Analysis and Efficient Computation for Nonlinear Eigenvalue Problems in Quantum Physics and Chemistry

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