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Quasi-exactly solvable models in quantum mechanics and Lie
algebras
Quasi-exactly solvable models in quantum mechanics and Lie
algebras
S. N. DolyaB. Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Sciences of Ukraine
S. N. DolyaB. Verkin Institute for Low Temperature Physics and Engineering
of the National Academy of Sciences of Ukraine
S. N. Dolya JMP, 50 (2009)S. N. Dolya JMP, 49 (2008).S. N. Dolya O. B. Zaslavskii J. Phys. A: Math. Gen. 34 (2001)S. N. Dolya O. B. Zaslavskii J. Phys. A: Math. Gen. 34 (2001)S. N. Dolya O. B. Zaslavskii J. Phys. A: Math. Gen. 33 (2000)
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OutlineOutline
1. QES-extension (A)1. QES-extension (A)2. quadratic QES - Lie algebras2. quadratic QES - Lie algebras3. physical applications3. physical applications4. 4. QES-extension (B)QES-extension (B)5. 5. cubic QES - Lie algebrascubic QES - Lie algebras
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sl2(R)-Hamiltonians
Representation:
Invariantsubspace
Turbiner et al
0
2
2x
x
x
nJ x
J
J x nx
(partial algebraization)
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What is being studied?• Hamiltonians are formulated in terms of QES Lie algebras.
• eigenvalues and eigenfunctions when possible.
• Invariant subspaces:
How this is being studied?
• Nonlinear QES Lie algebras
1 2 1V = span{ ( ), ( ),..., ( ), ( )}n n nf x f x f x f x
pqsspqji
ijpqqp cXcXXcXX ,
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QES-QES-extension:extension:
0.our strategy
1) We find a general form of the operator of the second order P2 for which subspace M2 = span{f1, f2} is preserved.
2) We make extension of the subspace M2 → M4 = span{f1, f2, f3, f4}
3) We find a general form of the operator of the second order P4 for which subspace M4 is preserved.
4) we obtain the explicit form of operator P2(N+1) that acts on the elements of the subspace M2(N+1) = {f1,f2,…, f2(N+1)}
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QES-extension:QES-extension:2 0 0M ={ , }f f
2
2( ) ( ) ( ) ( ) ( ) ( ) = 0
d dq x f x p x f x r x f x
dxdx
0 ( ) = ( )f x f x0 ( ) = ( ) = ( )'df x f x f x
dx
;
I. Select the minimal invariant subspace
2
2 3 2 02= ( ) ( ) ( )
d dP a x a x a x
dxdx Select the invariant
operator
2 0 1 0 2 0=P f c f c f
2 0 3 0 4 0=P f c f c f
Condition for the subspace M2
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QES-QES-extension:extension:
2 0 0M ={ , }f f 4 0 1 0 1M ={ , , , }f f f f
4 0 01 0 02 0 03 1 04 1=P f c f c f c f c f
4 1 11 0 12 0 13 1 14 1
......................................................................
......................................................................
=P f c f c f c f c f
II.extension for the minimal invariant subspace
Condition for the subspace M4
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QES-QES-extension:extension:
2 4 2(N+1)M M ... M
0 0 0 1 0 1 0 1 0 1{ , } { , , , } ... { , ,... , , ,..., }N Nf f f f f f f f f f f f
0 1 0 1, , , 0W f f f f
Conditions of the Conditions of the QES-QES-extension:extension:
2 4 2 1order ( ) order( ) ... order( ) 2.NP P P
1
2
Wronskian matrix
III.Extension for the minimal invariant subspace
Order of derivatives
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hypergeometric
function
2
021 ( ) = 0
d dx s f x
dxdx
0 0 1
0 0 1
_( ) = ;
_( ) = ;
1
f x F xs
f x F xs
Realization (special functions:
hypergeometric, Airy, Bessel ones)
2 0 0M ={ , }f f
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QES-extension:QES-extension:
2 4 2(N+1)M M ... M
0 0 0 1 0 1 0 1 0 1{ , } { , , , } ... { , ,... , , ,..., }N Nf f f f f f f f f f f f
1 0 1 0= = , = = ,n nn n n nf x f x f f x f x f
Particular choice of QES extension
act more
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QES-extension:QES-extension:Example 1Example 1
10 1 0 1= span{ , ,... , , ,..., }N N NR f f f f f f
