Bethe–Sommerfeld Conjecture and absolutely continuous spectrum of multi-dimensional quasi-periodic Schrödinger operators

We consider a polyharmonic operator \documentclass[12pt]{minimal}\begin{document}$H=(-\Delta)^l+V({\vec{x}})$\end{document}H=(−Δ)l+V(x⃗) in dimension two with l ⩾ 2, l being an integer, and a quasi-periodic potential \documentclass[12pt]{minimal}\begin{document}$V({\vec{x}})$\end{document}V(x⃗). 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 \documentclass[12pt]{minimal}\begin{document}$e^{i\langle {\vec{\varkappa }},{\vec{x}}\rangle }$\end{document}ei⟨ϰ⃗,x⃗⟩ at the high energy region. Second, the isoenergetic curves in the space of momenta \documentclass[12pt]{minimal}\begin{document}${\vec{\varkappa }}$\end{document}ϰ⃗ 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.

We consider a polyharmonic operator H = ( − Δ)l + V(x) in dimension two with l ⩾ 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 ei⟨k, x⟩ 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.

We prove the existence of ballistic transport for the Schr\"odinger operator with limit-periodic or quasi-periodic potential in dimension two. This is done under certain regularity assumptions on the potential which have been used in prior work to establish the existence of an absolutely continuous component and other spectral properties. The latter include detailed information on the structure of generalized eigenvalues and eigenfunctions. These allow to establish the crucial ballistic lower bound through integration by parts on an appropriate extension of a Cantor set in momentum space, as well as through stationary phase arguments.

Quasicrystals are long-range ordered but not periodic, representing an interesting middle ground between order and disorder. We experimentally and numerically study the localization transition in the ground state of noninteracting and weakly interacting bosons in an eightfold symmetric quasicrystalline optical lattice. In contrast to typically used real space insitu techniques, we probe the system in momentum space by recording matter wave diffraction patterns. Shallow lattices lead to extended states whereas we observe a localization transition at a critical lattice depth of V_{0}≈1.78(2)E_{rec} for the noninteracting system. Our measurements and Gross-Pitaevskii simulations demonstrate that in interacting systems the transition is shifted to deeper lattices, as expected from superfluid order counteracting localization. Quasiperiodic potentials, lacking conventional rare regions, provide the ideal testing ground to realize many-body localization in 2D.

Localization of waves by disorder is a fundamental physical problem encompassing a diverse spectrum of theoretical, experimental and numerical studies in the context of metal-insulator transition, quantum Hall effect, light propagation in photonic crystals, and dynamics of ultra-cold atoms in optical arrays. Large intensity light can induce nonlinear response, ultracold atomic gases can be tuned into an interacting regime, which leads again to nonlinear wave equations on a mean field level. The interplay between disorder and nonlinearity, their localizing and delocalizing effects is currently an intriguing and challenging issue in the field. We will discuss recent advances in the dynamics of nonlinear lattice waves in random potentials. In the absence of nonlinear terms in the wave equations, Anderson localization is leading to a halt of wave packet spreading. Nonlinearity couples localized eigenstates and, potentially, enables spreading and destruction of Anderson localization due to nonintegrability, chaos and decoherence. The spreading process is characterized by universal subdiffusive laws due to nonlinear diffusion. We review extensive computational studies for one- and two-dimensional systems with tunable nonlinearity power. We also briefly discuss extensions to other cases where the linear wave equation features localization: Aubry-Andre localization with quasiperiodic potentials, Wannier-Stark localization with dc fields, and dynamical localization in momentum space with kicked rotors.

Motivated by the stunning projections for future CMB surveys, we evaluate the amount of dark radiation produced in the early Universe by two-body decays or binary scatterings with thermal bath particles via a rigorous analysis in momentum space. We track the evolution of the dark radiation phase space distribution, and we use the asymptotic solution to evaluate the amount of additional relativistic energy density parameterized in terms of an effective number of additional neutrino species ΔN eff. Our approach allows for studying light particles that never reach equilibrium across cosmic history, and to scrutinize the physics of the decoupling when they thermalize instead. We incorporate quantum statistical effects for all the particles involved in the production processes, and we account for the energy exchanged between the visible and invisible sectors. Non-instantaneous decoupling is responsible for spectral distortions in the final distributions, and we quantify how they translate into the corresponding value for ΔN eff. Finally, we undertake a comprehensive comparison between our exact results and approximated methods commonly employed in the existing literature. Remarkably, we find that the difference can be larger than the experimental sensitivity of future observations, justifying the need for a rigorous analysis in momentum space.

