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.

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 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 TIME DELAY AND CALABI INVARIANT IN CLASSICAL SCATTERING THEORY

2012 ◽  
Vol 24 (09) ◽  
pp. 1250023
Author(s):  
A. GOURNAY ◽  
R. TIEDRA DE ALDECOA
Keyword(s):  
Quantum Mechanics ◽  
Time Delay ◽  
Explicit Expression ◽  
Scattering Theory ◽  
Scattering Matrix ◽  
Classical Particle ◽  
Hamiltonian Mechanics ◽  
Classical Version ◽  
Calabi Invariant ◽  
Scattering Map

We define, prove the existence and obtain explicit expressions for classical time delay defined in terms of sojourn times for abstract scattering pairs (H0, H) on a symplectic manifold. As a by-product, we establish a classical version of the Eisenbud–Wigner formula of quantum mechanics. Using recent results of Buslaev and Pushnitski on the scattering matrix in Hamiltonian mechanics, we also obtain an explicit expression for the derivative of the Calabi invariant of the Poincaré scattering map. Our results are applied to dispersive Hamiltonians, to a classical particle in a tube and to Hamiltonians on the Poincaré ball.

scholarly journals TIME DELAY FOR DISPERSIVE SYSTEMS IN QUANTUM SCATTERING THEORY

2009 ◽  
Vol 21 (05) ◽  
pp. 675-708 ◽  
Author(s):  
RAFAEL TIEDRA DE ALDECOA
Keyword(s):  
Time Delay ◽  
Scattering Theory ◽  
Integral Formula ◽  
Radial Component ◽  
Quantum Scattering ◽  
Scattering Operator ◽  
Localization Operators ◽  
Quantum Scattering Theory ◽  
One Dimensional ◽  
Klein Gordon

We consider time delay and symmetrized time delay (defined in terms of sojourn times) for quantum scattering pairs {H0 = h(P), H}, where h(P) is a dispersive operator of hypoelliptic-type. For instance, h(P) can be one of the usual elliptic operators such as the Schrödinger operator h(P) = P2 or the square-root Klein–Gordon operator [Formula: see text]. We show under general conditions that the symmetrized time delay exists for all smooth even localization functions. It is equal to the Eisenbud–Wigner time delay plus a contribution due to the non-radial component of the localization function. If the scattering operator S commutes with some function of the velocity operator ∇h(P), then the time delay also exists and is equal to the symmetrized time delay. As an illustration of our results, we consider the case of a one-dimensional Friedrichs Hamiltonian perturbed by a finite rank potential. Our study puts into evidence an integral formula relating the operator of differentiation with respect to the kinetic energy h(P) to the time evolution of localization operators.

scholarly journals SPECTRAL AND SCATTERING THEORY FOR THE AHARONOV–BOHM OPERATORS

2011 ◽  
Vol 23 (01) ◽  
pp. 53-81 ◽  
Author(s):  
KONSTANTIN PANKRASHKIN ◽  
SERGE RICHARD
Keyword(s):  
Scattering Theory ◽  
High Energy ◽  
Low Energy ◽  
Scattering Operator ◽  
Wave Operators ◽  
Bohm Model ◽  
Spectral And Scattering Theory ◽  
Aharonov Bohm

We review the spectral and the scattering theory for the Aharonov–Bohm model on ℝ2. New formulae for the wave operators and for the scattering operator are presented. The asymptotics at high energy and at low energy of the scattering operator are computed.

scholarly journals The three‐body time‐delay operator

1975 ◽  
Vol 16 (8) ◽  
pp. 1533-1549 ◽  
Author(s):  
T. A. Osborn ◽  
D. Bollé
Keyword(s):  
Time Delay ◽  
Three Body ◽  
Delay Operator
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