Singular components of spectral measures for ergodic Jacobi matrices

For ergodic 1d Jacobi operators we prove that the random singular components of any spectral measure are almost surely mutually disjoint as long as one restricts to the set of positive Lyapunov exponent. In the context of extended Harper's equation this yields the first rigorous proof of the Thouless' formula for the Lyapunov exponent in the dual regions.Comment: to appear in the Journal of Mathematical Physics, vol 52 (2011

Edge Currents for Quantum Hall Systems, I. One-Edge, Unbounded Geometries

Devices exhibiting the integer quantum Hall effect can be modeled by one-electron Schroedinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barrier create edge currents. In this, the first of two papers, we prove explicit lower bounds on the edge currents associated with one-edge, unbounded geometries formed by various confining potentials. This work extends some known results that we review. The edge currents are carried by states with energy localized between any two Landau levels. These one-edge geometries describe the electron confined to certain unbounded regions in the plane obtained by deforming half-plane regions. We prove that the currents are stable under various potential perturbations, provided the perturbations are suitably small relative to the magnetic field strength, including perturbations by random potentials. For these cases of one-edge geometries, the existence of, and the estimates on, the edge currents imply that the corresponding Hamiltonian has intervals of absolutely continuous spectrum. In the second paper of this series, we consider the edge currents associated with two-edge geometries describing bounded, cylinder-like regions, and unbounded, strip-like, regions.Comment: 68 page

Bound States at Threshold resulting from Coulomb Repulsion

The eigenvalue absorption for a many-particle Hamiltonian depending on a parameter is analyzed in the framework of non-relativistic quantum mechanics. The long-range part of pair potentials is assumed to be pure Coulomb and no restriction on the particle statistics is imposed. It is proved that if the lowest dissociation threshold corresponds to the decay into two likewise non-zero charged clusters then the bound state, which approaches the threshold, does not spread and eventually becomes the bound state at threshold. The obtained results have applications in atomic and nuclear physics. In particular, we prove that atomic ion with atomic critical charge $Z_{cr}$ and $N_e$ electrons has a bound state at threshold given that $Z_{cr} \in (N_e -2, N_e -1)$, whereby the electrons are treated as fermions and the mass of the nucleus is finite.Comment: This is a combined and updated version of the manuscripts arXiv:math-ph/0611075v2 and arXiv:math-ph/0610058v

Pauli-Fierz model with Kato-class potentials and exponential decays

Generalized Pauli-Fierz Hamiltonian with Kato-class potential \KPF in nonrelativistic quantum electrodynamics is defined and studied by a path measure. \KPF is defined as the self-adjoint generator of a strongly continuous one-parameter symmetric semigroup and it is shown that its bound states spatially exponentially decay pointwise and the ground state is unique.Comment: We deleted Lemma 3.1 in vol.

Multiscale Analysis in Momentum Space for Quasi-periodic Potential in Dimension Two

We consider a polyharmonic operator H=(-\Delta)^l+V(\x) in dimension two with $l\geq 2$, $l$ being an integer, and a quasi-periodic potential V(\x). We prove that the absolutely continuous spectrum of $H$ contains a semiaxis and there is a family of generalized eigenfunctions at every point of this semiaxis with the following properties. First, the eigenfunctions are close to plane waves $e^{i}$ at the high energy region. Second, the isoenergetic curves in the space of momenta \k corresponding to these eigenfunctions have a form of slightly distorted circles with holes (Cantor type structure). A new method of multiscale analysis in the momentum space is developed to prove these results.Comment: 125 pages, 4 figures. arXiv admin note: incorporates arXiv:1205.118

Effect of quasi-bound states on coherent electron transport in twisted nanowires

Quantum transmission spectra of a twisted electron waveguide expose the coupling between traveling and quasi-bound states. Through a direct numerical solution of the open-boundary Schr\"odinger equation we single out the effects of the twist and show how the presence of a localized state leads to a Breit-Wigner or a Fano resonance in the transmission. We also find that the energy of quasi-bound states is increased by the twist, in spite of the constant section area along the waveguide. While the mixing of different transmission channels is expected to reduce the conductance, the shift of localized levels into the traveling-states energy range can reduce their detrimental effects on coherent transport.Comment: 8 pages, 9 color figures, submitte

On the Geometry of Supersymmetric Quantum Mechanical Systems

We consider some simple examples of supersymmetric quantum mechanical systems and explore their possible geometric interpretation with the help of geometric aspects of real Clifford algebras. This leads to natural extensions of the considered systems to higher dimensions and more complicated potentials.Comment: 18 page

Existence of the Stark-Wannier quantum resonances

In this paper we prove the existence of the Stark-Wannier quantum resonances for one-dimensional Schrodinger operators with smooth periodic potential and small external homogeneous electric field. Such a result extends the existence result previously obtained in the case of periodic potentials with a finite number of open gaps.Comment: 30 pages, 1 figur

Superevolution

Usually, in supersymmetric theories, it is assumed that the time-evolution of states is determined by the Hamiltonian, through the Schr\"odinger equation. Here we explore the superevolution of states in superspace, in which the supercharges are the principal operators. The superevolution equation is consistent with the Schr\"odinger equation, but it avoids the usual degeneracy between bosonic and fermionic states. We discuss superevolution in supersymmetric quantum mechanics and in a simple supersymmetric field theory.Comment: 23 page

On Singularity formation for the L^2-critical Boson star equation

We prove a general, non-perturbative result about finite-time blowup solutions for the $L^2$-critical boson star equation $i\partial_t u = \sqrt{-\Delta+m^2} \, u - (|x|^{-1} \ast |u|^2) u$ in 3 space dimensions. Under the sole assumption that the solution blows up in $H^{1/2}$ at finite time, we show that $u(t)$ has a unique weak limit in $L^2$ and that $|u(t)|^2$ has a unique weak limit in the sense of measures. Moreover, we prove that the limiting measure exhibits minimal mass concentration. A central ingredient used in the proof is a "finite speed of propagation" property, which puts a strong rigidity on the blowup behavior of $u$. As the second main result, we prove that any radial finite-time blowup solution $u$ converges strongly in $L^2$ away from the origin. For radial solutions, this result establishes a large data blowup conjecture for the $L^2$-critical boson star equation, similar to a conjecture which was originally formulated by F. Merle and P. Raphael for the $L^2$-critical nonlinear Schr\"odinger equation in [CMP 253 (2005), 675-704]. We also discuss some extensions of our results to other $L^2$-critical theories of gravitational collapse, in particular to critical Hartree-type equations.Comment: 24 pages. Accepted in Nonlinearit