Solving the inverse problem of quantum scattering theory

K. Chadan, Pierre C. Sabatier, Roger G. Newton

B. M. Levitan

Tamás Pálmai

George L. Lamb

The amplification of a coherent optical pulse is analyzed by employing an inverse scattering method. In the extreme Doppler limit, one obtains an analytical expression that exhibits the expected evolution toward a $\ensuremath{\pi}$ pulse with pulse amplification and compression. In the sharp-line limit, one obtains justification for previous usage of a similarity solution to describe $\ensuremath{\pi}$-pulse propagation in an amplifier. The analysis is carried out for an initial pulse profile (truncated hyperbolic secant) which yields a sufficiently simple scattering problem that the Marchenko equation of inverse scattering theory may be solved analytically.

U. Mark Kac

Introduction.We give three applications of probability to scattering theory.The first question we consider is that of the "scattering length."We obtain a probabilistic expression for scattering length and use this expression to prove that in the case of an impenetrable obstacle ("hard" potential) the scattering length T is the same as the electrostatic capacity.Next, we briefly touch upon a problem which is inherently probabilistic-namely, scattering in a medium with random boundary.We find that if a large number of small hard spheres ("dust particles") are distributed at random in the medium, the effect is to shift all the eigenvalues by a constant.Our final topic is the so-called "inverse problem" of scattering theory: given the scattering operator, to compute the potential.For certain spherically symmetric potentials, this problem was solved in a famous paper of GelTand and Levitan.Here we show that their result can be included in the circle of potential-theoretic problems which can be solved by means of Brownian motion and Wiener integrals.We quickly sketch some of the physical background.We are concerned with the Schrodinger equation of non-relativistic quantum mechanics.In its stationary or time-independent form, this equation isThe "wave function" I/J depends on both the position vector fiEfi 3 and the wave number k.The real parameter k 2 is proportional to the energy.The function q(R) is called the "potential."It represents the interaction of the quantum-mechanical "particle" with some force field.The "scattering problem" is, roughly speaking, to determine the effect of this interaction asymptotically for large R.More precisely, we seek a solution of (1) which at |fl| = » is of the asymptotic form e iV2kz + fk(e)e iV ^k> ^l\\R\\ where e is the unit vector in the direction of R. The first term, e iV2kz , is an incident wave in the direction of the zaxis; the second term /fc(e)e iV ^I^H/||R||, is the "scattered wave."