pinecone/core-2020-05-10-deduplication huggingface dataset:
Flow-invariant hypersurfaces in semi-dispersing billiards

This work results from our attempts to solve Boltzmann-Sinai's hypothesis
about the ergodicity of hard ball gases. A crucial element in the studies of
the dynamics of hard balls is the analysis of special hypersurfaces in the
phase space consisting of degenerate trajectories (which lack complete
hyperbolicity). We prove that if a flow-invariant hypersurface $J$ in the phase
space of a semi-dispersing billiard has a negative Lyapunov function, then the
volume of the forward image of $J$ grows at least linearly in time. Our proof
is independent of the solution of the Boltzmann-Sinai hypothesis, and we
provide a complete and self-contained argument here
Scattering Poles Near the Real Axis for Two Strictly Convex Obstacles

To study the location of poles for the acoustic scattering matrix for two
strictly convex obstacles with smooth boundaries, one uses an approximation of
the quantized billiard operator $M$ along the trapped ray between the two
obstacles. Using this method Ikawa and G{\'e}rard established the existence of
parallel rows of poles in a strip $Im z\leq c$ as $Re z$ tends to infinity.
Assuming that the boundaries are analytic and the eigenvalues of Poincar{\'e}
map are non-resonant we use the Birkhoff normal form for $M$ to improve this
result and to get the complete asymptotic expansions for the poles in any
logarithmic neighborhood of real axis
On the quasi-static evolution of nonequilibrium steady states

The quasi-static evolution of steady states far from equilibrium is
investigated from the point of view of quantum statistical mechanics. As a
concrete example of a thermodynamic system, a two-level quantum dot coupled to
several reservoirs of free fermions at different temperatures is considered. A
novel adiabatic theorem for unbounded and nonnormal generators of evolution is
proven and applied to study the quasi-static evolution of nonequilibrium steady
states (NESS) of the coupled system.Comment: 39 pages. Some typos corrected. To appear in Ann. Henri Poincar
Non-isotropic cusp conditions and regularity of the electron density of
  molecules at the nuclei

We investigate regularity properties of molecular one-electron densities rho
near the nuclei. In particular we derive a representation rho(x)=mu(x)*(e^F(x))
with an explicit function F, only depending on the nuclear charges and the
positions of the nuclei, such that mu belongs to C^{1,1}(R^3), i.e., mu has
locally essentially bounded second derivatives. An example constructed using
Hydrogenic eigenfunctions shows that this regularity result is sharp. For
atomic eigenfunctions which are either even or odd with respect to inversion in
the origin, we prove that mu is even C^{2,\alpha}(R^3) for all alpha in (0,1).
Placing one nucleus at the origin we study rho in polar coordinates x=r*omega
and investigate rho'(r,omega) and rho''(r,omega) for fixed omega as r tends to
zero. We prove non-isotropic cusp conditions of first and second order, which
generalize Kato's classical result.Comment: 19 page
Continuity properties of integral kernels associated with Schroedinger
  operators on manifolds

For Schroedinger operators (including those with magnetic fields) with
singular (locally integrable) scalar potentials on manifolds of bounded
geometry, we study continuity properties of some related integral kernels: the
heat kernel, the Green function, and also kernels of some other functions of
the operator. In particular, we show the joint continuity of the heat kernel
and the continuity of the Green function outside the diagonal. The proof makes
intensive use of the Lippmann-Schwinger equation.Comment: 38 pages, major revision; to appear in Annales Henri Poincare (2007
Static vacuum solutions from convergent null data expansions at
  space-like infinity

We study formal expansions of asymptotically flat solutions to the static
vacuum field equations which are determined by minimal sets of freely
specifyable data referred to as `null data'. These are given by sequences of
symmetric trace free tensors at space-like infinity of increasing order. They
are 1:1 related to the sequences of Geroch multipoles. Necessary and sufficient
growth estimates on the null data are obtained for the formal expansions to be
absolutely convergent. This provides a complete characterization of all
asymptotically flat solutions to the static vacuum field equations.Comment: 65 page
Long Range Scattering and Modified Wave Operators for the
  Maxwell-Schr"odinger System II. The general case

