On a Heavy Quantum Particle
- 11 Downloads
We consider the Schrödinger equation for a particle on a flat n-torus with a bounded potential depending on time. The mass of the particle equals 1/μ2, where μ is a small parameter. We show that the Sobolev Hν-norms of the wave function grow approximately as tν on the time interval t ∈ [−t*, t*], where t* is slightly less than O(1/μ).
Unable to display preview. Download preview PDF.
- 2.D. Bambusi, B. Grebert, A. Maspero, and D. Robert, “Growth of Sobolev Norms for Abstract Linear Schrödinger Equations,” arXiv:1706.09708 [math.AP].Google Scholar
- 6.J. Bourgain, “Estimates on Green’s Functions, Localization and the Quantum Kicked Rotor Model,” Ann. of Math. 156 (249), (2002).Google Scholar
- 9.D. Fang, Z. Han, and W.–M. Wang, “Bounded Sobolev Norms for Klein–Gordon Equations under Non–Resonant Perturbation,” J. Math. Phys. 55 (12), (2014).Google Scholar
- 10.B. Grébert and E. Paturel, “KAM for the Klein–Gordon Equation on Sd,” ArXiv e–prints, arXiv: 1601.00610 (2016).Google Scholar
- 11.B. Grébert and E. Paturel, “On Reducibility of Quantum Harmonic Oscillator on ℝd with Quasiperiodic in Time Potential,” ArXiv e–prints, arXiv: 1603.07455 (2016).Google Scholar
- 13.F. M. Izrailev and D. L. Shepelyansky, “Quantum Resonans for a Rotator in Nonlinear Periodic Field,” Dokl. Phys. 43 (3), (1980).Google Scholar
- 17.A. Maspero, “Lower Bounds on the Growth of Sobolev Norms in Some Linear Time Dependent Schrödinger Equations,” arXiv:1801.06813v2 [math.AP].Google Scholar
- 18.R. Montalto, “A Reducibility Result for a Class of Linear Wave Equations on Td,” IMRN, rnx167, DOI: 10.1093/imrn/rnx167/ (2017).Google Scholar
- 20.M. E. Taylor, Partial Differential Equations (III Springer, 2012).Google Scholar