Joachim Asch

Motion in periodic potentials

We consider motion in a periodic potential in a classical, quantum, and semiclassical context. Various results on the distribution of asymptotic velocities are proven.

Stability of Driven Systems with Growing Gaps, Quantum Rings, and Wannier Ladders

We consider a quantum particle in a periodic structure submitted to a constant external electromotive force. The periodic background is given by a smooth potential plus singular point interactions and has the property that the gaps between its bands are growing with the band index. We prove that the spectrum is pure point—i.e., trajectories of wave packets lie in compact sets in Hilbert space—if the Bloch frequency is nonresonant with the frequency of the system and satisfies a Diophantine-type estimate, or if it is resonant. Furthermore, we show that the KAM method employed in the nonresonant case produces uniform bounds on the growth of energy for driven systems.

Quantum Transport on KAM Tori

Abstract:Although quantum tunneling between phase space tori occurs, it is suppressed in the semiclassical limit ћ ↘ 0 for the Schrödinger equation of a particle in ℝd under the influence of a smooth periodic potential.In particular this implies that the distribution of quantum group velocities near energy E converges to the distribution of the classical asymptotic velocities near E, up to a term of the order

Lower Bounds on the Width of Stark-Wannier Type Resonances

\mathcal{O}(1/\sqrt E )

Sojourn time for rank one perturbations

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Propagators weakly associated to a family of Hamiltonians and the adiabatic theorem for the Landau Hamiltonian with a time-dependent Aharonov–Bohm flux

We prove that the Schrödinger operator −d2/dx2+Fx+W(x) onL2(R) withW bounded and analytic in a strip has no resonances in a region ImE≥−exp(−C/F).

Dynamics of a classical Hall system driven by a time-dependent Aharonov-Bohm flux

We consider a self-adjoint, purely absolutely continuous operator M. Let P be a rank one operator Pu=⟨φ,u⟩φ such that for β0 Hβ0≔M+β0P has a simple eigenvalue E0 embedded in its absolutely continuous spectrum, with corresponding eigenvector ψ. Let Hω be a rank one perturbation of the operator Hβ0, namely, Hω=M+(β0+ω)P. Under suitable conditions, the operator Hω has no point spectrum in a neighborhood of E0, for ω small. Here, we study the evolution of the state ψ under the Hamiltonian Hω, in particular, we obtain explicit estimates for its sojourn time τω(ψ)=∫−∞∞∣⟨ψ,e−iHωtψ⟩∣2dt. By perturbation theory, we prove that τω(ψ) is finite for ω≠0, and that for ω small it is of order ω−2. Finally, by using an analytic deformation technique, we estimate the sojourn time for the Friedrichs model in Rn.

A constant of quantum motion in two dimensions in crossed magnetic and electric fields

We study the dynamics of a quantum particle moving in a plane under the influence of a constant magnetic field and driven by a slowly time-dependent singular flux tube through a puncture. The known standard adiabatic results do not cover directly these models as the Hamiltonian has time-dependent domain. We give a meaning to the propagator and prove an adiabatic theorem. To this end we introduce and develop the new notion of a propagator weakly associated to a time-dependent Hamiltonian.

Resonant cyclotron acceleration of particles by a time periodic singular flux tube

We study the dynamics of a classical particle moving in a punctured plane under the influence of a homogeneous magnetic field, an electric background, and driven by a time-dependent singular flux tube through the hole. We exhibit a striking (de)localization effect: when the electric background is absent we prove that a linearly time-dependent flux tube opposite to the homogeneous flux eventually leads to the usual classical Hall motion: the particle follows a cycloid whose center is drifting orthogonal to the electric field; if the fluxes are additive, the drifting center eventually gets pinned by the flux tube whereas the kinetic energy is growing with the additional flux.

On the Dynamics of Crystal Electrons, High Momentum Regime

We consider the quantum dynamics of a single particle in the plane under the influence of a constant perpendicular magnetic and a crossed electric potential field. For a class of smooth and small potentials we construct a non-trivial invariant of motion. Do to so we proof that the Hamiltonian is unitarily equivalent to an effective Hamiltonian which commutes with the observable of kinetic energy.