Quasi-exactly solvable models in quantum mechanics and Lie algebras

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Quasi-exactly solvable models in quantum
mechanics and Lie algebras

• S. N. Dolya
• B. 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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Outline

• 1. QES-extension (A)
• 2. quadratic QES - Lie algebras
• 3. physical applications
• 4. QES-extension (B)
• 5. cubic QES - Lie algebras

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sl2(R)-Hamiltonians
Turbiner et al
Representation
Invariant subspace
(partial algebraization)
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What is being studied?

• Hamiltonians are formulated in terms of QES Lie
  algebras.

• eigenvalues and eigenfunctions when possible.

How this is being studied?

• Invariant subspaces

• Nonlinear QES Lie algebras

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0.
QES-extension
our strategy

1. We find a general form of the operator of the
   second order P2 for which subspace M2 spanf1,
   f2 is preserved.
2. We make extension of the subspace M2 ? M4
   spanf1, 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(N1)
   that acts on the elements of the subspace M2(N1)
   f1,f2,, f2(N1)

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I.
QES-extension
Select the minimal invariant subspace
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II.
QES-extension
extension for the minimal invariant subspace
Condition for the subspace M4
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III.
QES-extension
Extension for the minimal invariant subspace
Conditions of the QES-extension
Wronskian matrix
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2
Order of derivatives
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hypergeometric function
Realization (special functions hypergeometric,
Airy, Bessel ones)
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QES-extension
act more
Particular choice of QES extension
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QES-extension Example 1
counter
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QES-extension The commutation relations of the
operators
Casimir operator
Casimir invariant
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QES-extension Example 2
counter
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QES-extension The commutation relations of the
operators
Casimir operator
Casimir invariant
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QES-extension Example 3
counter
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QES-extension The commutation relations of the
operators
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Two-photon Rabi Hamiltonian
Rabi Hamiltonian describes a two-level system
(atom) coupled to a single mode of radiation via
dipole interaction.
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Two-photon Rabi Hamiltonian
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The two-photon Rabi Hamiltonian
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The two-photon Rabi Hamiltonian
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The two-photon Rabi Hamiltonian
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Example
matrix representation
condition det(L1) 0
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QES-extension continuation Example 4 (QES qubic
Lie algebra )
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QES-extension continuation Example 4 (QES qubic
Lie algebra ) The commutation relations of the
operators
Casimir operator
Casimir invariant
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QES-extension continuation
1) Select the minimal invariant subspace
2) Select the minimal invariant subspace
Condition for the functions f(x), g(x)
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QES-extension continuation Example 5 ( QES Lie
algebra )
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QES-extension continuation Example 5 ( QES Lie
algebra )
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QES-extension continuation Example 6 ( QES Lie
algebra )
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QES-extension continuation Example 6 ( QES Lie
algebra )
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comparison
Angular Momentum
QES quadratic Lie algebra