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)

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

sl2(R)-Hamiltonians

Representation:

Invariantsubspace

Turbiner et al

0

2

2x

x

x

nJ x

J

J x nx

(partial algebraization)

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 ,

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)}

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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::

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)

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

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

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

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

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