We study the theory of scattering for the Maxwell-Schr"odinger system in
space dimension 3, in the Coulomb gauge. We prove the existence of modified
wave operators for that system with no size restriction on the Schr"odinger and
Maxwell asymptotic data and we determine the asymptotic behaviour in time of
solutions in the range of the wave operators. The method consists in partially
solving the Maxwell equations for the potentials, substituting the result into
the Schr"odinger equation, which then becomes both nonlinear and nonlocal in
time. The Schr"odinger function is then parametrized in terms of an amplitude
and a phase satisfying a suitable auxiliary system, and the Cauchy problem for
that system, with prescribed asymptotic behaviour determined by the asymptotic
data, is solved by an energy method, thereby leading to solutions of the
original system with prescribed asymptotic behaviour in time. This paper is the
generalization of a previous paper with the same title. However it is entirely
selfcontained and can be read without any previous knowledge of the latter.Comment: latex 96 page
Triviality of Bloch and Bloch-Dirac bundles

In the framework of the theory of an electron in a periodic potential, we
reconsider the longstanding problem of the existence of smooth and periodic
quasi-Bloch functions, which is shown to be equivalent to the triviality of the
Bloch bundle. By exploiting the time-reversal symmetry of the Hamiltonian and
some bundle-theoretic methods, we show that the problem has a positive answer
for any d < 4, thus generalizing a previous result by G. Nenciu. We provide a
general formulation of the result, aiming at the application to the Dirac
equation with a periodic potential and to piezoelectricity.Comment: 20 pages, no figure
On surface-symmetric spacetimes with collisionless and charged matter

Some future global properties of cosmological solutions for the
Einstein-Vlasov-Maxwell system with surface symmetry are presented. Global
existence is proved, the homogeneous spacetimes are future complete for causal
trajectories, and the same is true for inhomogeneous plane-symmetric solutions
with small initial data. In the latter case some decay properties are also
obtained at late times. Similar but slightly weaker results hold for hyperbolic
symmetry.Comment: 34 pages, version to be published in AH
The bipolaron in the strong coupling limit

The bipolaron are two electrons coupled to the elastic deformations of an
ionic crystal. We study this system in the Fr\"{o}hlich approximation. If the
Coulomb repulsion dominates, the lowest energy states are two well separated
polarons. Otherwise the electrons form a bound pair. We prove the validity of
the Pekar-Tomasevich energy functional in the strong coupling limit, yielding
estimates on the coupling parameters for which the binding energy is strictly
positive. Under the condition of a strictly positive binding energy we prove
the existence of a ground state at fixed total momentum $P$, provided $P$ is
not too large.Comment: 31 page
Distant perturbations of the Laplacian in a multi-dimensional space

We consider the Laplacian in $\mathbb{R}^n$ perturbed by a finite number of
distant perturbations those are abstract localized operators. We study the
asymptotic behaviour of the discrete spectrum as the distances between
perturbations tend to infinity. The main results are the convergence theorem
and the asymptotics expansions for the eigenelements. Some examples of the
possible distant perturbations are given; they are potential, second order
differential operator, magnetic Schrodinger operator, integral operator, and
$\d$-potential
Relative Haag Duality for the Free Field in Fock Representation

We consider a natural generalization of Haag duality to the case in which the
observable algebra is restricted to a subset of the space-time and is not
irreducible: the commutant and the causal complement have to be considered
relatively to the ambient space. We prove this relative form of Haag duality
under quite general conditions for the free scalar and electromagnetic field of
space dimension d>1 in the vacuum representation. Such property is interesting
in view of a theory of superselection sectors for the electromagnetic field.Comment: 22 page
Theory of Non-Equilibrium Sationary States as a Theory of Resonances

We study a small quantum system (e.g. a simplified model for an atom or
molecule) interacting with two bosonic or fermionic reservoirs (say, photon or
phonon fields). We show that the combined system has a family of stationary
states, parametrized by two numbers $T_1$, $T_2$ (``reservoir temperatures'').
If $T_1\neq T_2$, then these states are non-equilibrium, stationary states
(NESS). In the latter case we show that they have nonvanishing heat fluxes and
positive entropy production. Furthermore, we show that these states are
dynamically asymptotically stable. The latter means that the evolution with an
initial condition, normal with respect to any state where the reservoirs are in
equilibria at temperatures $T_1$ and $T_2$, converges to the corresponding
NESS. Our results are valid for the temperatures satisfying the bound
$\min(T_1, T_2) > g^{2+\alpha}$, where $g$ is the coupling constant and $0<
\alpha<1$ is a power related to the infra-red behaviour of the coupling
functions.Comment: 1 figure. To appear in Ann. H. Poincar
Rigidity and Positivity of Mass for Asymptotically Hyperbolic Manifolds

