109 research outputs found
Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W+/-. I
We consider a Hamiltonian with cutoffs describing the weak decay of spin one
massive bosons into the full family of leptons. The Hamiltonian is a
self-adjoint operator in an appropriate Fock space with a unique ground state.
We prove a Mourre estimate and a limiting absorption principle above the ground
state energy and below the first threshold for a sufficiently small coupling
constant. As a corollary, we prove absence of eigenvalues and absolute
continuity of the energy spectrum in the same spectral interval.Comment: Correction of minor misprint
Spectral theory for a mathematical model of the weak interaction: The decay of the intermediate vector bosons W+/-, II
We do the spectral analysis of the Hamiltonian for the weak leptonic decay of
the gauge bosons W+/-. Using Mourre theory, it is shown that the spectrum
between the unique ground state and the first threshold is purely absolutely
continuous. Neither sharp neutrino high energy cutoff nor infrared
regularization are assumed.Comment: To appear in Ann. Henri Poincar\'
Van der Waals–London interaction of atoms with pseudo-relativistic kinetic energy
We consider a multiatomic system where the nuclei are assumed to be point charges at fixed positions. Particles interact via Coulomb potential and electrons have pseudo–relativistic kinetic energy. We prove the van der Waals-London law, which states that the interaction energy between neutral atoms decays as the sixth power of the distance between the atoms. In the many atom case, we rigorously compute all the terms in the binding energy up to the order with error term of order . This yields the first proof of the famous Axilrod–Teller–Muto three–body correction to the van der Waals–London interaction, which plays an important role in atom physics. As intermediate steps we prove exponential decay of eigenfunctions of multiparticle Schrödinger operators with permutation symmetry imposed by the Pauli principle, and new estimates of the localization error
Quantum electrodynamics of relativistic bound states with cutoffs
We consider an Hamiltonian with ultraviolet and infrared cutoffs, describing
the interaction of relativistic electrons and positrons in the Coulomb
potential with photons in Coulomb gauge. The interaction includes both
interaction of the current density with transversal photons and the Coulomb
interaction of charge density with itself. We prove that the Hamiltonian is
self-adjoint and has a ground state for sufficiently small coupling constants.Comment: To appear in "Journal of Hyperbolic Differential Equation
Quantitative estimates on the Hydrogen ground state energy in non-relativistic QED
In this paper, we determine the exact expression for the hydrogen binding
energy in the Pauli-Fierz model up to the order ,
where denotes the finestructure constant, and prove rigorous bounds on
the remainder term of the order . As a consequence,
we prove that the binding energy is not a real analytic function of ,
and verify the existence of logarithmic corrections to the expansion of the
ground state energy in powers of , as conjectured in the recent
literature.Comment: AMS Latex, 51 page
Exponential localization of hydrogen-like atoms in relativistic quantum electrodynamics
We consider two different models of a hydrogenic atom in a quantized
electromagnetic field that treat the electron relativistically. The first one
is a no-pair model in the free picture, the second one is given by the
semi-relativistic Pauli-Fierz Hamiltonian. We prove that the no-pair operator
is semi-bounded below and that its spectral subspaces corresponding to energies
below the ionization threshold are exponentially localized. Both results hold
true, for arbitrary values of the fine-structure constant, , and the
ultra-violet cut-off, , and for all nuclear charges less than the
critical charge without radiation field, . We obtain
similar results for the semi-relativistic Pauli-Fierz operator, again for all
values of and and for nuclear charges less than .Comment: 37 page
Intermixture of extended edge and localized bulk energy levels in macroscopic Hall systems
We study the spectrum of a random Schroedinger operator for an electron
submitted to a magnetic field in a finite but macroscopic two dimensional
system of linear dimensions equal to L. The y direction is periodic and in the
x direction the electron is confined by two smooth increasing boundary
potentials. The eigenvalues of the Hamiltonian are classified according to
their associated quantum mechanical current in the y direction. Here we look at
an interval of energies inside the first Landau band of the random operator for
the infinite plane. In this energy interval, with large probability, there
exist O(L) eigenvalues with positive or negative currents of O(1). Between each
of these there exist O(L^2) eigenvalues with infinitesimal current
O(exp(-cB(log L)^2)). We explain what is the relevance of this analysis to the
integer quantum Hall effect.Comment: 29 pages, no figure
Quantum interference in nanofractals and its optical manifestation
We consider quantum interferences of ballistic electrons propagating inside
fractal structures with nanometric size of their arms. We use a scaling
argument to calculate the density of states of free electrons confined in a
simple model fractal. We show how the fractal dimension governs the density of
states and optical properties of fractal structures in the RF-IR region. We
discuss the effect of disorder on the density of states along with the
possibility of experimental observation.Comment: 19 pages, 6 figure
Norm estimates of complex symmetric operators applied to quantum systems
This paper communicates recent results in theory of complex symmetric
operators and shows, through two non-trivial examples, their potential
usefulness in the study of Schr\"odinger operators. In particular, we propose a
formula for computing the norm of a compact complex symmetric operator. This
observation is applied to two concrete problems related to quantum mechanical
systems. First, we give sharp estimates on the exponential decay of the
resolvent and the single-particle density matrix for Schr\"odinger operators
with spectral gaps. Second, we provide new ways of evaluating the resolvent
norm for Schr\"odinger operators appearing in the complex scaling theory of
resonances
Unique Solutions to Hartree-Fock Equations for Closed Shell Atoms
In this paper we study the problem of uniqueness of solutions to the Hartree
and Hartree-Fock equations of atoms. We show, for example, that the
Hartree-Fock ground state of a closed shell atom is unique provided the atomic
number is sufficiently large compared to the number of electrons. More
specifically, a two-electron atom with atomic number has a unique
Hartree-Fock ground state given by two orbitals with opposite spins and
identical spatial wave functions. This statement is wrong for some , which
exhibits a phase segregation.Comment: 18 page
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