Trajectories of the Poles of the S-Matrix and Resonance Scattering

This article discusses some applications of random matrix theory (RMT) to quantum or wave chaotic resonance scattering. It first provides an overview of selected topics on universal statistics of resonances and scattering observables, with emphasis on theoretical results obtained via non-perturbative methods starting from the mid-1990s. It then considers the statistical properties of scattering observables at a given fixed value of the scattering energy, taking into account the maximum entropy approach as well as quantum transport and the Selberg integral. It also examines the correlation properties of the S-matrix at different values of energy and concludes by describing other characteristics and applications of RMT to resonance scattering of waves in chaotic systems, including those relating to time delays, quantum maps and sub-unitary random matrices, and microwave cavities at finite absorption.

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The analytic behaviour of Heisenberg’s S matrix

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1949.0137 ◽

1949 ◽

Vol 199 (1057) ◽

pp. 256-271 ◽

Cited By ~ 4

Keyword(s):

Excited States ◽

Threshold Energy ◽

Resonance Scattering ◽

Spectral Lines ◽

Matrix Elements ◽

S Matrix ◽

The Matrix ◽

Unitary Condition ◽

Analytic Behaviour ◽

Energy Plane

The S matrix is considered for a system made up of an elementary particle being scattered on a fixed centre which has internal excited states. At threshold energy values for inelastic scattering, the S matrix undergoes abrupt changes of behaviour. A method of representing these as non-analytic changes in the matrix elements as functions of the total energy is suggested, and some of the implications investigated. It is shown that it may still be possible for the eigenvalues of S to be analytic functions of energy at the threshold values. The usual perturbation theory of quantum mechanics is used to consider a resonance scattering system of this type, and it is shown that the non-analytic changes in the matrix elements correspond to a non-analytic change in the unitary condition of S . When the incident particle is a photon, the excited states of the scatterer are necessarily unstable, and the S -matrix elements have singularities in the complex energy plane which correspond to these unstable levels. These singular points show clearly the connexion between the line widths for resonance scattering and Einstein’s coefficients for spontaneous emission. It is shown that relative intensities of spectral lines may be obtained from the S matrix for scattering of light on an atom.

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Resonance Scattering by a Small Particle above a Perfectly Reflecting Periodic Rough Surface

Telecommunications and Radio Engineering ◽

10.1615/telecomradeng.v64.i10.40 ◽

2005 ◽

Vol 64 (10) ◽

pp. 819-831

Author(s):

F.G. Bass ◽

D. V. Mikhaylova ◽

V. Prosentsov ◽

L. Resnick

Keyword(s):

Rough Surface ◽

Small Particle ◽

Resonance Scattering

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POLES OF S-MATRIX: 1D WELL/BARRIER POTENTIAL

Anais da XXI Semana da Física ◽

10.29327/xxifisica.128766 ◽

2018 ◽

Author(s):

Alexandre Drinko ◽

Fabiano M. Andrade

Keyword(s):

Barrier Potential ◽

S Matrix

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1H NMR characterisation of the diffusional properties of methanogenic granular sludge

Water Science & Technology ◽

10.2166/wst.1999.0358 ◽

1999 ◽

Vol 39 (7) ◽

pp. 187-194 ◽

Cited By ~ 9

Author(s):

P. Lens ◽

F. Vergeldt ◽

G. Lettinga ◽

H. Van As

Keyword(s):

Diffusion Coefficient ◽

Relaxation Time ◽

Bacterial Cell ◽

Granular Sludge ◽

T2 Relaxation ◽

Self Diffusion ◽

Diffusion Analysis ◽

S Matrix ◽

Granular Matrix ◽

Pfg Nmr

The diffusive properties of mesophilic methanogenic granular sludge have been studied using diffusion analysis by relaxation time separated pulsed field gradient nuclear magnetic resonance (DARTS PFG NMR) spectroscopy. NMR measurements were performed at 22°C with 10 ml granular sludge at a magnetic field strength of 0.5 T (20 MHz resonance frequency for protons). Spin-spin relaxation (T2) time measurements indicate that three 1H populations can be distinguished in methanogenic granular sludge beds, corresponding to water in three different environments. The T2 relaxation time measurements clearly differentiate the extragranular water (T2 ≈ 1000 ms) from the water present in the granular matrix (T2 = 40-100 ms) and bacterial cell associated water (T2 = 10-15 ms). Self-diffusion coefficient measurements at 22°C of the different 1H-water populations as the tracer show that methanogenic granular sludge does not contain one unique diffusion coefficient. The observed distribution of self-diffusion coefficients varies between 1.1 × 10−9 m2/s (bacterial cell associated water) and 2.1 × 10−9 m2/s (matrix associated water).

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