scholarly journals WAVE PROPAGATION AND SCATTERING FOR THE RS2 BRANE COSMOLOGY MODEL

2009 ◽  
Vol 06 (04) ◽  
pp. 809-861 ◽  
Author(s):  
ALAIN BACHELOT
Keyword(s):  
Cauchy Problem ◽  
Scattering Theory ◽  
Scattering Matrix ◽  
Asymptotic Completeness ◽  
Decay Estimates ◽  
Brane Cosmology ◽  
Dispersive Waves ◽  
Wave Operators ◽  
Global Cauchy Problem

We study the wave equation for the gravitational fluctuations in the Randall–Sundrum brane cosmology model. We solve the global Cauchy problem and we establish that the solutions are the sum of a slowly decaying massless wave localized near the brane, and a superposition of massive dispersive waves. We compute the kernel of the truncated resolvent. We prove some L1-L∞, L2-L∞ decay estimates and global Lp Strichartz type inequalities. We develop the complete scattering theory: existence and asymptotic completeness of the wave operators, computation of the scattering matrix, determination of the resonances on the logarithmic Riemann surface.

scholarly journals SPECTRAL ANALYSIS AND TIME-DEPENDENT SCATTERING THEORY ON MANIFOLDS WITH ASYMPTOTICALLY CYLINDRICAL ENDS

2013 ◽  
Vol 25 (02) ◽  
pp. 1350003 ◽  
Author(s):  
S. RICHARD ◽  
R. TIEDRA DE ALDECOA
Keyword(s):  
Spectral Analysis ◽  
Time Delay ◽  
Hilbert Spaces ◽  
Scattering Theory ◽  
Asymptotic Completeness ◽  
Time Dependent ◽  
Resolvent Estimates ◽  
Wave Operators ◽  
Use Of Time ◽  
Stationary Scattering

We review the spectral analysis and the time-dependent approach of scattering theory for manifolds with asymptotically cylindrical ends. For the spectral analysis, higher order resolvent estimates are obtained via Mourre theory for both short-range and long-range behaviors of the metric and the perturbation at infinity. For the scattering theory, the existence and asymptotic completeness of the wave operators is proved in a two-Hilbert spaces setting. A stationary formula as well as mapping properties for the scattering operator are derived. The existence of time delay and its equality with the Eisenbud–Wigner time delay is finally presented. Our analysis mainly differs from the existing literature on the choice of a simpler comparison dynamics as well as on the complementary use of time-dependent and stationary scattering theories.

scholarly journals On the Lax-Phillips scattering theory

1981 ◽  
Vol 87 (3-4) ◽  
pp. 219-230 ◽  
Author(s):  
W. O. Amrein ◽  
M. Wollenberg
Keyword(s):  
Time Delay ◽  
Scattering Theory ◽  
Scattering Matrix ◽  
Scattering Operator ◽  
Simple Description ◽  
Wave Operators ◽  
Scattering Systems ◽  
Delay Operator

SynopsisWe give a simple description of the wave operators appearing in the Lax-Phillips scattering theory. This is used to derive a relation between the scattering matrix and a kind of time delay operator and to characterize all scattering systems having the same scattering operator.

ASYMPTOTIC COMPLETENESS FOR THE SPIN-BOSON MODEL WITH A PARTICLE NUMBER CUTOFF

1996 ◽  
Vol 08 (04) ◽  
pp. 549-589 ◽  
Author(s):  
C. GÉRARD
Keyword(s):  
Ground State ◽  
Scattering Theory ◽  
Radiative Decay ◽  
Particle Number ◽  
Physical Phenomenon ◽  
Asymptotic Completeness ◽  
Simplified Model ◽  
Wave Operators ◽  
Channel Wave ◽  
Boson Model

We study the spin-boson model with a particle number cutoff. The spin-boson model is a simplified model of an atom interacting with a quantized photon field. An important physical phenomenon that one would like to understand rigorously on this model is the phenomenon of radiative decay, where the atom asymptotically relaxes to its ground state by emitting photons. One of the possible approaches to radiative decay is through scattering theory. For the cutoff spin-boson hamiltonian, we prove the existence and asymptotic completeness of the channel wave operators, which have natural interpretation in terms of the radiative decay property.

scholarly journals Mathematical scattering theory in quantum waveguides

2019 ◽  
Vol 489 (2) ◽  
pp. 142-146
Author(s):  
B. A. Plamenevskii ◽  
A. S. Poretskii ◽  
O. V. Sarafanov
Keyword(s):  
Continuous Spectrum ◽  
Scattering Theory ◽  
Scattering Matrix ◽  
Stationary Problem ◽  
Variable Coefficients ◽  
Scattering Operator ◽  
Wave Operators ◽  
Order Elliptic Operator ◽  
The Laplace Operator ◽  
Quantum Waveguides

