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 URL: Collaborators: Fong Ying Lim (IHPC, Singapore), Yanzhi Zhang (FSU) Ming-Huang Chai (NUSHS); Yongyong Cai (NUS)

Outline Motivation Singularly perturbed nonlinear eigenvalue problems Existence, uniqueness & nonexistence Asymptotic approximations Numerical methods & results Extension to systems Conclusions

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

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

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!!!

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

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!!!!

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Harmonic Oscillator Potential in 1D –Matched asymptotic approximation –Width of interior layer:

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

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

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

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’)

Ground states Numerical results (Bao&W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, 06’) –Box potential 1D-states 1D-energy 2D-surface 2D-contour1D-states 1D-energy 2D-surface 2D-contour –Harmonic oscillator potential: 1D 2D-surface 2D-contour1D2D-surface 2D-contour –Optical lattice potential: 1D 2D-surface 2D-contour 3D1D2D-surface 2D-contour 3D next

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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 latticerotationalfastoptical lattice –In 3D: With a fast rotationfast next

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

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

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

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

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

Quantum phase transition Ferromagnetic g s <0 Antiferromagnetic g s > 0

Dipolar Quantum Gas Experimental setup –Molecules meet to form dipoles –Cool down dipoles to ultracold –Hold in a magnetic trap –Dipolar condensation –Degenerate dipolar quantum gas Experimental realization –Chroimum (Cr52) Stuttgart, Germany –PRL, 94 (2005) Big-wave in theoretical study A. Griesmaier,et al., PRL, 94 (2005)160401

Mathematical Model Gross-Pitaevskii equation (re-scaled) –Trap potential –Interaction constants –Long-range dipole-dipole interaction kernel References: –L. Santos, et al. PRL 85 (2000), –S. Yi & L. You, PRA 61 (2001), (R); D. H. J. O’Dell, PRL 92 (2004),

Mathematical Model Mass conservation (Normalization condition) Energy conservation Long-range interaction kernel: –It is highly singular near the origin !! At singularity near the origin !! –Its Fourier transform reads No limit near origin in phase space !! Bounded & no limit at far field too !! Physicists simply drop the second singular term in phase space near origin!! Locking phenomena in computation !!

A New Formulation Using the identity (O’Dell et al., PRL 92 (2004), , Parker et al., PRA 79 (2009), ) Dipole-dipole interaction becomes Gross-Pitaevskii-Poisson type equations (Bao,Cai & Wang, JCP, 10’) Energy

Ground State Results Theorem (Existence, uniqueness & nonexistence) (Bao, Cai & Wang, JCP, 10’) –Assumptions –Results There exists a ground state if Positive ground state is uniqueness Nonexistence of ground state, i.e. –Case I: –Case II:

Conclusions Analytical study –Leading asymptotics of energy and chemical potential –Existence, uniqueness & quantum phase transition!! –Thomas-Fermi approximation –Matched asymptotic approximation –Boundary & interior layers and their widths Numerical study –Normalized gradient flow –Numerical results Extension to rotating, multi-component, spin-1, dipolar cases.