The Witten spinorial argument has been adapted in several works over the
years to prove positivity of mass in the asymptotically AdS and asymptotically
hyperbolic settings in arbitrary dimensions. In this paper we prove a scalar
curvature rigidity result and a positive mass theorem for asymptotically
hyperbolic manifolds that do not require a spin assumption. The positive mass
theorem is reduced to the rigidity case by a deformation construction near the
conformal boundary. The proof of the rigidity result is based on a study of
minimizers of the BPS brane action.Comment: 42 pages, 2 figure
Projecting Massive Scalar Fields to Null Infinity

It is known that, in an asymptotically flat spacetime, null infinity cannot
act as an initial-value surface for massive real scalar fields. Exploiting
tools proper of harmonic analysis on hyperboloids and global norm estimates for
the wave operator, we show that it is possible to circumvent such obstruction
at least in Minkowski spacetime. Hence we project norm-finite solutions of the
Klein-Gordon equation of motion in data on null infinity and, eventually, we
interpret them in terms of boundary free field theory.Comment: 26 page
On the volume of nodal sets for eigenfunctions of the Laplacian on the
  torus

We study the volume of nodal sets for eigenfunctions of the Laplacian on the
standard torus in two or more dimensions. We consider a sequence of eigenvalues
$4\pi^2\eigenvalue$ with growing multiplicity $\Ndim\to\infty$, and compute the
expectation and variance of the volume of the nodal set with respect to a
Gaussian probability measure on the eigenspaces. We show that the expected
volume of the nodal set is $const \sqrt{\eigenvalue}$. Our main result is that
the variance of the volume normalized by $\sqrt{\eigenvalue}$ is bounded by
$O(1/\sqrt{\Ndim})$, so that the normalized volume has vanishing fluctuations
as we increase the dimension of the eigenspace.Comment: 20 pages, Was accepted for publication in the Annales Henri Poincar
Spectral analysis of magnetic Laplacians on conformally cusp manifolds

We consider an open manifold which is the interior of a compact manifold with
boundary. Assuming gauge invariance, we classify magnetic fields with compact
support into being trapping or non-trapping. We study spectral properties of
the associated magnetic Laplacian for a class of Riemannian metrics which
includes complete hyperbolic metrics of finite volume. When $B$ is
non-trapping, the magnetic Laplacian has nonempty essential spectrum. Using
Mourre theory, we show the absence of singular continuous spectrum and the
local finiteness of the point spectrum. When $B$ is trapping, the spectrum is
discrete and obeys the Weyl law. The existence of trapping magnetic fields with
compact support depends on cohomological conditions, indicating a new and very
strong long-range effect.
  In the non-gauge invariant case, we exhibit a strong Aharonov-Bohm effect. On
hyperbolic surfaces with at least two cusps, we show that the magnetic
Laplacian associated to every magnetic field with compact support has purely
discrete spectrum for some choices of the vector potential, while other choices
lead to a situation of limit absorption principle.
  We also study perturbations of the metric. We show that in the Mourre theory
it is not necessary to require a decay of the derivatives of the perturbation.
This very singular perturbation is then brought closer to the perturbation of a
potential.Comment: 52 pages. Revised version: references added. To appear in Annales
  Henri Poincar\'
On spectral properties of translationally invariant magnetic
  Schr\"odinger operators

We consider a class of translationally invariant magnetic fields such that
the corresponding potential has a constant direction.
  Our goal is to study basic spectral properties of the Schr\"odinger operator
${\bf H}$ with such a potential. In particular, we show that the spectrum of
${\bf H}$ is absolutely continuous and we find its location. Then we study the
long-time behaviour of solutions $\exp(-i {\bf H} t)f$ of the time dependent
Schr\"odinger equation. It turnes out that a quantum particle remains localized
in the plane orthogonal to the direction of the potential. Its propagation in
this direction is determined by group velocities. It is to a some extent
similar to a evolution of a one-dimensional free particle but "exits" to
$+\infty$ and $-\infty$ might be essentially different
Infrared problem and spatially local observables in electrodynamics

