in strongly correlated electron physics a dmft 在强关联电子物理的龋齿问题课件

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),Outline,1. QES-extension (A) 2. quadratic QES - Lie algebras 3. physical applications 4. QES-extension (B) 5. cubic QES - Lie algebras,sl2(R)-Hamiltonians,Representation:,Invariant subspace,Turbiner et al,(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,QES-extension:,0.,our strategy,We find a general form of the operator of the second order P2 for which subspace M2 = spanf1, f2 is preserved. We make extension of the subspace M2 M4 = spanf1, 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),QES-extension:,I.,Select the minimal invariant subspace,QES-extension:,II.,extension for the minimal invariant subspace,Condition for the subspace M4,QES-extension:,Conditions of the QES-extension:,1,2,Wronskian matrix,III.,Extension for the minimal invariant subspace,Order of derivatives,hypergeometric function,Realization (special functions: hypergeometric, Airy, Bessel ones),QES-extension:,Particular choice of QES extension,act more,QES-extension: Example 1,counter,QES-extension: The commutation relations of the operators,Casimir operator:,Casimir invariant,QES-extension: Example 2,counter,QES-extension: The commutation relations of the operators,Casimir invariant,Casimir operator:,QES-extension: Example 3,counter,QES-extension: The commutation relations of the operators,Two-photon Rabi Hamiltonian,Rabi Hamiltonian describes a two-level system (atom) coupled to a single mode of radiation via dipole interaction.,Two-photon Rabi Hamiltonian,The two-photon Rabi Hamiltonian,The two-photon Rabi Hamiltonian,The two-photon Rabi Hamiltonian,Example,matrix representation,condition det(L1) = 0,QES-extension: continuation Example 4 (QES qubic Lie algebra ),QES-extension: continuation Example 4 (QES qubic Lie algebra ) The commutation relations of the operators,Casimir invariant,Casimir operator:,QES-extension: continuation,1) Select the minimal invariant subspace:,2) Select the minimal invariant subspace:,Condition for the functions f(x), g(x),QES-extension: continuation Example 5 ( QES Lie algebra ),QES-extension: continuation Example 5 ( QES Lie algebra ),QES-extension: cont