Analysis and Efficient Computation for Nonlinear Eigenvalue Problems in Quantum Physics and...
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Transcript of Analysis and Efficient Computation for Nonlinear Eigenvalue Problems in Quantum Physics and...
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: [email protected]
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.