An algebra previously proposed as an asymptotic field structure in
electrodynamics is considered in respect of localization properties of fields.
Fields are 'spatially local' -- localized in regions resulting as unions of two
intersecting (solid) lightcones: a future- and a past-lightcone. This
localization remains in concord with the usual idealizations connected with the
scattering theory. Fields thus localized naturally include infrared
characteristics normally placed at spacelike infinity and form a structure
respecting Gauss law. When applied to the description of the radiation of an
external classical current the model is free of 'infrared catastrophe'.Comment: 30 pages; accepted for publication in Ann. Henri Poincare; a few
  minor correction
Quasi-static Limits in Nonrelativistic Quantum Electrodynamics

We consider a system of N nonrelativistic particles of spin 1/2 interacting
with the quantized Maxwell field (mass zero and spin one) in the limit when the
particles have a small velocity, imposing to the interaction an ultraviolet
cutoff, but no infrared cutoff.
  Two ways to implement the limit are considered: c going to infinity with the
velocity v of the particles fixed, the case for which rigorous results have
already been discussed in the literature, and v going to 0 with c fixed. The
second case can be rephrased as the limit of heavy particles, m_{j} -->
epsilon^{-2}m_{j}, observed over a long time, t --> epsilon^{-1}t, epsilon -->
0^{+}, with kinetic energy E_{kin} = Or(1).
  Focusing on the second approach we construct subspaces which are invariant
for the dynamics up to terms of order epsilon sqrt{log(epsilon^{-1})} and
describe effective dynamics, for the particles only, inside them. At the lowest
order the particles interact through Coulomb potentials. At the second one,
epsilon^{2}, the mass gets a correction of electromagnetic origin and a
velocity dependent interaction, the Darwin term, appears.
  Moreover, we calculate the radiated piece of the wave function, i. e., the
piece which leaks out of the almost invariant subspaces and calculate the
corresponding radiated energy.Comment: 46 pages, no figures. Minor changes in the introduction and
  correction of some typos. Version accepted for publication in Annales Henri
  Poincare
Singular Yamabe metrics and initial data with exactly
  Kottler-Schwarzschild-de Sitter ends

We construct large families of initial data sets for the vacuum Einstein
equations with positive cosmological constant which contain exactly Delaunay
ends; these are non-trivial initial data sets which coincide with those for the
Kottler-Schwarzschild-de Sitter metrics in regions of infinite extent. From the
purely Riemannian geometric point of view, this produces complete, constant
positive scalar curvature metrics with exact Delaunay ends which are not
globally Delaunay. The ends can be used to construct new compact initial data
sets via gluing constructions. The construction provided applies to more
general situations where the asymptotic geometry may have non-spherical
cross-sections consisting of Einstein metrics with positive scalar curvature.Comment: Minor changes, updated references. Final version. To appear in
  Annales Henri Poincar
Semiclassical resolvent estimates for Schroedinger operators with
  Coulomb singularities

Consider the Schroedinger operator with semiclassical parameter h, in the
limit where h goes to zero. When the involved long-range potential is smooth,
it is well known that the boundary values of the operator's resolvent at a
positive energy E are bounded by O(1/h) if and only if the associated Hamilton
flow is non-trapping at energy E. In the present paper, we extend this result
to the case where the potential may possess Coulomb singularities. Since the
Hamilton flow then is not complete in general, our analysis requires the use of
an appropriate regularization.Comment: 39 pages, no figures, corrected versio
Structure and Classification of Superconformal Nets

