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.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
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
1 2
2
i  t ( x , t )      V ( x )   | | 
2
– t : time & x (  R ) : spatial coordinate (d=1,2,3)
–  ( x , t ) : complex-valued wave function
– V ( x ) : real-valued external potential
d
4 as ( N  1)
)
: interaction
a0
• =0: linear; >0: repulsive interaction
• <0: attractive interaction
–  (e.g., 
constant
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
N ( ) : 
2
2
2
2
   ( x , t ) d x    ( x ,0) d x    0 ( x ) d x : N ( 0 ) (  1),



2
2
4
1
E ( ) :    ( x , t )  V ( x )  ( x , t )  2  ( x , t )  d x  E ( 0 )
2


Motivation
Stationary states (ground & excited states)
 ( x, t )   ( x ) e
it 
Nonlinear eigenvalue problems: Find (  ,  ) s.t.
1 2
  ( x )     ( x )  V ( x ) ( x )   |  ( x ) |2  ( x ), x    R d
2
 ( x )  0,
x    ;
 :  |  (x) |2 dx  1
2

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

    ( )  E ( )   |  (x) |4 dx

2
– With energy
E  ( )  

1

2
2
[ |  ( x ) | V ( x )|  ( x ) |  | ( x ) |4 ] dx
2
2
Special solutions
– Soliton in 1D with attractive interaction
– Vortex states in 2D
im
( x)  fm (r)e
Motivation
Ground state: Non-convex minimization problem
E (g )  min E ( )
S
S   |   1,  |x  0, E ( )  
– Euler-Lagrange equation  Nonlinear eigenvalue problem
Theorem (Lieb, etc, PRA, 02’)
–
–
–
–
–
V ( x)  
Existence d-dimensions (d=1,2,3):   0 & |lim
x |
Positive minimizer is unique in d-dimensions (d=1,2,3)!!
No minimizer in 3D (and 2D) when   0 (   cr  0)
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
1
  ( x )    ''( x )  V ( x ) ( x )   |  ( x ) |2  ( x ), with
2
V ( x)  U ( x 2  a 2 )2
&

2
|

(
x
)
|
dx  1


 : positive  0  negative
Motivation
Excited states:
1, 2 , 3 , 
Open question: (Bao & W. Tang, JCP, 03’; Bao, F. Lim & Y. Zhang, Bull Int. Math, 06’)
g ,
1 ,
2 ,

E ( g )  E (1 )  E ( 2 )  
  ( g )    (1 )    ( 2 )   ???????
Continuous normalized gradient flow:
 ( (., t ))
1 2
2
t ( x, t )     V ( x )    |  |  
 , t  0,
2
2
||  (., t ) ||
 ( x,0)  0 ( x )
with
|| 0 ( x ) || 1.
– Mass conservation & energy diminishing
Singularly Perturbed NEP
For bounded  with box potential for 

 :
,      ,     


1

2
– Singularly perturbed NEP
    ( x)  
  ( x )  0,
2
2
1
  |   ( x ) |2 dx  1

 2  ( x )  |   ( x ) |2   ( x ), x ,
2
x   
– Eigenvalue or chemical potential
1
    (  )  E (  )   |   (x) | dx  O(1)
2
4

2
 2
1
E ( )       |  
2
2



|4 dx  O(1),

0
1
– Leading asymptotics of the previous NEP
   ( )    (  )  O( ) & E ( )   E (  )  O( ), 
1
Singularly Perturbed NEP
For whole space with harmonic potential for  1
x   x,  ( x)    ( x ),     ,  : 
    |  ( x) |
1/ 2


d / 4
1
 d /( d  2)
 2

2
dx  1
d
– Singularly perturbed NEP
    ( x)  
2
2
 2  ( x )  V ( x )  ( x )  |   ( x ) |2   ( x ), x d
– Eigenvalue or chemical potential
1
    (  )  E (  )   |   (x) | dx  O(1)
2
4

d
2
 2
1
E ( )       V ( x ) |   |2  |  
2
2
d 


|4 dx  O(1),

0
1
– Leading asymptotics of the previous NEP
   ( )   1  (  )  O( 1 )  O( d /(d 2) ) & E ( )   1 E (  )  O( d /(d 2) ), 
1
General Form of NEP
  (x)  
 ( x )  0,
2
2
 2 ( x )  V ( x ) ( x )   |  ( x ) |2  ( x ),
x    ;
x   Rd
 :  |  ( x ) |2 dx  1
2

