: (Bao & W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, Bull Int. Math, 06’) • Continuous normalized gradient flow: • Mass conservation & energy diminishing

10. Singularly Perturbed NEP • For bounded with box potential for • Singularly perturbed NEP • Eigenvalue or chemical potential • Leading asymptotics of the previous NEP

11. Singularly Perturbed NEP • For whole space with harmonic potential for • Singularly perturbed NEP • Eigenvalue or chemical potential • Leading asymptotics of the previous NEP

12. General Form of NEP • Eigenvalue or chemical potential • Energy • Three typical parameter regimes: • Linear: • Weakly interaction: • Strongly repulsive interaction:

13. Box Potential in 1D • The potential: • The nonlinear eigenvalue problem • Case I: no interaction, i.e. • A complete set of orthonormal eigenfunctions

14. Box Potential in 1D • Ground state & its energy: • j-th-excited state & its energy • Case II: weakly interacting regime, i.e. • Ground state & its energy: • j-th-excited state & its energy

15. Box Potential in 1D • Case III: Strongly interacting regime, i.e. • Thomas-Fermi approximation, i.e. drop the diffusion term • Boundary condition is NOT satisfied, i.e. • Boundary layer near the boundary

16. Box Potential in 1D • Matched asymptotic approximation • Consider near x=0, rescale • We get • The inner solution • Matched asymptotic approximation for ground state

17. Box Potential in 1D • Approximate energy • Asymptotic ratios: • Width of the boundary layer:

18. Box Potential in 1D • Matched asymptotic approximation for excited states • Approximate chemical potential & energy • Boundary layers • Interior layers

19. Harmonic Oscillator Potential in 1D • The potential: • The nonlinear eigenvalue problem • Case I: no interaction, i.e. • A complete set of orthonormal eigenfunctions

20. Harmonic Oscillator Potential in 1D • Ground state & its energy: • j-th-excited state & its energy • Case II: weakly interacting regime, i.e. • Ground state & its energy: • j-th-excited state & its energy

21. Harmonic Oscillator Potential in 1D • Case III: Strongly interacting regime, i.e. • Thomas-Fermi approximation, i.e. drop the diffusion term • No boundary and interior layer • It is NOT differentiable at

22. Harmonic Oscillator Potential in 1D • Thomas-Fermi approximation for first excited state • Jump at x=0! • Interior layer at x=0

23. Harmonic Oscillator Potential in 1D • Matched asymptotic approximation • Width of interior layer:

24. Thomas-Fermi (or semiclassical) limit • In 1D with strongly repulsive interaction • Box potential • Harmonic potential • In 1D with strongly attractive interaction

25. Numerical methods • Runge-Kutta method: (M. Edwards and K. Burnett, Phys. Rev. A, 95’) • Analytical expansion: (R. Dodd, J. Res. Natl. Inst. Stan., 96’) • Explicit imaginary time method: (S. Succi, M.P. Tosi et. al., PRE, 00’) • Minimizing by FEM: (Bao & W. Tang, JCP, 02’) • Normalized gradient flow:(Bao & Q. Du, SIAM Sci. Comput., 03’) • Backward-Euler + finite difference (BEFD) • Time-splitting spectral method (TSSP) • Gauss-Seidel iteration method: (W.W. Lin et al., JCP, 05’) • Continuation method: W. W. Lin, etc., C. S. Chien, etc

26. Imaginary time method • Idea: Steepest decent method + Projection • The first equation can be viewed as choosing in GPE • For linear case: (Bao & Q. Du, SIAM Sci. Comput., 03’) • For nonlinear case with small time step, CNGF

27. Normalized gradient glow • Idea: letting time step go to 0 (Bao & Q. Du, SIAM Sci. Comput., 03’) • Energy diminishing • Numerical Discretizations • BEFD: Energy diminishing & monotone (Bao & Q. Du, SIAM Sci. Comput., 03’) • TSSP: Spectral accurate with splitting error (Bao & Q. Du, SIAM Sci. Comput., 03’) • BESP: Spectral accuracy in space & stable (Bao, I. Chern & F. Lim, JCP, 06’) • Uniformly convergent method (Bao&Chai, Comm. Comput. Phys, 07’)

28. Ground states • Numerical results (Bao&W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, 06’) • Box potential • 1D-states 1D-energy 2D-surface 2D-contour • Harmonic oscillator potential: • 1D2D-surface 2D-contour • Optical lattice potential: • 1D2D-surface 2D-contour 3D next

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40. Extension to rotating BEC • BEC in rotation frame(Bao, H. Wang&P. Markowich,Comm. Math. Sci., 04’) • Ground state: existence & uniqueness, quantized vortex • In 2D: In a rotational frame &With a fast rotation & optical lattice • In 3D: With a fast rotation next

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45. Extension to two-component • Two-component (Bao, MMS, 04’) • Ground state • Existence & uniqueness • Quantized vortices & fractional index • Numerical methods & results: Crarter & domain wall

46. Results • Theorem • Assumptions • No rotation & Confining potential • Repulsive interaction • Results • Existence & Positive minimizer is unique • No minimizer in 3D when • Nonuniquness in attractive interaction in 1D • Quantum phase transition in rotating frame

48. Two-component with an external driving field • Two-component (Bao & Cai, 09’) • Ground state • Existence & uniqueness (Bao & Cai, 09’) • Limiting behavior & Numerical methods • Numerical results: Crarter & domain wall

49. Theorem (Bao & Cai, 09’) • No rotation & confining potential & • Existence of ground state!! • Uniqueness in the form under • At least two different ground states under– quantum phase transition • Limiting behavior

50. Extension to spin-1 • Spin-1 BEC (Bao & Wang, SINUM, 07’; Bao & Lim, SISC 08’, PRE 08’) • Continuous normalized gradient flow(Bao & Wang, SINUM, 07’) • Normalized gradient flow (Bao & Lim, SISC 08’) • Gradient flow + third projection relation