= 0,1,2,...N 1dim = 2 1NR N
0 1
_= ;n
nf x F xs
0 1
_= ;
1n
nf x F xs
= 0,1,.., 1,n N N
2 2
21 12 2
= 1 , = 2d d d d
J x s J x s N x xdx dx dx dx
11
11
1 1
21
=1 2 1
1 2=
2
nn n nn
nn n n n
n n nn
nn n n
Bn A s f ff
J sf n s A f s B f
nf n n s f ff
J sf f n n s f ns f
= 2nA n N =nB n N
counter
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2
1 1 1 1 1 1 1 1
1 1
2 21
4 4 4
2 4 2 2
2 2 6 2 1
CasimirF S J J J S J J J
N s N s J J
s s N sN J s
QES-extension: QES-extension: The commutation relations of the operators The commutation relations of the operators
1 1 1
1 1 1 1 1 1 6 1 7
2
1 1 1 1
, =
, = 4 2 2
, = 2 2
J J S
J S J J S J c J c
J S J J
6 2 2 2c N s s N
7 1 2 2c s N s
Casimir Casimir operatoroperator::
Casimir invariant
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QES-extension:QES-extension:Example 2Example 2
20 1 0 1= span{ , ,... , , ,..., }N N NR f f f f f f
= 0,1,2,...N 2dim = 2 1NR N = 0,1,.., 1,n N N
= 2nA n N =nB n N
11= ;nnf t F t
s
11
1= ;
1n
nf t F ts
2 2
22 22 2
= 1 , = 2d d d d
J t s t J t s N t t t Ndt dt dt dt
1 12
1
12
1 1
1 2 /=
1 2 1
1 2 /=
1 2
n n n n n nn
n n n n n nn
n n nn
n n nn
n A s f B f B s ffJ
s n A f B f s B ff
n f n n s f n s ffJ
n f n n s f ns ff
counter
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QES-extension: QES-extension: The commutation relations of the operators The commutation relations of the operators
2 2 2
2 2 2 2 2 5 2 6 2 7
2
2 2 2 2 5 2
, =
, = 4 2
, = 2 1
J J S
J S J J S c J c J c
J S J J c J N s
5 2 1c s 6 2 2 2c N s s N 7 2 2 1c N N s s
Casimir invariant
Casimir Casimir operatoroperator::
2
2 2 2 2 2 2 5 2 2
2
2 2 2
5 2 2 7 5 2
4 4 2
2 4 2 2
2 2 1 1 2 2
1 3 6
CasimirF S J J J S J c J J
J N s N s J J
c S s s N J c c J
s N N s N
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QES-extension:QES-extension:Example 3Example 3
30 1= span{ , ,... }N NR f f f
= 0,1,2,...N 3dim = 1NR N = 0,1,.., 1,n N N
=nB n N
11= ;n
nf F t
s
2 2
23 32 2
= , =d d d d
J t s t J t s N t t tdt dt dt dt
3
3 1 1
=
=
n n
n n n n n n n n
J f n f
J f s n C f B f n s f
=n n = 2n nC N n
counter
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QES-extension: QES-extension: The commutation relations of the operators The commutation relations of the operators
3 3 3
3 3 3 3 3 5 3 6 3 7
2
3 3 3 3 5 3
, =
, = 4 2
, = 2
J J S
J S J J S c J c J c
J S J J c J s
5 2c s N
6 2c N s N s
7 2c s N s
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Two-photon Rabi HamiltonianRabi Hamiltonian describes a two-level system (atom) coupled to a single mode of radiation via dipole interaction.
220R =
2 zH a a g a a
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Two-photon Rabi
Hamiltonian 220
R =2 zH a a g a a
00
1 0 0 0 1 4= , = , = ; , , .
0 1 0 1 0 2x y z
i g Eg E
i
0 1 1
0 2 2
= ,L
EL
2 2/2L a a g a a
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The two-photon Rabi
Hamiltonian
1 1
2 21 0
= 0
=
L
L L L E L L E
= , =2 2
x xx xa a
0 1 1
0 2 2
= ,L
EL
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The two-photon Rabi
Hamiltonian 22 2 2 2 2
1 2 2 2 2 0 0 2== 4 4 2 4c x c x
t xe L e g J g S g J J a
2= 3/4, = 1/2 (the parameters of subspace )Ns
1 1= (the parameter of gauge transformation)
2 1
gc
g
2= (the parameter of change of variable)
1
g
g
21= 2 1 1 (the energy of the Hamiltonian).
2E N g
2 20 = 3 4 4 1 3 5 /4a N N g g
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The two-photon Rabi
Hamiltonian 22 2 2 2
1 3 3 3 3 0 0 2== 4 4 2 4c x c x
t xe L e J g S g J J a
33= , = 1/2 (the parameters of subspaces )
4 2 N
Ns
1 1= (the parameter of gauge transformation)
2 1
gc
g
2= (the parameter of change of variable)
1
g
g
21= 1 1 (the energy of the Hamiltonian).
2E N g
2 20 = 1 2 2 1 1 3 /4a N N g g
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Example 20
2 2
0 0 3 61 1,
1 1 0 12 4J J
matrix representation
20
1 2 20
3 9/4 3 2 1
2 1 1/4 3 4
g g gL
g g g
2 2
0 0
0 00
4 9 1 4 1 3= , < < .