– Eigenvalue or chemical potential

   ( )  E ( )   |  (x) |4 dx
2 
– Energy
2
E  ( )   [


2
|  ( x ) | V ( x )|  ( x ) | 
2
2
Three typical parameter regimes:
– Linear:   1&   0
– Weakly interaction:   1& |  | 1
– Strongly repulsive interaction:

2
| ( x ) |4 ] dx
  1& 0  
1
Box Potential in 1D
 0, 0  x  1,
V ( x)  
, otherwise.
The potential:
The nonlinear eigenvalue problem
  ( x)  
2
2
 ( x)   |  ( x) |2  ( x),
 (0)   (1)  0
0  x  1,
1
2
|

(
x
)
|
dx  1

with
Case I: no interaction, i.e.   1&   0
0
– A complete set of orthonormal eigenfunctions
l ( x)  2 sin(l x),
1
2
l  l 2 2 ,
l  1, 2,3,
Box Potential in 1D
– Ground state & its energy:
g ( x)   ( x)  2 sin( x),
0
g
Eg : E ( ) 
0
g
2
2
– j-th-excited state & its energy
 j ( x)   ( x)  2 sin(( j  1) x),
0
j
  g :  (g0 )
( j  1)2  2
E j : E ( ) 
  j :  ( 0j )
2
0
j
Case II: weakly interacting regime, i.e.   1& |  | o(1)
– Ground state & its energy:
3
2
0
g ( x)   ( x)  2 sin( x), Eg : E (g )  E ( ) 

,  g :  (g )   (g ) 
 3
2
2
2
0
g
0
g
2
– j-th-excited state & its energy
( j  1) 2  2 3
 j ( x)   ( x)  2 sin(( j  1) x), E j : E ( j )  E ( ) 

,
2
2
( j  1) 2  2
0
 j :  ( j )   ( j ) 
 3
2
0
j
0
j
Box Potential in 1D
Case III: Strongly interacting regime, i.e.   1& 0  
– Thomas-Fermi approximation, i.e. drop the diffusion term
 gTF gTF ( x)  | gTF ( x) |2 gTF ( x),
0  x  1,
 gTF ( x)   gTF
1

TF
2
|

(
x
)
|
dx  1
g

0
g (x)  gTF ( x)  1,
1
E g  E TF

,
g
2
g   gTF  1,
• Boundary condition is NOT satisfied, i.e.
• Boundary layer near the boundary
gTF (0)  gTF (1)  1  0
1
Box Potential in 1D
– Matched asymptotic approximation
• Consider near x=0, rescale
• We get
x

X,
g
1
( X )   ( X )   3 ( X ), 0  X  ;
2
 ( x)  g ( x)
(0)  0,
• The inner solution
( X )  tanh( X ), 0  X  
 g ( x)  g tanh(
g

lim ( X )  1
X 
x), 0  x  o(1)
• Matched asymptotic approximation for ground state
g ( x)  
MA
g
( x)  
MA
g

 gMA
 gMA
 gMA 
 tanh(
x)  tanh(
(1  x))  tanh(
) , 0  x  1


 



1
1   | gMA ( x) |2 dx   g   gMA  1  2 1   2  2 2   gTF  2 1   2  2 2 , 0  
0
1.
Box Potential in 1D
• Approximate energy
Eg  E
MA
g
1 4
   1   2  2 2
2 3
• Asymptotic ratios:
Eg
1
 ,
 0 
2
g
lim
• Width of the boundary layer:
O( )
Box Potential in 1D
• Matched asymptotic approximation for excited states
 j ( x)   jMA ( x)   MA
j
[( j 1) / 2]
[

tanh(
l 0
 gMA

(x 
2l
)) 
j 1
 gMA 2l  1
 gMA
tanh(
(
 x))  C j tanh(
)]


j 1

l 0
[ j / 2]
• Approximate chemical potential & energy
 j   MA
 1  2( j  1) 1  ( j  1) 2  2  2( j  1) 2  2 ,
j
E j  E MA

j
1 4
 ( j  1) 1  ( j  1) 2  2  2( j  1) 2  2 ,
2 3
• Boundary layers
• Interior layers
O( )
Harmonic Oscillator Potential in 1D
The potential:
The nonlinear eigenvalue problem
x2
V ( x) 
2
  ( x)  
2
2
 ( x)  V ( x) ( x)   |  ( x) |2  ( x), with

