arXiv subjects
Publications and source records attributed to Joachim Asch.
We establish limiting absorption principles for contractions on a Hilbert space. Our sufficient conditions are based on positive commutator estimates. We discuss the dynamical implications of this principle to the corresponding discrete-time semigroup and provide several applications. Notably to Toeplitz operators and contractive quantum walks.
We provide a mathematical analysis of two models advocated in the theoretical and experimental condensed matter literature: the two dimensional spin-$1/2$ Quantum Walk and the Kagome quantum network; they apply to occurrence of stable quantum currents. This illustrates the theory of stable absolutely continuous spectrum and stable currents developed in \cite{ABJ5}.
We study transport properties of discrete quantum dynamical systems on the lattice, in particular Coined Quantum Walks and the Chalker--Coddington model. We prove existence of a non trivial charge transport and that the absolutely continuous spectrum covers the whole unit circle under mild assumptions. For Quantum Walks we exhibit explicit constructions of coins which imply existence of stable directed quantum currents along classical curves. The results are of topological nature and independent of the details of the model.
We consider the dynamical properties of Quantum Walks defined on the d-dimensional cubic lattice, or the homogeneous tree of coordination number 2d, with site dependent random phases, further characterised by transition probabilities between neighbouring sites equal to 1/(2d). We show that the localisation length for these Balanced Random Quantum Walks can be expressed as a combinatorial expression involving sums over weighted paths on the considered graph. This expression provides lower bounds on the localisation length by restriction to paths with weight 1, which allows us to prove the localisation length diverges on the tree as d^2. On the cubic lattice, the method yields the lower bound 1/ln(2) for all d, and allows us to bound the localisation length from below by the correlation length of self-avoiding walks computed at 1/(2d)
We study transport properties of a Chalker-Coddington type model in the plane which presents asymptotically pure anti-clockwise rotation on the left and clockwise rotation on the right. We prove delocalisation in the sense that the absolutely continuous spectrum covers the whole unit circle. The result is of topological nature and independent of the details of the model.
One manifestation of quantum resonances is a large sojourn time, or autocorrelation, for states which are initially localized. We elaborate on Lavine's time-energy uncertainty principle and give an estimate on the sojourn time. For the case of perturbed embedded eigenstates the bound is explicit and involves Fermi's Golden Rule. It is valid for a very general class of systems. We illustrate the theory by applications to resonances for time dependent systems including the AC Stark effect as well as multistate systems.
We consider the magnetic AC Stark effect for the quantum dynamics of a single particle in the plane under the influence of an oscillating homogeneous electric and a constant perpendicular magnetic field. We prove that the electron cyclotron resonance is insensitive to impurity potentials.
We review various unitary network models used in quantum computing, spectral analysis or condensed matter physics and establish relationships between them. We show that symmetric one dimensional quantum walks are universal, as are CMV matrices. We prove spectral stability and propagation properties for general asymptotically uniform models by means of unitary Mourre theory.
We study a quantum network percolation model which is numerically pertinent to the understanding of the delocalization transition of the quantum Hall effect. We show dynamical localization for parameters corresponding to edges of Landau bands, away from the expected transition point.
We study the dynamics of a classical nonrelativistic charged particle moving on a punctured plane under the influence of a homogeneous magnetic field and driven by a periodically time-dependent singular flux tube through the hole. We observe an effect of resonance of the flux and cyclotron frequencies. The particle is accelerated to arbitrarily high energies even by a flux of small field strength which is not necessarily encircled by the cyclotron orbit; the cyclotron orbits blow up and the particle oscillates between the hole and infinity. We support this observation by an analytic study of an approximation for small amplitudes of the flux which is obtained with the aid of averaging methods. This way we derive asymptotic formulas that are afterwards shown to represent a good description of the accelerated motion even for fluxes which are not necessarily small. More precisely, we argue that the leading asymptotic terms may be regarded as approximate solutions of the original system in the asymptotic domain as the time tends to infinity.
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.
The Chalker Coddington quantum network percolation model is numerically pertinent to the understanding of the delocalization transition of the quantum Hall effect. We study the model restricted to a cylinder of perimeter 2M. We prove firstly that the Lyapunov exponents are simple and in particular that the localization length is finite; secondly that this implies spectral localization. Thirdly we prove a Thouless formula and compute the mean Lyapunov exponent which is independent of M.
Although quantum tunneling between phase space tori occurs, it is suppressed in the semiclassical limit $\hbar\searrow 0$ for the Schrödinger equation of a particle in $\bR^d$ 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 $\cO(1/\sqrt{E})$.
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 non-resonant with the frequency of the system and satisfies a Diophantine type estimate, or if it is resonant. Furthermore it is shown that the KAM method employed in the non-resonant case produces uniform bounds on the growth of energy for driven systems.