Quantum similarity for atoms is investigated using electron densities in position and momentum spaces. Contrary to the results in position space, the analysis in the momentum space shows how the momentum density carries fundamental information about periodicity and structure of the system and reveals the pattern of Mendeleev's table. A global analysis in the joint r-p space keeps this result.

Chemical bonding analysis in position space

We prove that a class of discrete Schrodinger operators with a quasi- periodic potential taking only a finite number of values, exhibits purely continuous spectrum; in particular they cannot have localized eigenvectors.

Integrable models form pillars of theoretical physics because they allow for full analytical understanding. Despite being rare, many realistic systems can be described by models that are close to integrable. Therefore, an important question is how small perturbations influence the behavior of solvable models. This is particularly true for many-body interacting quantum systems where no general theorems about their stability are known. Here, we show that no such theorem can exist by providing an explicit example of a one-dimensional many-body system in a quasiperiodic potential whose transport properties discontinuously change from localization to diffusion upon switching on interaction. This demonstrates an inherent instability of a possible many-body localization in a quasiperiodic potential at small interactions. We also show how the transport properties can be strongly modified by engineering potential at only a few lattice sites.

An imaginary gauge transformation is at the core of the non-Hermitian skin effect. Here, we show that such a transformation can be performed in momentum space as well, which reveals that certain gain- and loss-modulated systems in their parity-time (PT) symmetric phases are equivalent to Hermitian systems with real potentials. Our analysis in momentum space also distinguishes two types of exceptional points (EPs) in the same system. Besides the conventional type that leads to a PT transition upon the continuous increase of gain and loss, we find real-valued energy bands connected at a Dirac EP in hybrid dimensions, consisting of a spatial dimension and a synthetic dimension for the gain and loss strength.

Alongside conventional domain‐averaged Fermi hole (DAFH) analysis in position space, we examine an alternative representation in momentum space. As examples, we consider the processes of splitting the chemical bonds in the simple diatomic molecules H2, N2, and LiH, as representatives of nonpolar single, nonpolar multiple, and polar bonding. We believe that the additional information provided by the complementary description in momentum space contributes to a better understanding of the phenomenon of chemical bonding. © 2009 Wiley Periodicals, Inc. Int J Quantum Chem, 2009

The ground-state electron density $\ensuremath{\rho}(r)$ in a model of the He-like sequence of atomic ions, in which the Coulomb repulsion is replaced by the $s$-wave component, has recently been obtained by Howard and March [Phys. Rev. A 71, 042501 (2005)]. Their result is here generalized to derive the one-particle density matrix yielding the Howard and March $\ensuremath{\rho}(r)$ on the diagonal. Its momentum space counterpart is also presented and the momentum density is studied in detail, with particular emphasis on the asymptotic behavior of momentum distribution. Attention is then focussed on the kinetic energy density and on the spectral decomposition of the ground-state wave function. A natural orbital analysis in both direct and momentum space is also presented. Finally, the relevance for density functional theory of this exactly solvable model is briefly considered.

An extension of the Kunz–Souillard approach to localization in one dimension and applications to almost-periodic Schrödinger operators

A new multi-dimensional Hierarchical Structure Finder (HSF) to study the phase-space structure of dark matter in N-body cosmological simulations is presented. The algorithm depends mainly on two parameters, which control the level of connectivity of the detected structures and their significance compared to Poisson noise. By working in 6D phase-space, where contrasts are much more pronounced than in 3D position space, our HSF algorithm is capable of detecting subhaloes including their tidal tails, and can recognise other phase-space structures such as pure streams and candidate caustics. If an additional unbinding criterion is added, the algorithm can be used as a self-consistent halo and subhalo finder. As a test, we apply it to a large halo of the Millennium Simulation, where 19 % of the halo mass are found to belong to bound substructures, which is more than what is detected with conventional 3D substructure finders, and an additional 23-36 % of the total mass belongs to unbound HSF structures. The distribution of identified phase-space density peaks is clearly bimodal: high peaks are dominated by the bound structures and low peaks belong mostly to tidal streams. In order to better understand what HSF provides, we examine the time evolution of structures, based on the merger tree history. Bound structures typically make only up to 6 orbits inside the main halo. Still, HSF can identify at the present time at least 80 % of the original content of structures with a redshift of infall as high as z <= 0.3, which illustrates the significant power of this tool to perform dynamical analyses in phase-space.