A waveguide occupies a domain G with several cylindrical ends. The waveguide is described by a nonstationary equation of the form it f = Af ,where A is a selfadjoint second order elliptic operator with variable coefficients (in particular, for A = -, where stands for the Laplace operator, the equation coincides with the Schrodinger equation). For the corresponding stationary problem with spectral parameter, we define continuous spectrum eigenfunctions and a scattering matrix. The limiting absorption principle provides expansion in the continuous spectrum eigenfunctions. We also calculate wave operators and prove their completeness. Then we define a scattering operator and describe its connections with the scattering matrix.

scholarly journals Scattering theory for mathematical models of the weak interaction

2019 ◽  
Vol 32 (01) ◽  
pp. 2050002
Author(s):  
Benjamin Louis Alvarez ◽  
Jérémy Faupin
Keyword(s):  
Mathematical Models ◽  
Particle Physics ◽  
Scattering Theory ◽  
Adjoint Operator ◽  
High Energy ◽  
Weak Decay ◽  
Asymptotic Completeness ◽  
Coupling Constant ◽  
Wave Operators ◽  
The Masses

We consider mathematical models of the weak decay of the vector bosons [Formula: see text] into leptons. The free quantum field Hamiltonian is perturbed by an interaction term from the standard model of particle physics. After the introduction of high energy and spatial cut-offs, the total quantum Hamiltonian defines a self-adjoint operator on a tensor product of Fock spaces. We study the scattering theory for such models. First, the masses of the neutrinos are supposed to be positive: for all values of the coupling constant, we prove asymptotic completeness of the wave operators. In a second model, neutrinos are treated as massless particles and we consider a simpler interaction Hamiltonian: for small enough values of the coupling constant, we prove again asymptotic completeness, using singular Mourre’s theory, suitable propagation estimates and the conservation of the difference of some number operators.

scholarly journals Generic nature of asymptotic completeness in dissipative scattering theory

2020 ◽  
pp. 2060001
Author(s):  
Jérémy Faupin
Keyword(s):  
Scattering Theory ◽  
Optical Model ◽  
Asymptotic Completeness ◽  
Quantum Systems ◽  
The Self ◽  
Wave Operators ◽  
Effective Dynamics ◽  
Long Time ◽  
Dissipative Quantum Systems ◽  
Pure States

We review recent results obtained in the scattering theory of dissipative quantum systems representing the long-time evolution of a system [Formula: see text] interacting with another system [Formula: see text] and susceptible of being absorbed by [Formula: see text]. The effective dynamics of [Formula: see text] is generated by an operator of the form [Formula: see text] on the Hilbert space of the pure states of [Formula: see text], where [Formula: see text] is the self-adjoint generator of the free dynamics of [Formula: see text], [Formula: see text] is symmetric and [Formula: see text] is bounded. The main example is a neutron interacting with a nucleus in the nuclear optical model. We recall the basic objects of the scattering theory for the pair [Formula: see text], as well as the results, proven in [10, 11], on the spectral singularities of [Formula: see text] and the asymptotic completeness of the wave operators. Next, for the nuclear optical model, we show that asymptotic completeness generically holds.

scholarly journals Spectral and scattering theory for Schrödinger operators on perturbed topological crystals

2018 ◽  
Vol 30 (04) ◽  
pp. 1850009 ◽  
Author(s):  
D. Parra ◽  
S. Richard
Keyword(s):  
Scattering Theory ◽  
Short Range ◽  
Schrödinger Operators ◽  
Multiplication Operator ◽  
Asymptotic Completeness ◽  
Schrodinger Operators ◽  
Wave Operators ◽  
Range Function ◽  
Range Type ◽  
Local Wave

In this paper, we investigate the spectral and the scattering theory of Schrödinger operators acting on perturbed periodic discrete graphs. The perturbations considered are of two types: either a multiplication operator by a short-range or a long-range function, or a short-range type modification of the measure defined on the vertices and on the edges of the graph. Mourre theory is used for describing the nature of the spectrum of the underlying operators. For short-range perturbations, existence and asymptotic completeness of local wave operators are also proved.

Determination of a part of boundary for the Cauchy problem for the Laplace equation

2010 ◽  
Vol 18 (4) ◽  
pp. 535-548 ◽  
Author(s):  
Ji-Chuan Liu ◽  
Ting Wei
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
The Cauchy Problem

scholarly journals Decay estimates for the Cauchy problem for the damped extensible beam equation

2015 ◽  
Vol 95 (5) ◽  
pp. 1118-1136 ◽  
Author(s):  
Reinhard Racke ◽  
Shuji Yoshikawa
Keyword(s):  
Cauchy Problem ◽  
Beam Equation ◽  
Decay Estimates ◽  
The Cauchy Problem
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