8 2 2g
condition det(L1) = 0
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QES-extension: continuationQES-extension: continuationExample 4 (Example 4 (QES qubic Lie algebra ))
= 0,1,2,...N
2dim = 2 1NV N
= 0,1,.., 1,n N N
3 2
2 26 6 3 2
= , = 2d d d d
J x J x x N x xdx dx dx dx
6 1
6 1 1
=
= 1
n n n
n n n n n n n n n
J f n f f
J f B f f n B f f
20 1 0 1= span{ , ,... , , ,..., }N N NV f f f f f f
1 11 1= ; ;1 1nf F x F x
n n
=nB n N= ;n n
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QES-extension: continuation QES-extension: continuation Example 4 (Example 4 (QES qubic Lie algebra ) ) The commutation relations of the operators The commutation relations of the operators
6 6 6
3 2
6 6 6 6 6 6
6 6 6
, =
, = 8 6
, =
J J S
J S J N J c J
J S J
6 2 2c N N
4 2 32
6 6 6 6
2
6 6 6
4 4
4 2 0
CasimirF S J J N J
c J N J
Casimir invariant
Casimir Casimir operatoroperator::
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QES-extension: continuation
2 23M ={ ( ) , ( ) ( ), ( ) }f x f x f x f x
1) Select the minimal invariant subspace:
4M ={ ( ) ( ), ( ) ( ), ( ) ( ), ( ) ( )}f x g x f x g x f x g x f x g x
2) Select the minimal invariant subspace:
2
2
( ) ( ) 0
( ) ( ) ( )
L f x L g x
d dL q x p x r x
dx dx
Condition for the functions f(x), g(x)
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QES-extension: continuationQES-extension: continuationExample 5 ( Example 5 ( QES Lie algebra )) = 0,1,2,...N
3dim = 3 1NV N
= 0,1,.., 1,n N N
3 * * *0 1 0 1 0 1= span{ , ,... , , ,..., , , ,..., }N N N NV f f f f f f f f f
2
211
1 11 1
2
* 1 2 111
= ( ) ;1/ 2
1= ( ) ( ) ; ;
1/ 2 3/ 2
1= ( ) ;
3 / 2
n nn
n nn
n nn
f x f x x F x
f x f x f x x F x F x
f x f x x F x
3NV
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3 22
10 03 2
4 3 22
10 1 04 3 2
20 1
9= 6 1 6 3 1 2
2 2
15= 5 2 14
4
276 4 , 2 2 , 9 6 8 10 2 ;
2
d x d N dJ x N x x x a x
dx dx dx
d d d dJ x x x x x a x a x
dx dx dx dx
N a x x N a x N x xN x
QES-extension: continuationQES-extension: continuationExample 5 ( Example 5 ( QES Lie algebra ))
3NV
10 1 2 1 3 1 4 2 5 2=n n n n n nJ C C C C C
5 4 4*
2 10 0
42 3
, 0, 0, 1 4 ;4
2 10 0
4
n
n n n
n
n
fn
f C C C
f n
1 2 1n n n n
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QES-extension: continuationQES-extension: continuationExample 6 ( Example 6 ( QES Lie algebra )) = 0,1,2,...N
4dim = 4 1NV N
= 0,1,.., 1,n N N40 1 0 1
* * *0 1 0 1
= span{ , ,... , , ,..., ,
, , ,..., , , ,..., }
N N N
N N
V f f f f f f
f f f f f f
1 11 1
11 11 1
*1 11 1
= ( ) ( ) ; ; ; 1/ 2 1/ 2
1 1 = ( ) ( ) ; ; ;
3 / 2 3/ 2
1= ( ) ( ) ; ; ;
1/ 2 3/ 2
= ( )
n nn
n nn
n nn
nn
f x f x g x x F x F x
f x x f x g x x F x F x
f x f x g x x F x F x
f x f x
1 11 1
1( ) ; ;
3 / 2 1/ 2ng x x F x F x
4NV
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QES-extension: continuationQES-extension: continuationExample 6 ( Example 6 ( QES Lie algebra ))
3 22 2
12 3 2
4 3 22 2
12 4 3 2
9 9= 1 3 4
4 2
15= 5 4 2 3 2
4
d d dJ x x N N x x xN
dx dx dx
d d d dJ x x x x x xN
dx dx dx dx
12 1 2 1 3 1 4 2 5 2=n n n n n nJ C C C C C
4NV
5 4 4*
2 10 0 0
42 1
0 0 04, 0, 0, 1 2 1 ;
2 31 4 0
42 3
1 4 04
n
nn
n
n
n
f nf
C C C n n nnf
fn
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Angular Momentum
ip
prL
kijkji LiLL ],[
0, 1, 2,......,lm l
QES quadratic Lie algebra 3NR
3 3 3, ,J J S
, ...ijp q pq i jX X c X X
0,1,2,....l = 0,1,2,....N
30,1,2,...,dim = 1N NR N
comparison
L L
CasimirF