44 research outputs found

### Completeness of the Coulomb scattering wave functions

Completeness of the eigenfunctions of a self-adjoint Hamiltonian, which is
the basic ingredient of quantum mechanics, plays an important role in nuclear
reaction and nuclear structure theory. However, until now, there was no a
formal proof of the completeness of the eigenfunctions of the two-body
Hamiltonian with the Coulomb interaction. Here we present the first formal
proof of the completeness of the two-body Coulomb scattering wave functions for
repulsive unscreened Coulomb potential. To prove the completeness we use the
Newton's method [R. Newton, J. Math Phys., 1, 319 (1960)]. The proof allows us
to claim that the eigenfunctions of the two-body Hamiltonian with the potential
given by the sum of the repulsive Coulomb plus short-range (nuclear) potentials
also form a complete set. It also allows one to extend the Berggren's approach
of modification of the complete set of the eigenfunctions by including the
resonances for charged particles. We also demonstrate that the resonant Gamow
functions with the Coulomb tail can be regularized using Zel'dovich's
regularization method.Comment: 12 pages and 1 figur

### Resonance Theory of Decoherence and Thermalization

We present a rigorous analysis of the phenomenon of decoherence for general
$N-$level systems coupled to reservoirs. The latter are described by free
massless bosonic fields. We apply our general results to the specific cases of
the qubit and the quantum register. We compare our results with the explicitly
solvable case of systems whose interaction with the environment does not allow
for energy exchange (non-demolition, or energy conserving interactions). We
suggest a new approach which applies to a wide variety of systems which are not
explicitly solvable

### Uniqueness of the ground state in the Feshbach renormalization analysis

In the operator theoretic renormalization analysis introduced by Bach,
Froehlich, and Sigal we prove uniqueness of the ground state.Comment: 10 page

### Solitary Wave Dynamics in an External Potential

We study the behavior of solitary-wave solutions of some generalized
nonlinear Schr\"odinger equations with an external potential. The equations
have the feature that in the absence of the external potential, they have
solutions describing inertial motions of stable solitary waves.
We construct solutions of the equations with a non-vanishing external
potential corresponding to initial conditions close to one of these solitary
wave solutions and show that, over a large interval of time, they describe a
solitary wave whose center of mass motion is a solution of Newton's equations
of motion for a point particle in the given external potential, up to small
corrections corresponding to radiation damping.Comment: latex2e, 41 pages, 1 figur

### Scattering theory for Klein-Gordon equations with non-positive energy

We study the scattering theory for charged Klein-Gordon equations:
\{{array}{l} (\p_{t}- \i v(x))^{2}\phi(t,x) \epsilon^{2}(x,
D_{x})\phi(t,x)=0,[2mm] \phi(0, x)= f_{0}, [2mm] \i^{-1} \p_{t}\phi(0, x)=
f_{1}, {array}. where: \epsilon^{2}(x, D_{x})= \sum_{1\leq j, k\leq
n}(\p_{x_{j}} \i b_{j}(x))A^{jk}(x)(\p_{x_{k}} \i b_{k}(x))+ m^{2}(x),
describing a Klein-Gordon field minimally coupled to an external
electromagnetic field described by the electric potential $v(x)$ and magnetic
potential $\vec{b}(x)$. The flow of the Klein-Gordon equation preserves the
energy: h[f, f]:= \int_{\rr^{n}}\bar{f}_{1}(x) f_{1}(x)+
\bar{f}_{0}(x)\epsilon^{2}(x, D_{x})f_{0}(x) - \bar{f}_{0}(x) v^{2}(x) f_{0}(x)
\d x. We consider the situation when the energy is not positive. In this
case the flow cannot be written as a unitary group on a Hilbert space, and the
Klein-Gordon equation may have complex eigenfrequencies. Using the theory of
definitizable operators on Krein spaces and time-dependent methods, we prove
the existence and completeness of wave operators, both in the short- and
long-range cases. The range of the wave operators are characterized in terms of
the spectral theory of the generator, as in the usual Hilbert space case

### On Spectra of Linearized Operators for Keller-Segel Models of Chemotaxis

We consider the phenomenon of collapse in the critical Keller-Segel equation
(KS) which models chemotactic aggregation of micro-organisms underlying many
social activities, e.g. fruiting body development and biofilm formation. Also
KS describes the collapse of a gas of self-gravitating Brownian particles. We
find the fluctuation spectrum around the collapsing family of steady states for
these equations, which is instrumental in derivation of the critical collapse
law. To this end we develop a rigorous version of the method of matched
asymptotics for the spectral analysis of a class of second order differential
operators containing the linearized Keller-Segel operators (and as we argue
linearized operators appearing in nonlinear evolution problems). We explain how
the results we obtain are used to derive the critical collapse law, as well as
for proving its stability.Comment: 22 pages, 1 figur

### Dynamics of Collective Decoherence and Thermalization

We analyze the dynamics of N interacting spins (quantum register)
collectively coupled to a thermal environment. Each spin experiences the same
environment interaction, consisting of an energy conserving and an energy
exchange part.
We find the decay rates of the reduced density matrix elements in the energy
basis. We show that if the spins do not interact among each other, then the
fastest decay rates of off-diagonal matrix elements induced by the energy
conserving interaction is of order N^2, while that one induced by the energy
exchange interaction is of the order N only. Moreover, the diagonal matrix
elements approach their limiting values at a rate independent of N.
For a general spin system the decay rates depend in a rather complicated (but
explicit) way on the size N and the interaction between the spins.
Our method is based on a dynamical quantum resonance theory valid for small,
fixed values of the couplings. We do not make Markov-, Born- or weak coupling
(van Hove) approximations

### Proof of the ionization conjecture in a reduced Hartree-Fock model

The ionization conjecture for atomic models states that the ionization energy and maximal excess charge are bounded by constants independent of the nuclear charge. We prove this for the Hartree-Fock model without the exchange term.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/46574/1/222_2005_Article_BF01245077.pd

### Asymptotic stability of breathers in some Hamiltonian networks of weakly coupled oscillators

We consider a Hamiltonian chain of weakly coupled anharmonic oscillators. It
is well known that if the coupling is weak enough then the system admits
families of periodic solutions exponentially localized in space (breathers). In
this paper we prove asymptotic stability in energy space of such solutions. The
proof is based on two steps: first we use canonical perturbation theory to put
the system in a suitable normal form in a neighborhood of the breather, second
we use dispersion in order to prove asymptotic stability. The main limitation
of the result rests in the fact that the nonlinear part of the on site
potential is required to have a zero of order 8 at the origin. From a technical
point of view the theory differs from that developed for Hamiltonian PDEs due
to the fact that the breather is not a relative equilibrium of the system

### On scattering of solitons for the Klein-Gordon equation coupled to a particle

We establish the long time soliton asymptotics for the translation invariant
nonlinear system consisting of the Klein-Gordon equation coupled to a charged
relativistic particle. The coupled system has a six dimensional invariant
manifold of the soliton solutions. We show that in the large time approximation
any finite energy solution, with the initial state close to the solitary
manifold, is a sum of a soliton and a dispersive wave which is a solution of
the free Klein-Gordon equation. It is assumed that the charge density satisfies
the Wiener condition which is a version of the ``Fermi Golden Rule''. The proof
is based on an extension of the general strategy introduced by Soffer and
Weinstein, Buslaev and Perelman, and others: symplectic projection in Hilbert
space onto the solitary manifold, modulation equations for the parameters of
the projection, and decay of the transversal component.Comment: 47 pages, 2 figure