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 SingaporeEmail: 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)

Outline

MotivationSingularly perturbed nonlinear eigenvalue problemsExistence, uniqueness & nonexistenceAsymptotic approximationsNumerical methods & resultsExtension to systemsConclusions

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

2 21( , ) ( ) | |

2ti x t V x

( R )dx

( , )x t

( )V x

0

4 ( 1)( . ., )sa Ne g

a

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 laws2 2 22

0 0

2 2 4

02

( ) : ( , ) ( ,0) ( ) : ( ) ( 1),

1( ) : ( , ) ( ) ( , ) ( , ) ( )

2

N x t d x x d x x d x N

E x t V x x t x t d x E

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!!!

2 2

2 2

1( ) ( ) ( ) ( ) | ( ) | ( ), R

2

( ) 0, ; : | (x) | 1

dx x V x x x x x

x x dx

( , ) s.t. ( , ) ( ) i tx t x e

Motivation

The eigenvalue is also called as chemical potential

– With energy

Special solutions– Soliton in 1D with attractive interaction– Vortex states in 2D

4( ) ( ) | (x) |2

E dx

2 2 41( ) [ | ( ) | ( )| ( ) | | ( ) | ]

2 2x V x x x dxE

( ) ( ) immx f r e

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!!!!

| |0 & lim ( )

xV x

( ) min ( ) | 1, | 0, ( )g xS

E E S E

cr0 ( 0)

Symmetry breaking in ground state

Attractive interaction with double-well potential2 2

2 2 2

1( ) ''( ) ( ) ( ) | ( ) | ( ), with | ( ) | 1

2

( ) ( ) & : positive 0 negative

x x V x x x x x dx

V x U x a

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

,,, 321

???????)()()(

)()()(

,,,

21

21

21

g

g

g

EEE

2 22

0 0

( (., ))1( , ) ( ) | | , 0,

2 || (., ) ||

( ,0) ( ) with || ( ) || 1.

t

tx t V x t

t

x x x

Singularly Perturbed NEP

For bounded with box potential for

– Singularly perturbed NEP

– Eigenvalue or chemical potential

– Leading asymptotics of the previous NEP

22 21

: , , | ( ) | 1x dx

22 2( ) ( ) | ( ) | ( ), ,

2

( ) 0,

x x x x x

x x

1

4

22 4

1( ) ( ) | (x) | (1)

2

1( ) | | (1), 0 1

2 2

E dx O

E dx O

( ) ( ) ( ) & ( ) ( ) ( ), 1O E E O

Singularly Perturbed NEP

For whole space with harmonic potential for

– Singularly perturbed NEP

– Eigenvalue or chemical potential

– Leading asymptotics of the previous NEP

21/ 2 / 4 1 /( 2) 2, ( ) ( ), , : | ( ) | 1d

d d dx x x x x dx

22 2( ) ( ) ( ) ( ) | ( ) | ( ),

2dx x V x x x x x

1

4

22 2 4

1( ) ( ) | (x) | (1)

2

1( ) ( ) | | | | (1), 0 1

2 2

d

d

E dx O

E V x dx O

1 1 /( 2) 1 /( 2)( ) ( ) ( ) ( ) & ( ) ( ) ( ), 1d d d dO O E E O

General Form of NEP

– Eigenvalue or chemical potential

– Energy

Three typical parameter regimes:– Linear: – Weakly interaction: – Strongly repulsive interaction:

22 2

2 2

( ) ( ) ( ) ( ) | ( ) | ( ), R2

( ) 0, ; : | ( ) | 1

dx x V x x x x x

x x x dx

4( ) ( ) | (x) |2

E dx

22 2 4( ) [ | ( ) | ( )| ( ) | | ( ) | ]

2 2x V x x x dxE

1& 0 1& | | 1

1& 0 1

Box Potential in 1D

The potential:The nonlinear eigenvalue problem

Case I: no interaction, i.e. – A complete set of orthonormal eigenfunctions

0, 0 1,( )

