Abstract:
We consider quantum analogues of several theorems on Birkhoff normal forms and prove a quantum analogue of the theorem on reducing a Hamiltonian to the normal form and the analogue of the theorem on reducing a Hamiltonian to the real normal form. We obtain the normal form explicitly in the nonresonant case. We consider the uniqueness problems of the normal form and of the normalizing transformation in the nonresonant and resonant cases.
Keywords:
Birkhoff normal form, quantum analogue, algebra of observables, automorphism.
This publication is cited in the following 5 articles:
E. M. Novikova, “New Approach to the Procedure of Quantum Averaging for the Hamiltonian of a Resonance Harmonic Oscillator with Polynomial Perturbation for the Example of the Spectral Problem for the Cylindrical Penning Trap”, Math. Notes, 109:5 (2021), 777–793
Novikova E.M., “On Calculating the Coefficients in the Quantum Averaging Procedure For the Hamiltonian of the Resonance Harmonic Oscillator Perturbed By a Differential Operator With Polynomial Coefficients”, Russ. J. Math. Phys., 28:3 (2021), 406–410
E. M. Novikova, “Algebra of Symmetries of Three-Frequency Hyperbolic Resonance”, Math. Notes, 106:6 (2019), 940–956
Anikin A., “Non-commutative normal form, spectrum and inverse problem”, Asymptotic Anal., 101:4 (2017), 207–225
Novikova, EM, “Minimal basis of the symmetry algebra for three-frequency resonance”, Russian Journal of Mathematical Physics, 16:4 (2009), 518