2015/10/23 by Mariagiovanna Gianfreda, Gianfreda, Mariagiovanna, Giulio Landolfi +1
Mathematics · Physics and Astronomy · #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Physical sciences #Mathematical Physics (math-ph) #Quantum Mechanics and Non-Hermitian Physics #Quantum Physics (quant-ph) #Quantum chaos and dynamical systems #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.1510.06893
openalex publication_date 2015/10/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We address the problem of integrating operator equations concomitant with the\ndynamics of non autonomous quantum systems by taking advantage of the use of\ntime-dependent canonical transformations. In particular, we proceed to a\ndiscussion in regard to basic examples of one-dimensional non-autonomous\ndynamical systems enjoying the property that their Hamiltonian can be mapped\nthrough a time-dependent linear canonical transformation into an autonomous\nform, up to a time-dependent multiplicative factor. The operator equations we\nprocess essentially reproduce at the quantum level the classical integrability\ncondition for these systems. Operator series form solutions in the Bender-Dunne\nbasis of pseudo-differential operators for one dimensional quantum system are\nsought for such equations. The derivation of generating functions for the\ncoefficients involved in the \minimal representation of the series\nsolutions to the operator equations under consideration is particularized. We\nalso provide explicit form of operators that implement arbitrary linear\ntransformations on the Bender-Dunne basis by expressing them in terms of the\ninitial Weyl ordered basis elements. We then remark that the matching of the\nminimal solutions obtained independently in the two basis, i.e. the basis prior\nand subsequent the action of canonical linear transformation, is perfectly\nachieved by retaining only the lowest order contribution in the expression of\nthe transformed Bender-Dunne basis elements.\n