, otherwise.

xV x

22

12

0

( ) ( ) | ( ) | ( ), 0 1,2

(0) (1) 0 with | ( ) | 1

x x x x x

x dx

1& 0

2 21( ) 2 sin( ), , 1, 2,3,

2l lx l x l l

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

20 0 0( ) ( ) 2 sin( ), : ( ) : ( )

2g g g g g gx x x E E

2 20 0 0( 1)

( ) ( ) 2 sin(( 1) ), : ( ) : ( )2j j j j j j

jx x j x E E

1& | | (1)o

2 20 0 03

( ) ( ) 2 sin( ), : ( ) ( ) , : ( ) ( ) 32 2 2g g g g g g g gx x x E E E

2 20 0

2 20

( 1) 3( ) ( ) 2 sin(( 1) ), : ( ) ( ) ,

2 2

( 1): ( ) ( ) 3

2

j j j j j

j j j

jx x j x E E E

j

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

1& 0 1

TF TF TF 2 TF TF TF

1TF 2

0

TF TF TFg g g

( ) | ( ) | ( ), 0 1, ( )

| ( ) | 1

1 (x) ( ) 1, E E , 1,

2

g g g g g g

g

g g g

x x x x x

x dx

x

TF TF(0) (1) 1 0g g

Box Potential in 1D

– Matched asymptotic approximation• Consider near x=0, rescale• We get

• The inner solution

• Matched asymptotic approximation for ground state

, ( ) ( )g

g

x X x x

31( ) ( ) ( ), 0 ; (0) 0, lim ( ) 1

2 XX X X X X

( ) tanh( ), 0 ( ) tanh( ), 0 (1)g

g gX X X x x x o

MA MA MA

MA MA

1MA 2 MA 2 2 TF 2 2

0

( ) ( ) tanh( ) tanh( (1 )) tanh( ) , 0 1

1 | ( ) | 1 2 1 2 2 1 2 , 0 1.

g g g

g g g

g g g g

x x x x x

x dx

Box Potential in 1D

• Approximate energy

• Asymptotic ratios:

• Width of the boundary layer:

MA 2 21 41 2

2 3g gE E

( )O

0

1lim ,

2g

g

E

Box Potential in 1D

• Matched asymptotic approximation for excited states

• Approximate chemical potential & energy

• Boundary layers • Interior layers

MA[( 1) / 2]MA MA

0

MA MA[ / 2]

0

2( ) ( ) [ tanh( ( ))

1

2 1tanh( ( )) tanh( )]

1

jg

j j jl

jg g

jl

lx x x

j

lx C

j

MA 2 2 2 2

MA 2 2 2 2

1 2( 1) 1 ( 1) 2( 1) ,

1 4( 1) 1 ( 1) 2( 1) ,

2 3

j j

j j

j j j

E E j j j

( )O

Harmonic Oscillator Potential in 1D

The potential:The nonlinear eigenvalue problem

Case I: no interaction, i.e. – A complete set of orthonormal eigenfunctions

2

( )2

xV x

22 2( ) ( ) ( ) ( ) | ( ) | ( ), with | ( ) | 1

2x x V x x x x x dx

1& 0

2

2

2

1/ 2 / 21/ 4

20 1 2

1 1( ) (2 !) ( ), , 0,1,2,3,

2

( ) ( 1) : Hermite polynomials with

( ) 1, ( ) 2 , ( ) 4 2,

l xl l l

l xl x

l l

lx l e H x l

d eH x e

dx

H x H x x H x x

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

20 / 2 0 01/ 4

1 1( ) ( ) , : ( ) : ( )

2x

g g g g g gx x e E E

20 1/ 2 / 2 0 00 01/ 4

1 ( 1)( ) ( ) (2 !) ( ), : ( ) : ( )

