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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 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
back
back
back
back
back
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.
Recommended