Giovanni Alberto Viano

The Stability of Inverse Problems

Many inverse problems arising in optics and other fields like geophysics, medical diagnostics and remote sensing, present numerical instability: the noise affecting the data may produce arbitrarily large errors in the solutions. In other words, these problems are ill-posed in the sense of Hadamard.

Reconstructing the thermal Green functions at real times from those at imaginary times

Abstract: By exploiting the analyticity and boundary value properties of the thermal Green functions that result from the KMS condition in both time and energy complex variables, we treat the general (non-perturbative) problem of recovering the thermal functions at real times from the corresponding functions at imaginary times, introduced as primary objects in the Matsubara formalism. The key property on which we rely is the fact that the Fourier transforms of the retarded and advanced functions in the energy variable have to be the “unique Carlsonian analytic interpolations” of the Fourier coefficients of the imaginary-time correlator, the latter being taken at the discrete Matsubara imaginary energies, respectively in the upper and lower half-planes. Starting from the Fourier coefficients regarded as “data set”, we then develop a method based on the Pollaczek polynomials for constructing explicitly their analytic interpolations.

On the problems of object restoration and image extrapolation in optics

In this paper we consider the problems of object restoration and image extrapolation, according to the regularization theory of improperly posed problems. In order to take into account the stochastic nature of the noise and to introduce the main concepts of information theory, great attention is devoted to the probabilistic methods of regularization. The kind of the restored continuity is investigated in detail; in particular we prove that, while the image extrapolation presents a Holder type stability, the object restoration has only a logarithmic continuity.

The source identification problem in electromagnetic theory

The problem of the identification of the electromagnetic source which produces an assigned radiation pattern is ill-posed: the solution is, in general, not unique and it does not depend continuously on the data. In this paper we treat in detail these two aspects of the problem. First of all we reconsider the radiation problem in the very general setting of the Sobolev spaces in order to make more acceptable, from a physical viewpoint, the conditions which have to be imposed on the electromagnetic sources. Then by the use of the Euclidean character of the Hilbert spaces we decompose the sources into a radiating and a non radiating component. We determine the subspace of the radiating sources and we find the basis spanning this subspace. Particular attention is then devoted to the case of the linear antenna. In this case the solution of the problem is unique but it does not depend continuously on the data. We may, however, implement the problem taking into account a bound on the ohmic losses. This is suffic...

Resolution beyond the diffraction limit for regularized object restoration

We propose a new formulation of Millers regularization theory, which is particularly suitable for object restoration problems. By means of simple geometrical arguments, we obtain upper and lower bounds for the errors on regularized solutions. This leads to distinguish between ‘Holder continuity’ which is quite good for practical computations and ‘logarithmic continuity’ which is very poor. However, in the latter case, one can reconstruct local weighted averages of the solution. This procedure allows for precise valuations of the resolution attainable in a given problem. Numerical computations, made for object restoration beyond the diffraction limit in Fourier optics, show that, when logarithmic continuity holds, the resolution is practically independent of the data noise level.

Further analysis of α-C12 scattering process by regge representation

SummaryAn analysis of two resonances in the low-energy α-l2C scattering is made by Regge’s representation. It is shown that by one- and two-pole approximations all the relevant features of the process are satisfactorily described. The background integral appears to be negligible except near the forward direction. Some discrepancies met in previous analysis of this type are overcome. The results indicate that a Regge description for this process can be made successfully when there is no interference of too many resonances belonging to different trajectories.RiassuntoMediante la rappresentazione di Regge vengono analizzate due risonanze nello scattering α-12C a bassa energia. Si mostra che median te approssimazione ad nno e due poli si deaorivono soddisfacentemente tutte le prinoipali caratteristiche del processo. L’integrale di background è trascurabile tranne che per angoli molto vicini alia direzione in avanti. Vengono superate alcune discrepanze inoontrate in precedenti analisi di questo tipo ed i risultati indicano one si può descrivere alia Regge questo processo quando non C’è interferenza di troppe risonanze appartenenti a traiettorie diverse.

The Born series for nonlocal potentials (S-wave)

SummaryA sufficient condition for the convergence of the Born series for nonlocal potentials is derived. The analysis is restricted to theS-wave Schrödinger equation. In order to have some information on the general structure of the scattering solution as a function of the potential strengthg and of the linear momentumk, the separable potentials are reconsidered.RiassuntoSi dimostra una condizione sufficiente per la convergenza della serie di Born nel caso di potenziali non locali. Si riconsiderano i potenziali separabili allo scopo di estrarne informazioni sulla dipendenza generale della soluzione di scattering dalla costante d’accoppiamentog e dell’impulsok.

Restoration of optical objects using regularization

Using the regularization theory for improperly posed problems, we discuss object restoration beyond the diffraction limit in the presence of noise. Only the case of one-dimensional coherent objects is considered. We focus attention n the estimation of the error on the restored objects, and we show that, in most realistic cases, it is at best proportional to an inverse power of |In epsilon|, where epsilon is the error on the data (logarithmic continuity). Finally we suggest the extension of this result to other inverse problems.

Bound states and Levinson's theorem for a class of non-local potentials (s-wave)

Abstract We present an improvement of the results, obtained in a previous paper, for a subclass of the class of non-local potentials considered there. With our method we obtain all the bound state solutions of Schrodingers equation, and we show that the total number of bound states and “spurious” bound states (i.e. bound states with positive energy) is finite. We prove also a representation of the S-matrix in terms of Fredholms determinants (this representation has been already deduced elsewhere in a formal way); as a consequence we deduce Levinsons theorem.

Volterra algebra and the Bethe–Salpeter equation

The diagonalization of the Bethe–Salpeter equation for the absorptive part of the amplitude is reconsidered. In particular the mathematical tools required by the diagonalization, like the Volterra algebra and the related spherical Laplace transform, are investigated in detail.