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: [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)
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Analysis and Efficient Computation for Nonlinear Eigenvalue Problems in Quantum Physics and Chemistry Weizhu Bao Department of Mathematics & Center of.
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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
In quantum physics & nonlinear optics: – Interaction between particles with quantum effect– Bose-Einstein condensation (BEC): bosons at low temperature
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
Experimental setup – Molecules meet to form dipoles – Cool down dipoles to ultracold – Hold in a magnetic trap – Dipolar condensation – Degenerate dipolar quantum gas
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’)
– 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