Many-photon ionization, resonance scattering on a nonstationary potential, and complex poles of the S matrix

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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