2j x

j j j j j j j

jx x j e H x E E

1& | | (1)o

20 / 2 0 00 01/ 4

1 1 1( ) ( ) , : ( ) ( ) , : ( ) ( )

2 2 2x

g g g g g g g gx x e E E E C C

0 0

0 0 4j j

-

( 1)( ) ( ), : ( ) ( ) ,

2 2

( 1): ( ) ( ) with C = | ( ) |

2

j j j j j j

j j j j

jx x E E E C

jC x dx

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

1& 0 1

TF 2 TFTF TF TF TF 2 TF TF

TF 3/ 2TF 2 TF 2/3

-

/ 2, | | 2( ) ( ) ( ) | ( ) | ( ) ( )

0, otherwise

2(2 ) 1 3 1 | ( ) | ( )

3 2 2

g gg g g g g g

gg g g

x xx V x x x x x

x dx

TF2 gx

Harmonic Oscillator Potential in 1D

– Thomas-Fermi approximation for first excited state

• Jump at x=0!• Interior layer at x=0

TF TF TF TF 2 TF1 1 1 1 1

TF 2 TFTF 1 1

1

TF 3/ 2TF 2 TF 2/31

1 1 1

-

( ) ( ) ( ) | ( ) | ( )

sign( ) / 2, 0 | | 2( )

0, otherwise

2(2 ) 1 3 1 | ( ) | ( )

3 2 2

x V x x x x

x x xx

x dx

Harmonic Oscillator Potential in 1D

– Matched asymptotic approximation

– Width of interior layer:

MA1MA MA

1 1MA MA 2 MA1 1 1

| |tanh( ) 0 | | 2

( ) 2 / 2

0 otherwise

x xx x

x x

( )O

Thomas-Fermi (or semiclassical) limit

In 1D with strongly repulsive interaction– Box potential

– Harmonic potential

In 1D with strongly attractive interaction

0

1 ??? ??? : ( ) ???g g g gE

0 1,11 0 1 1( ) 1

0 0,1 2g g g g

xx W E

x

0 2 00

0 2/3 0 2/3

/ 2, | | 2( ) ( ) [0,0.5)

0, otherwise

3 3 1 3( ) , ( )

10 2 2 2

g gg g

g g g g

x xx x C

E E

1 0 1/2 2

0

0

( ) ( )

( ) ( )g g g gx x x L E

V x V x x

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

( )E

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

22 2

1

11

1

0 0

( )1( , ) ( ) | | ,

2 2

( , )( , ) , 0,1,2,

|| ( , ) ||

( ,0) (x) with || ( ) || 1.

t n n

nn

n

Ex t V x t t t

xx t n

x

x x

tt

1( (., ) ) ( (., ) ) ( (., 0) )n nE t E t E

it

0

1

2 1̂

??)()(

)()ˆ(

)()ˆ(

01

11

01

EE

EE

EE

g

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’)

22 2

2

0 0

( (., ))( , ) ( ) | | , 0,

2 || (., ) ||

( ,0) ( ) with || ( ) || 1.

t

tx t V x t

t

x x x

0|| (., ) || || || 1, ( (., )) 0, 0d

t E t td t

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:

• 1D 2D-surface 2D-contour – Optical lattice potential:

• 1D 2D-surface 2D-contour 3D next

otherwise;100)( xxV

2/xV(x) 2

2 2( ) / 2 12sin (4 )V x x x

back

back

back

back

back

back

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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 lattice

– In 3D: With a fast rotationnext

2 2

2 2

1( ) [ ( ) | | ] ,

2

: | (x) | 1d

dzx V x L x

dx

: ( ) , ,z y x y xL xp yp i x y i L x P P i

: ( ) min ( ), | 1, ( )g gS

E E E S E

back

back

back

back

Extension to two-component

Two-component (Bao, MMS, 04’)

Ground state

– Existence & uniqueness– Quantized vortices & fractional index– Numerical methods & results: Crarter & domain wall

