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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.sgbao@math.nus.edu.sg URL: http://www.math.nus.edu.sg/~baohttp://www.math.nus.edu.sg/~bao Collaborators: Fong Ying Lim (IHPC, Singapore), Yanzhi Zhang (FSU) Ming-Huang Chai (NUSHS); Yongyong Cai (NUS)
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Outline Motivation Singularly perturbed nonlinear eigenvalue problems Existence, uniqueness & nonexistence Asymptotic approximations Numerical methods & results Extension to systems Conclusions
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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
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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
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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!!!
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Motivation The eigenvalue is also called as chemical potential –With energy Special solutions –Soliton in 1D with attractive interaction –Vortex states in 2D
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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!!!!
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Symmetry breaking in ground state Attractive interaction with double-well potential
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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
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Singularly Perturbed NEP For bounded with box potential for –Singularly perturbed NEP –Eigenvalue or chemical potential –Leading asymptotics of the previous NEP
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Singularly Perturbed NEP For whole space with harmonic potential for –Singularly perturbed NEP –Eigenvalue or chemical potential –Leading asymptotics of the previous NEP
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General Form of NEP –Eigenvalue or chemical potential –Energy Three typical parameter regimes: –Linear: –Weakly interaction: –Strongly repulsive interaction:
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Box Potential in 1D The potential: The nonlinear eigenvalue problem Case I: no interaction, i.e. –A complete set of orthonormal eigenfunctions
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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
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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
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Box Potential in 1D –Matched asymptotic approximation Consider near x=0, rescale We get The inner solution Matched asymptotic approximation for ground state
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Box Potential in 1D Approximate energy Asymptotic ratios: Width of the boundary layer:
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Box Potential in 1D Matched asymptotic approximation for excited states Approximate chemical potential & energy Boundary layers Interior layers
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Harmonic Oscillator Potential in 1D The potential: The nonlinear eigenvalue problem Case I: no interaction, i.e. –A complete set of orthonormal eigenfunctions
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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
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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
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Harmonic Oscillator Potential in 1D –Thomas-Fermi approximation for first excited state Jump at x=0! Interior layer at x=0
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Harmonic Oscillator Potential in 1D –Matched asymptotic approximation –Width of interior layer:
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Thomas-Fermi (or semiclassical) limit In 1D with strongly repulsive interaction –Box potential –Harmonic potential In 1D with strongly attractive interaction
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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
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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
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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’)
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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
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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
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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
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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
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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
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Quantum phase transition Ferromagnetic g s <0 Antiferromagnetic g s > 0
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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) –2005@Univ. Stuttgart, Germany –PRL, 94 (2005) 160401 Big-wave in theoretical study A. Griesmaier,et al., PRL, 94 (2005)160401
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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), 1791-1797 –S. Yi & L. You, PRA 61 (2001), 041604(R); D. H. J. O’Dell, PRL 92 (2004), 250401
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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 !!
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A New Formulation Using the identity (O’Dell et al., PRL 92 (2004), 250401, Parker et al., PRA 79 (2009), 013617) Dipole-dipole interaction becomes Gross-Pitaevskii-Poisson type equations (Bao,Cai & Wang, JCP, 10’) Energy
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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:
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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.
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