Improved energy estimates for a class of time-dependent perturbed Hamiltonians

Abstract

We consider time-dependent perturbations which are relatively bounded with respect to the square root of an unperturbed Hamiltonian operator, and whose commutator with the latter is controlled by the full perturbed Hamiltonian. The perturbation is modulated by two auxiliary parameters, one regulates its intensity as a prefactor and the other one controls its time-scale via a regular function, whose derivative is compactly supported in a finite interval. We introduce a natural generalization of energy conservation in the case of time-dependent Hamiltonians: the boundedness of the two-parameter unitary propagator for the physical evolution with respect to the $n/2$-th power energy norm for all $n\in \mathbb{Z}$. We provide bounds of the $n/2$-th power energy norms, uniformly in time and in the time-scale parameter, for the unitary propagators, generated by the time-dependent perturbed Hamiltonian and by the unperturbed Hamiltonian in the interaction picture. The physically interesting model of Landau-type Hamiltonians with an additional weak and time-slowly-varying electric potential of unit drop is included in this framework.

22 pages; in the Introduction and in Sect. 5 presentation of the results improved, Remark 2.6 added

References in corpus (4)

• Justifying Kubo's formula for gapped systems at zero temperature: a brief review and some new results
• A new approach to transport coefficients in the quantum spin Hall effect
• From charge to spin: analogies and differences in quantum transport coefficients
• Purely linear response of the quantum Hall current to space-adiabatic perturbations

Cited by in corpus (5)

• Justifying Kubo's formula for gapped systems at zero temperature: a brief review and some new results
• Localization of generalized Wannier bases implies Chern triviality in non-periodic insulators
• From charge to spin: analogies and differences in quantum transport coefficients
• On the self-consistent Landauer-Büttiker formalism
• Purely linear response of the quantum Hall current to space-adiabatic perturbations