2
|

(
x
)
|
dx  1


Case I: no interaction, i.e.   1&   0
– A complete set of orthonormal eigenfunctions
l ( x)  (2l l !) 1/ 2
1
 1/ 4
e x
2
/2
H l ( x),
l 
l 1
,
2
l x
l x d e
H l ( x)  (1) e
: Hermite polynomials with
l
dx
H 0 ( x)  1, H1 ( x)  2 x, H 2 ( x)  4 x 2  2,
2
2
l  0,1, 2,3,
Harmonic Oscillator Potential in 1D
– Ground state & its energy:
g ( x)  g0 ( x) 
1
 1/ 4
e x
2
/2
Eg : E (g0 ) 
,
– j-th-excited state & its energy
 j ( x)   j0 ( x)  (2 j j !) 1/ 2
1
 1/ 4
e x
2
/2
H j ( x),
1
  g :  (g0 )
2
E j : E0 ( 0j ) 
( j  1)
  j : 0 ( j0 )
2
Case II: weakly interacting regime, i.e.   1& |  | o(1)
– Ground state & its energy:
g ( x)  g0 ( x) 
1
 1/ 4
e x
2
/2
, Eg : E (g )  E (g0 ) 
1 
1
 C0 ,  g :  (g )   (g0 )    C0
2 2
2
– j-th-excited state & its energy
 j ( x)   j0 ( x), E j : E ( j )  E ( j0 ) 
( j  1)
 j :  ( j )   ( ) 
 Cj
2
0
j
( j  1) 
 Cj,
2
2

with C j =  | j0 ( x) |4 dx
-
Harmonic Oscillator Potential in 1D
Case III: Strongly interacting regime, i.e.   1& 0  
– Thomas-Fermi approximation, i.e. drop the diffusion term

TF
g
  TF  x 2 / 2, | x | 2  TF
g
 ( x)  V ( x)  ( x)  |  ( x) |  ( x)   ( x)   g
0,
otherwise


2(2  gTF )3/ 2
1 3
TF
2
1   | g ( x) | dx 

 g   gTF  ( ) 2 / 3
3
2 2
-
TF
g
TF
g
TF
g
2
TF
g
TF
g
– No boundary and interior layer
– It is NOT differentiable at x  
2  gTF
1
Harmonic Oscillator Potential in 1D
– Thomas-Fermi approximation for first excited state
1TF 1TF ( x)  V ( x) 1TF ( x)  | 1TF ( x) |2 1TF ( x)
sign( x)  TF  x 2 / 2, 0 | x | 2  TF
1
1
 ( x)  
0,
otherwise

TF
1

2(21TF )3/ 2
1   |  ( x) | dx 
3
-
TF
1
2
• Jump at x=0!
• Interior layer at x=0

1 3
2 2
1  1TF  ( ) 2 / 3
Harmonic Oscillator Potential in 1D
– Matched asymptotic approximation

1MA
| x| x
MA


tanh(
x
)

 1

1MA ( x)  
2  1MA  x 2 / 2  1MA 



0


– Width of interior layer:
O( )
0 | x | 21MA
otherwise
Thomas-Fermi (or semiclassical) limit   0
g  ??? Eg  ??? g : (g )  ???
In 1D with strongly repulsive interaction
– Box potential
1 0  x  1

0
g  g ( x)  
W 1,1
– Harmonic potential
x  0,1
0
 1
1
Eg 
2

  0  x 2 / 2, | x | 2  0
g
g ( x )   ( x )   g
 C
0,
otherwise

3 3 2/3
1 3 2/3

0

0
Eg  Eg  ( ) ,
g  g  ( )
10 2
2 2

0
g
In 1D with strongly attractive interaction
g  g0 ( x )   1/2 ( x  x0 )  L2
V ( x0 )  V ( x )
x
Eg  
g  1
  [0,0.5)
  1
g  
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 E ( ) 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
1  E ( )  2 2
t ( x , t )  
    V ( x )   |  |2  , tn  t  tn 1
2 
2

