1. 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 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. 33 (2000)

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

3. sl 2 (R)-Hamiltonians Representation: Invariant subspace Turbiner et al (partial algebraization)

4. 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

5. QES-extension: 0. our strategy 1)We find a general form of the operator of the second order P 2 for which subspace M 2 = span{f1, f2} is preserved. 2)We make extension of the subspace M 2 → M 4 = span{f 1, f 2, f 3, f 4 } 3)We find a general form of the operator of the second order P 4 for which subspace M 4 is preserved. 4)we obtain the explicit form of operator P 2(N+1) that acts on the elements of the subspace M 2(N+1) = {f1,f2,…, f 2(N+1) }

6. QES-extension: ; I. Select the minimal invariant subspace Select the invariant operator Condition for the subspace M 2

7. QES-extension: II. extension for the minimal invariant subspace Condition for the subspace M 4

8. QES-extension: Conditions of the QES-extension: 1 2 Wronskian matrix III. Extension for the minimal invariant subspace Order of derivatives

9. hypergeometric function Realization (special functions: hypergeometric, Airy, Bessel ones )

10. QES-extension: Particular choice of QES extension act more

11. QES-extension: Example 1 counter

12. QES-extension: The commutation relations of the operators Casimir operator: Casimir invariant

13. QES-extension: Example 2 counter

14. QES-extension: The commutation relations of the operators Casimir invariant Casimir operator:

15. QES-extension: Example 3 counter

16. QES-extension: The commutation relations of the operators

17. Two-photon Rabi Hamiltonian Rabi Hamiltonian describes a two-level system (atom) coupled to a single mode of radiation via dipole interaction.

18. Two-photon Rabi Hamiltonian

19. The two-photon Rabi Hamiltonian

22. Example matrix representation condition det(L 1 ) = 0

23. QES-extension: continuation Example 4 () Example 4 ( QES qubic Lie algebra )

24. QES-extension: continuation Example 4 ( ) The commutation relations of the operators QES-extension: continuation Example 4 ( QES qubic Lie algebra ) The commutation relations of the operators Casimir invariant Casimir operator:

25. QES-extension: continuation 1) Select the minimal invariant subspace: 2) Select the minimal invariant subspace: Condition for the functions f(x), g(x)

26. QES-extension: continuation Example 5 ( ) Example 5 ( QES Lie algebra )

27. QES-extension: continuation Example 5 ( ) Example 5 ( QES Lie algebra )

28. QES-extension: continuation Example 6 ( ) Example 6 ( QES Lie algebra )

29. QES-extension: continuation Example 6 ( ) Example 6 ( QES Lie algebra )

30. Angular Momentum QES quadratic Lie algebra comparison