2 2 21 1 11 1 12 2 1

2 2 22 2 21 1 22 2 2

2 22 21 1 2 2

1( ) [ ( ) | | | | ]

21

( ) [ ( ) | | | | ]2

| ( ) | , | ( ) | 1 0 1d d

z

z

x V x L

x V x L

x dx x dx

1 2 1 2: ( ) min ( ), ( , ) | , 1 , ( )g gS

E E E S E

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

| |lim ( )x

V x

11 220 or 0

211 12 22 11 11 22 12, , 0 or 0 & 0

0

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

2 2 21 11 1 12 2 1 2

2 2 22 21 1 22 2 2 1

2 2 2 21 2 1 2

1( ) [ ( ) | | | | ]

21

( ) [ ( ) | | | | ]2

| ( ) | | ( ) | 1d

z

z

x V x L

x V x L

x x dx

1 2: ( ) min ( ), ( , ) | 1, ( )g g SE E E S E

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

211 11 22 12 12 11 220 & 0 or & 0

12 11 22 0 0 00 & 0 & ( , ) for 0

211 12 22 11 11 22 12, , 0 or 0 & 0

1 2(| |, sign( ) | |)g gg

1 2

1 2

1 2

| | | | & | |

| | 0 & | |

| | & | | 0

g g g

g g g

g g g

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

2 * 21 1 1 0 1 1 1 0

2 *0 0 1 1 0 1 1 0

2 * 21 1 1 0 1 1 1 0

2 2 2 2 2 21 0 1 1 0 1

1( ) [ ( ) ] ( )

21

2 [ ( ) ] ( ) 22

1( ) [ ( ) ] ( )

2

[| ( ) | ( ) | ( ) | ]

n s s

n s s

n s s

V x g g g

V x g g g

V x g g g

x x x

2 2 2 21 1 1 1

1,

[| ( ) | | ( ) | ] ( 1 1)

d

d

dx

x x dx M M

Quantum phase transition

Ferromagnetic gs <0 Antiferromagnetic gs > 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)– 2005@Univ. Stuttgart, Germany– PRL, 94 (2005) 160401

Big-wave in theoretical studyA. 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), 1791-1797– S. Yi & L. You, PRA 61 (2001), 041604(R); D. H. J. O’Dell, PRL 92 (2004), 250401

2 2ext dip

1( , ) ( ) | | | | ( , )

2i x t V x U x tt

3( , )x t x

2 2 2 2 2 2ext

1( )

2 x y zV z x y z 2

0 dip

20 0

4 (short-range), (long-range)

3s

mNN a

a a

2 2 23

dip 3 3

3 1 3( ) / | | 3 1 3cos ( )( ) , fixed & satisfies | | 1

4 | | 4 | |

n x xU x n n

x x

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 !!

3 3

2 22( ) : ( , ) ( , ) ( ,0) 1N t t x t d x x d x

3

2 2 4 2 2ext dip 0

1( ( , )) : | | ( ) | | | | ( | | ) | | ( )

2 2 2E t V x U d x E

23

dip 2

3( )( ) 1

| |

nU

3

1

| |O

x

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

2ext

2

| |

1( , ) ( ) ( ) | | 3 ( , )

2

( , ) | ( , ) | , lim ( , ) 0

n n

x

i x t V x x tt

x t x t x t

2 2

dip dip3 2 2

3 3( ) 1 3( )( ) 1 ( ) 3 ( ) 1

4 4 | |n n

n x nU x x U

r r r

2 2 2 2dip

1| | | | 3 & | | | |

4n nUr

3

2 2 4 2ext

1 3( ( , )) : | | ( ) | | | | | |

2 2 2 nE t V x d x

| | & & ( )n n n n nr x n

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:

3ext ext

| |( ) 0, & lim ( ) (confinement potential)

xV x x V x

g S 0 &2

00| | with i

g ge

lim ( )SE

0 0 & or

2

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.