0

 ( x , tn1 ) 
 ( x , t n1)

||  ( x , t n 1) ||
 ( x , 0)  0 (x)
with
,
n  0,1, 2,
1
2
ˆ1
E (ˆ1 )  E (0 )
E (ˆ )  E ( )
1
|| 0 ( x) || 1.
1
E (1 )  E (0 ) ??
g
– The first equation can be viewed as choosing t  i  in GPE
– For linear case: (Bao & Q. Du, SIAM Sci. Comput., 03’)
E (  (., tn1 ) )  E (  (., tn ) ) 
 E (  (., 0) )
– 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’)
t ( x, t ) 
2
2
2  V ( x )    |  |2  
 ( x, 0)  0 ( x )
with
 ( (., t ))
 , t  0,
2
||  (., t ) ||
|| 0 ( x ) || 1.
– Energy diminishing
||  (., t ) |||| 0 || 1,
d
E ( (., t ))  0,
dt
t 0
– 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
V ( x)  0 0  x  1;  otherwise
1D-energy
2D-surface
– Harmonic oscillator potential:
• 1D
2D-surface
2D-surface
V(x)  x 2 / 2
2D-contour
– Optical lattice potential:
• 1D
2D-contour
V ( x)  x2 / 2  12sin 2 (4 x)
2D-contour
3D
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Extension to rotating BEC
BEC in rotation frame(Bao, H. Wang&P. Markowich,Comm. Math. Sci., 04’)
1
  ( x )  [  2  V ( x )   Lz   |  |2 ]  , x d
2
 : 
2
d
|  (x) |2 dx  1
Lz : xpy  ypx  i( x y  y x )  i , L  x  P, P  i
Ground state: existence & uniqueness, quantized vortex
Eg : E (g )  min E ( ),
S   |   1, E ( )  
S
– In 2D: In a rotational frame &With a fast rotation & optical lattice
– In 3D: With a fast rotation
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Extension to two-component
Two-component (Bao, MMS, 04’)
1 2
1 1 ( x )  [   V ( x )   Lz  11 | 1 |2  12 | 2 |2 ] 1
2
1 2
2 2 ( x )  [   V ( x )   Lz   21 | 1 |2   22 | 2 |2 ] 2
2
1 
2
 | 1 ( x ) | dx   ,
2
d
Ground state
Eg : E ( g )  min E (),
S
2 
2
2
|

(
x
)
|
dx  1  
 2
0  1
d
S    (1, 2 ) | 1   , 2  1   , E ()  
– Existence & uniqueness
– Quantized vortices & fractional index
– Numerical methods & results: Crarter & domain wall
Results
Theorem
– Assumptions
V ( x)  
• No rotation   0 & Confining potential |lim
x |
2
• Repulsive interaction 11, 12 , 22  0 or 11  0 & 1122  12  0
– Results
• Existence & Positive minimizer is unique
– No minimizer in 3D when 11  0 or 22  0
Nonuniquness in attractive interaction in 1D
Quantum phase transition in rotating frame
Two-component with an external driving field
Two-component (Bao & Cai, 09’)
1
2
1 2
 2 ( x )  [   V ( x )   Lz   21 | 1 |2   22 | 2 |2 ] 2  1
2
 1 ( x )  [ 2  V ( x )     Lz  11 | 1 |2  12 | 2 |2 ] 1  2
1  2 
2
2
2
2
|