We study the general structure of Fermi conformal nets of von Neumann
algebras on the circle, consider a class of topological representations, the
general representations, that we characterize as Neveu-Schwarz or Ramond
representations, in particular a Jones index can be associated with each of
them. We then consider a supersymmetric general representation associated with
a Fermi modular net and give a formula involving the Fredholm index of the
supercharge operator and the Jones index. We then consider the net associated
with the super-Virasoro algebra and discuss its structure. If the central
charge c belongs to the discrete series, this net is modular by the work of F.
Xu and we get an example where our setting is verified by considering the
Ramond irreducible representation with lowest weight c/24. We classify all the
irreducible Fermi extensions of any super-Virasoro net in the discrete series,
thus providing a classification of all superconformal nets with central charge
less than 3/2.Comment: 49 pages. Section 8 has been removed. More details concerning the
  diffeomorphism covariance are give
Edge Currents for Quantum Hall Systems, II. Two-Edge, Bounded and
  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 bounded or
unbounded subsets of the plane by confining potential barriers. The edges of
the confining potential barriers create edge currents. This is the second of
two papers in which we review recent progress and prove explicit lower bounds
on the edge currents associated with one- and two-edge geometries. In this
paper, we study various unbounded and bounded, two-edge geometries with soft
and hard confining potentials. These two-edge geometries describe the electron
confined to unbounded regions in the plane, such as a strip, or to bounded
regions, such as a finite length cylinder. We prove that the edge currents are
stable under various perturbations, provided they are suitably small relative
to the magnetic field strength, including perturbations by random potentials.
The existence of, and the estimates on, the edge currents are independent of
the spectral type of the operator.Comment: 57 page
Agmon-Type Estimates for a Class of Difference Operators

We analyze a general class of self-adjoint difference operators
$H_\varepsilon = T_\varepsilon + V_\varepsilon$ on
$\ell^2(\varepsilon\mathbb{Z}^d)$, where $V_\varepsilon$ is a one-well
potential and $\varepsilon$ is a small parameter. We construct a Finslerian
distance $d$ induced by $H_\varepsilon$ and show that short integral curves are
geodesics. Then we show that Dirichlet eigenfunctions decay exponentially with
a rate controlled by the Finsler distance to the well. This is analog to
semiclassical Agmon estimates for Schr\"odinger operators.Comment: 27 page
Asymptotic Behavior of Spherically Symmetric Marginally Trapped Tubes

We give conditions on a general stress-energy tensor T_{\alpha \beta} in a
spherically symmetric black hole spacetime which are sufficient to guarantee
that the black hole will contain a (spherically symmetric) marginally trapped
tube which is eventually achronal, connected, and asymptotic to the event
horizon. Price law decay per se is not required for this asymptotic result, and
in this general setting, such decay only implies that the marginally trapped
tube has finite length with respect to the induced metric. We do, however,
impose a smallness condition (B1) which one may obtain in practice by imposing
decay on the T_{vv} component of the stress-energy tensor. We give two
applications of the theorem to self-gravitating Higgs field spacetimes, one
using weak Price law decay, the other certain strong smallness and monotonicity
assumptions.Comment: 43 pages, 7 figures. Updated to agree with published version; Theorem
  1 strengthened slightly, minor issues fixe
Topological features of massive bosons on two dimensional Einstein
  space-time

In this paper we tackle the problem of constructing explicit examples of
topological cocycles of Roberts' net cohomology, as defined abstractly by
Brunetti and Ruzzi. We consider the simple case of massive bosonic quantum
field theory on the two dimensional Einstein cylinder. After deriving some
crucial results of the algebraic framework of quantization, we address the
problem of the construction of the topological cocycles. All constructed
cocycles lead to unitarily equivalent representations of the fundamental group
of the circle (seen as a diffeomorphic image of all possible Cauchy surfaces).
The construction is carried out using only Cauchy data and related net of local
algebras on the circle.Comment: 41 pages, title changed, minor changes, typos corrected, references
  added. Accepted for publication in Ann. Henri Poincare
Hilbert Lattice Equations

There are five known classes of lattice equations that hold in every infinite
dimensional Hilbert space underlying quantum systems: generalised
orthoarguesian, Mayet's E_A, Godowski, Mayet-Godowski, and Mayet's E equations.
We obtain a result which opens a possibility that the first two classes
coincide. We devise new algorithms to generate Mayet-Godowski equations that
allow us to prove that the fourth class properly includes the third. An open
problem related to the last class is answered. Finally, we show some new
results on the Godowski lattices characterising the third class of equations.Comment: 24 pages, 3 figure