The existence of wave operators in scattering theory

1976 ◽  
Vol 146 (1) ◽  
pp. 69-91 ◽  
Author(s):  
Lars H�rmander
Keyword(s):  
Scattering Theory ◽  
Wave Operators

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.

Towards a Scattering Theory

2012 ◽  
Author(s):  
G. F. Roach ◽  
I. G. Stratis ◽  
A. N. Yannacopoulos
Keyword(s):  
Time Domain ◽  
Scattering Theory ◽  
Initial Boundary ◽  
Integral Transforms ◽  
Explicit Construction ◽  
Initial Boundary Value Problems ◽  
Evolution Operators ◽  
Wave Operators ◽  
The Time Domain ◽  
Initial Boundary Value

This chapter indicates how scattering theories can be developed when wave motions in chiral media are studied. To begin, this chapter remarks that a scattering process describes the effects of a perturbation on a system about which everything is known in the absence of the perturbation. It then presents a general formulation of the class of problems thus described in terms of evolution operators, and outlines an approach to scattering theory in the time domain. The chapter also shows how the use of spectral theory allows the explicit construction of solutions to abstract initial boundary value problems in terms of generalised integral transforms, and how these generalised integral transforms can be used for the construction of the wave operators and the scattering operator. Furthermore, this chapter explores the extension of these ideas to the study of electromagnetics of complex media.

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.

SCATTERING THEORY FOR ZAKHAROV EQUATIONS IN THREE-DIMENSIONAL SPACE WITH LARGE DATA

2004 ◽  
Vol 06 (06) ◽  
pp. 881-899 ◽  
Author(s):  
AKIHIRO SHIMOMURA
Keyword(s):  
Scattering Theory ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Large Data ◽  
Zakharov Equation ◽  
Unique Existence ◽  
Wave Operators ◽  
The Fourier Transform ◽  
The Given ◽  
Three Dimensional Space

We study the scattering theory for the Zakharov equation in three-dimensional space. We show the unique existence of the solution for this equation which tends to the given free profile with no restriction on the size of the scattered states and on the support of the Fourier transform of them. This yields the existence of the pseudo wave operators.

A transmission problem for the plate equation

1985 ◽  
Vol 99 (3-4) ◽  
pp. 285-312 ◽  
Author(s):  
R. Leis ◽  
G. F. Roach
Keyword(s):  
Wave Equation ◽  
Scattering Theory ◽  
Large Time ◽  
Asymptotic Methods ◽  
Transmission Problem ◽  
Plate Equation ◽  
Wave Operators ◽  
Transmission Problems ◽  
Regularity Of Solution ◽  
Dispersion Velocity

SynopsisA scattering theory is developed for transmission problems associated with the plate equation. Asymptotic methods of solution for large time are examined as are questions concerning regularity of solution, nature of the associated spectrum and existence of appropriate wave operators. It is shown that in contrast to solutions of the wave equation, signals can propagate with an infinite dispersion velocity.

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.

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