Computed Tomography by L. A. Shepp (ed.)

By L. A. Shepp (ed.)

During this quantity, the gathering of articles through Shepp, Helgason, Radon, and others, supplies mathematicians unexpected with utilized arithmetic a slightly complete spectrum of versions of computed tomography. integrated are great difficulties either proper and of intrinisic curiosity instructed through all of the papers

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231) and also the background phase shift δbg can be regarded as constants over the whole width of the resonance. For a broader resonance, however, the unique definition of its position and width can become a difficult problem (see also Sect. 3). The derivative of the phase shift with respect to energy is also a measure for the strength of the closed-channel component in the solution of the coupledchannel equations. 229). The 48 1 Review of Quantum Mechanics strength of the closed-channel admixture is quantitatively given by the square of the amplitude A cos δ in front of the (bound) wave function φ0 , which is normalized to unity.

Then the motion of the particle approaches that of a free particle asymptotically, provided the energy is large enough, E > V+ , E > V− . A particle incident from the left and travelling in the direction of the positive x-axis with a welldefined energy E > V− can be described by a monochromatic wave function, x→−∞ ¯ k = 2µ(E − V− ). The solution of the time-independent ψ ∝ eikx , with h Schr¨ odinger equation may also contain a leftward travelling contribution describing a part of the wave function reflected through the influence of the x→−∞ −ikx ∝ e .

5) G(r, r ) = −π φreg (r)φirr (r ) for r ≤ r , φreg (r )φirr (r) for r ≤ r . 220) is always smaller than r, because φ0 (r ) is a bound wave function so that the integrand vanishes for large r . 227) for G(r, r ) and perform the integration over r . 224) this leads to the following asymptotic form of φ1 (r): φ1 (r) = φreg (r) + tan δ φirr (r) 2µ sin(kr + δbg + δ) , π¯ h2 k and the angle δ ist given by = 1 cos δ tan δ = −π | φ0 |V2,1 |φreg |2 . 229) Being solutions of a homogeneous system of differential equations, the two-channel wave functions are determined only to within multiplication by a common arbitrary constant.

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