49 research outputs found
A new proof of the analyticity of the electronic density of molecules
We give a new, short proof of the regularity away from the nuclei of the
electronic density of a molecule obtained in [1,2]. The new argument is based
on the regularity properties of the Coulomb interactions underlined in [3,4]
and on well-known elliptic technics. [1] S. Fournais, M. Hoffmann-Ostenhof, T.
Hoffmann-Ostenhof, T. Oe stergaard Soerensen: The electron density is smooth
away from the nuclei. Comm. Math. Phys. 228, no. 3 (2002), 401-415. [2] S.
Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, T. Oestergaard Soerensen:
Analyticity of the density of electronic wave functions. Ark. Mat. 42, no. 1
(2004), 87-106. [3] W. Hunziker: Distortion analyticity and molecular
resonances curves. Ann. Inst. H. Poincar\'e, s. A, t. 45, no 4, 339-358 (1986).
[4] M. Klein, A. Martinez, R. Seiler, X.P. Wang: On the Born-Oppenheimer
expansion for polyatomic molecules. Comm. Math. Phys. 143, no. 3, 607-639
(1992). The paper is published in Letters in Mathematical Physics 93, number 1,
pp. 73-83, 2010. The original publication is available at "
www.springerlink.com "
Strong resonant tunneling, level repulsion and spectral type for one-dimensional adiabatic quasi-periodic Schr\"{o}dinger operators
In this paper, we consider one dimensional adiabatic quasi-periodic
Schr\"{o}dinger operators in the regime of strong resonant tunneling. We show
the emergence of a level repulsion phenomenon which is seen to be very
naturally related to the local spectral type of the operator: the more singular
the spectrum, the weaker the repulsion
Absence of Embedded Mass Shells: Cerenkov Radiation and Quantum Friction
We show that, in a model where a non-relativistic particle is coupled to a
quantized relativistic scalar Bose field, the embedded mass shell of the
particle dissolves in the continuum when the interaction is turned on, provided
the coupling constant is sufficiently small. More precisely, under the
assumption that the fiber eigenvectors corresponding to the putative mass shell
are differentiable as functions of the total momentum of the system, we show
that a mass shell could exist only at a strictly positive distance from the
unperturbed embedded mass shell near the boundary of the energy-momentum
spectrum.Comment: Revised version: a remark added at the end of Section
Localization for Random Unitary Operators
We consider unitary analogs of dimensional Anderson models on
defined by the product where is a deterministic
unitary and is a diagonal matrix of i.i.d. random phases. The
operator is an absolutely continuous band matrix which depends on a
parameter controlling the size of its off-diagonal elements. We prove that the
spectrum of is pure point almost surely for all values of the
parameter of . We provide similar results for unitary operators defined on
together with an application to orthogonal polynomials on the unit
circle. We get almost sure localization for polynomials characterized by
Verblunski coefficients of constant modulus and correlated random phases
Limiting absorption principle for the dissipative Helmholtz equation
Adapting Mourre's commutator method to the dissipative setting, we prove a
limiting absorption principle for a class of abstract dissipative operators. A
consequence is the resolvent estimates for the high frequency Helmholtz
equation when trapped trajectories meet the set where the imaginary part of the
potential is non-zero. We also give the resolvent estimates in Besov spaces
Localization criteria for Anderson models on locally finite graphs
We prove spectral and dynamical localization for Anderson models on locally
finite graphs using the fractional moment method. Our theorems extend earlier
results on localization for the Anderson model on \ZZ^d. We establish
geometric assumptions for the underlying graph such that localization can be
proven in the case of sufficiently large disorder
On the Two Spectra Inverse Problem for Semi-Infinite Jacobi Matrices
We present results on the unique reconstruction of a semi-infinite Jacobi
operator from the spectra of the operator with two different boundary
conditions. This is the discrete analogue of the Borg-Marchenko theorem for
Schr{\"o}dinger operators in the half-line. Furthermore, we give necessary and
sufficient conditions for two real sequences to be the spectra of a Jacobi
operator with different boundary conditions.Comment: In this slightly revised version we have reworded some of the
theorems, and we updated two reference
On the structure of eigenfunctions corresponding to embedded eigenvalues of locally perturbed periodic graph operators
The article is devoted to the following question. Consider a periodic
self-adjoint difference (differential) operator on a graph (quantum graph) G
with a co-compact free action of the integer lattice Z^n. It is known that a
local perturbation of the operator might embed an eigenvalue into the
continuous spectrum (a feature uncommon for periodic elliptic operators of
second order). In all known constructions of such examples, the corresponding
eigenfunction is compactly supported. One wonders whether this must always be
the case. The paper answers this question affirmatively. What is more
surprising, one can estimate that the eigenmode must be localized not far away
from the perturbation (in a neighborhood of the perturbation's support, the
width of the neighborhood determined by the unperturbed operator only).
The validity of this result requires the condition of irreducibility of the
Fermi (Floquet) surface of the periodic operator, which is expected to be
satisfied for instance for periodic Schroedinger operators.Comment: Submitted for publicatio
Quantum harmonic oscillator systems with disorder
We study many-body properties of quantum harmonic oscillator lattices with
disorder. A sufficient condition for dynamical localization, expressed as a
zero-velocity Lieb-Robinson bound, is formulated in terms of the decay of the
eigenfunction correlators for an effective one-particle Hamiltonian. We show
how state-of-the-art techniques for proving Anderson localization can be used
to prove that these properties hold in a number of standard models. We also
derive bounds on the static and dynamic correlation functions at both zero and
positive temperature in terms of one-particle eigenfunction correlators. In
particular, we show that static correlations decay exponentially fast if the
corresponding effective one-particle Hamiltonian exhibits localization at low
energies, regardless of whether there is a gap in the spectrum above the ground
state or not. Our results apply to finite as well as to infinite oscillator
systems. The eigenfunction correlators that appear are more general than those
previously studied in the literature. In particular, we must allow for
functions of the Hamiltonian that have a singularity at the bottom of the
spectrum. We prove exponential bounds for such correlators for some of the
standard models
Krein-Space Formulation of PT-Symmetry, CPT-Inner Products, and Pseudo-Hermiticity
Emphasizing the physical constraints on the formulation of a quantum theory
based on the standard measurement axiom and the Schroedinger equation, we
comment on some conceptual issues arising in the formulation of PT-symmetric
quantum mechanics. In particular, we elaborate on the requirements of the
boundedness of the metric operator and the diagonalizability of the
Hamiltonian. We also provide an accessible account of a Krein-space derivation
of the CPT-inner product that was widely known to mathematicians since 1950's.
We show how this derivation is linked with the pseudo-Hermitian formulation of
PT-symmetric quantum mechanics.Comment: published version, 17 page