Quasi-exactly solvable models in quantum mechanics and Lie algebras

1. Quasi-exactly solvable models in quantum mechanics andLie 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)

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

3. sl2(R)-Hamiltonians Turbiner et al Representation: Invariant subspace (partial algebraization)

4. What is being studied? • eigenvalues and eigenfunctions when possible. • Hamiltonians are formulated in terms of QES Lie algebras. How this is being studied? • Invariant subspaces: • Nonlinear QES Lie algebras

5. 0. QES-extension: our strategy • We find a general form of the operator of the second order P2 for which subspace M2 = span{f1, f2}is preserved. • We make extension of the subspace M2 → M4 = span{f1, f2, f3, f4} • We find a general form of the operator of the second order P4 for which subspace M4 is preserved. • 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)}

6. ; Select the invariant operator Condition for the subspace M2 I. QES-extension: Select the minimal invariant subspace

7. II. QES-extension: extension for the minimal invariant subspace Condition for the subspace M4

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

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

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

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 operator: Casimir invariant

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

20. The two-photon Rabi Hamiltonian

21. The two-photon Rabi Hamiltonian

22. Example matrix representation condition det(L1) = 0

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

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

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 ( QES Lie algebra)

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

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

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

30. comparison Angular Momentum QES quadratic Lie algebra