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

Monetary Policy under the Microscope: Intra-bank Transmission of Asset Purchase Programs of the ECB

With a unique loan portfolio maintained by a top-20 universal bank in Germany, this study tests whether unconventional monetary policy by the European Central Bank (ECB) reduced corporate borrowing costs. We decompose corporate lending rates into refinancing costs, as determined by money markets, and markups that the bank is able to charge its customers in regional markets. This decomposition reveals how banks transmit monetary policy within their organizations. To identify policy effects on loan rate components, we exploit the co-existence of eurozone-wide security purchase programs and regional fiscal policies at the district level. ECB purchase programs reduced refinancing costs significantly, even in an economy not specifically targeted for sovereign debt stress relief, but not loan rates themselves. However, asset purchases mitigated those loan price hikes due to additional credit demand stimulated by regional tax policy and enabled the bank to realize larger economic margins

Monetary policy under the microscope: Intra-bank transmission of asset purchase programs of the ECB

Based on detailed loan portfolio data of a top-20 universal bank in Germany, we investigate the effect of unconventional monetary policy on corporate loan pricing. We can decompose corporate lending rates, thereby shedding light on intra-bank transmission of monetary policy. We identify policy effects on contracted customer rates, refinancing rates charged internally, markups earned by the bank, and loan volumes by exploiting the co-existence of eurozone-wide security purchase programs by the European Central Bank (ECB) and local fiscal policies that are determined autonomously at the district level where bank customers reside between August 2011 until December 2013. The purchase programs of the ECB reduced refinancing costs significantly. Local fiscal stimuli increased loan prices and margins earned. The differential effect of unconventional expansionary monetary policy given local tax environments is significantly negative. Lending volumes do not respond significantly though

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

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

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

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

Completeness of the set of scattering amplitudes

Let $f\in L^2(S^2)$ be an arbitrary fixed function with small norm on the unit sphere $S^2$, and $D\subset \R^3$ be an arbitrary fixed bounded domain. Let $k>0$ and $\alpha\in S^2$ be fixed. It is proved that there exists a potential $q\in L^2(D)$ such that the corresponding scattering amplitude $A(\alpha')=A_q(\alpha')=A_q(\alpha',\alpha,k)$ approximates $f(\alpha')$ with arbitrary high accuracy: \|f(\alpha')-A_q(\alpha')_{L^2(S^2)}\|\leq\ve where \ve>0 is an arbitrarily small fixed number. This means that the set $\{A_q(\alpha')\}_{\forall q\in L^2(D)}$ is complete in $L^2(S^2)$. The results can be used for constructing nanotechnologically "smart materials"

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