(
x
)
|

|

(
x
)
|
dx  1
2
 1
d
Ground state
Eg : E ( g )  min E ( ),
S
S    (1, 2 ) |   1, E ( )  
– Existence & uniqueness (Bao & Cai, 09’)
– Limiting behavior & Numerical methods
– Numerical results: Crarter & domain wall
Theorem (Bao & Cai, 09’)
– No rotation & confining potential &
11, 12 , 22  0 or 11  0 & 1122  122  0
– Existence of ground state!!
g
g
– Uniqueness in the form g  (| 1 |, sign( ) | 2 |) under
11  0 & 1122  122  0 or 12  11  22 &   0
– At least two different ground states under– quantum phase transition
  0 & 12  11  22  0 &   (0 , 0 ) for 0  0
– Limiting behavior
|  |   | 1g | & | 2g |   g
    | 1g |  0 & | 2g |  g
    | 1g |   g
& | 2g | 0
Extension to spin-1
Spin-1 BEC (Bao & Wang, SINUM, 07’; Bao & Lim, SISC 08’, PRE 08’)
1
(    ) 1  [  2  V ( x )  g n  ] 1  g s ( 1   0   1 )1  g s*102
2
1
2  0  [  2  V ( x )  g n  ] 0  g s ( 1   1 )0  2 g s110*
2
1
(    ) 1  [  2  V ( x )  g n  ] 1  g s (  1   0  1 )1  g s1*02
2
1  0  1 
2
2
2
2
2
2
[|

(
x
)
|


(
x
)
|


(
x
)
|
]dx  1,
1
0

1

d
1  1 
2
2
2
2
[|

(
x
)
|

|

(
x
)
|
]dx  M
1

1

( 1  M  1)
d
– Continuous normalized gradient flow (Bao & Wang, SINUM, 07’)
– Normalized gradient flow (Bao & Lim, SISC 08’)
• Gradient flow + third projection relation
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 study
A. Griesmaier,et al., PRL, 94 (2005)160401
Mathematical Model
3



(
x
,
t
)
x

Gross-Pitaevskii equation (re-scaled)

 1

i  ( x , t )      Vext ( x )   | |2  U dip  | |2  ( x , t )
t
 2

– Trap potential Vext ( z )   x2 x 2   y2 y 2   z2 z 2 
2
– Interaction constants   4 N a (short-range),   mN  
a
3 a
– Long-range dipole-dipole interaction kernel
1
0
3 1  3(n  x )2 / | x |2
3 1  3cos2 ( )
Udip ( x ) 

,
3
3
4
|x|
4
|x|
2
dip
0
2
s
(long-range)
0
n
3
fixed & satisfies | n | 1
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
Mathematical Model
Mass conservation (Normalization condition)
N (t ) :  (, t ) 
2
  ( x, t )
3
2
dx
  ( x,0)
2
d x 1
3
Energy conservation
E ( (, t )) :


1
2
2
4
2
|


|

V
(
x
)
|

|

|

|

(
U

|

|
) |
ext
dip
3  2
2
2
Long-range interaction kernel:

|2  d x  E( 0 )

 1 
– It is highly singular near the origin !! At O  3 singularity near the origin !!
| x | 
– Its Fourier transform reads
2
3(n   )
• No limit near origin in phase space !! U dip ( )  1 
2
|

|
• Bounded & no limit at far field too !!

• Physicists simply drop the second singular term in phase space near origin!!
• Locking phenomena in computation !!
3
A New Formulation
r | x | & n  n  & n n  n (n )
Using the identity (O’Dell et al., PRL 92 (2004), 250401, Parker et al., PRA 79 (2009), 013617)
3  3(n  x )2 
 1 
U dip ( x ) 
1




(
x
)

3

nn 



4 r 3 
r2 
 4 r 
3(n   )2
 U dip ( )  1 
|  |2
Dipole-dipole interaction becomes
1
U dip  | |2   | |2 3 n n &  
 | |2   | |2
4 r
Gross-Pitaevskii-Poisson type equations (Bao,Cai & Wang, JCP, 10’)
i

 1
 ( x , t )      Vext ( x )  (    ) | |2 3 n n
t
 2
  ( x , t ) | ( x , t ) |2 ,
Energy
E ( (, t )) :

 ( x , t )
lim  ( x , t )  0
|x |
 
3
1
2
2
4
2
|


|

V
(
x
)
|

|

|

|

|



|
ext
n
3  2
 d x
2
2
Ground State Results
Theorem (Existence, uniqueness & nonexistence) (Bao, Cai & Wang, JCP, 10’)
– Assumptions
Vext ( x )  0, x 
3
&
– Results
lim Vext ( x )   (confinement potential)
|x|
• There exists a ground state g  S if   0
& 

2
 
i
• Positive ground state is uniqueness g  e 0 | g | with 0 
• Nonexistence of ground state, i.e. lim E ( )  
– Case I:
– Case II:
 0
S
  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.