We present a method able to recover location and residue of poles of functions meromorphic in a half--plane from samples of the function on the real positive semi-axis. The function is assumed to satisfy appropriate asymptotic conditions including, i
n particular, that required by Carlsons theorem. The peculiar features of the present procedure are: (i) it does not make use of the approximation of meromorphic functions by rational functions; (ii) it does not use the standard methods of regularization of ill-posed problems. The data required for the determination of the pole parameters (i.e., location and residue) are the approximate values of the meromorphic function on a finite set of equidistant points on the real positive semi-axis. We show that this method is numerically stable by proving that the algorithm is convergent as the number of data points tends to infinity and the noise on the input data goes to zero. Moreover, we can also evaluate the degree of approximation of the estimates of pole location and residue which we obtain from the knowledge of a finite number of noisy samples.
The inverse problem in optics, which is closely related to the classical question of the resolving power, is reconsidered as a communication channel problem. The main result is the evaluation of the maximum number $M_epsilon$ of $epsilon$-distinguish
able messages ($epsilon$ being a bound on the noise of the image) which can be conveyed back from the image to reconstruct the object. We study the case of coherent illumination. By using the concept of Kolmogorovs $epsilon$-capacity, we obtain: $M_epsilon ~ 2^{S log(1/epsilon)} to infty$ as $epsilon to 0$, where S is the Shannon number. Moreover, we show that the $epsilon$-capacity in inverse optical imaging is nearly equal to the amount of information on the object which is contained in the image. We thus compare the results obtained through the classical information theory, which is based on the probability theory, with those derived from a form of topological information theory, based on Kolmogorovs $epsilon$-entropy and $epsilon$-capacity, which are concepts related to the evaluation of the massiveness of compact sets.
In this article we study resonances and surface waves in $pi^+$--p scattering. We focus on the sequence whose spin-parity values are given by $J^p = {3/2}^+,{7/2}^+, {11/2}^+, {15/2}^+,{19/2}^+$. A widely-held belief takes for granted that this seque
nce can be connected by a moving pole in the complex angular momentum (CAM) plane, which gives rise to a linear trajectory of the form $J = alpha_0+alpha m^2$, $alphasim 1/(mathrm{GeV})^2$, which is the standard expression of the Regge pole trajectory. But the phenomenology shows that only the first few resonances lie on a trajectory of this type. For higher $J^p$ this rule is violated and is substituted by the relation $Jsim kR$, where $k$ is the pion--nucleon c.m.s.-momentum, and $Rsim 1$ fm. In this article we prove: (a) Starting from a non-relativistic model of the proton, regarded as composed by three quarks confined by harmonic potentials, we prove that the first three members of this $pi^+$-p resonance sequence can be associated with a vibrational spectrum of the proton generated by an algebra $Sp(3,R)$. Accordingly, these first three members of the sequence can be described by Regge poles and lie on a standard linear trajectory. (b) At higher energies the amplitudes are dominated by diffractive scattering, and the creeping waves play a dominant role. They can be described by a second class of poles, which can be called Sommerfelds poles, and lie on a line nearly parallel to the imaginary axis of the CAM-plane. (c) The Sommerfeld pole which is closest to the real axis of the CAM-plane is dominant at large angles, and describes in a proper way the backward diffractive peak in both the following cases: at fixed $k$, as a function of the scattering angle, and at fixed scattering angle $theta=pi$, as a function of $k$. (d) The evolution of this pole, as a function of $k$, is given in first approximation by $Jsimeq kR$.
In this article, the ray tracing method is studied beyond the classical geometrical theory. The trajectories are here regarded as geodesics in a Riemannian manifold, whose metric and topological properties are those induced by the refractive index (o
r, equivalently, by the potential). First, we derive the geometrical quantization rule, which is relevant to describe the orbiting bound-states observed in molecular physics. Next, we derive properties of the diffracted rays, regarded here as geodesics in a Riemannian manifold with boundary. A particular attention is devoted to the following problems: (i) modification of the classical stationary phase method suited to a neighborhood of a caustic; (ii) derivation of the connection formulae which enable one to obtain the uniformization of the classical eikonal approximation by patching up geodesic segments crossing the axial caustic; (iii) extension of the eikonal equation to mixed hyperbolic-elliptic systems, and generation of complex-valued rays in the shadow of the caustic. By these methods, we can study the creeping waves in diffractive scattering, describe the orbiting resonances present in molecular scattering beside the orbiting bound-states, and, finally, describe the generation of the evanescent waves, which are relevant in the nuclear rainbow.
We present a simple and fast algorithm for the computation of the coefficients of the expansion of a function f(cos u) in ultraspherical (Gegenbauer) polynomials. We prove that these coefficients coincide with the Fourier coefficients of an Abel-type
transform of the function f(cos u). This allows us to fully exploit the computational efficiency of the Fast Fourier Transform, computing the first N ultraspherical coefficients in just O (N log_2 N) operations.
The purpose of this paper is to establish meromorphy properties of the partial scattering amplitude T(lambda,k) associated with physically relevant classes N_{w,alpha}^gamma of nonlocal potentials in corresponding domains D_{gamma,alpha}^delta of the
space C^2 of the complex angular momentum lambda and of the complex momentum k (namely, the square root of the energy). The general expression of T as a quotient Theta(lambda,k)/sigma(lambda,k) of two holomorphic functions in D_{gamma,alpha}^delta is obtained by using the Fredholm-Smithies theory for complex k, at first for lambda=l integer, and in a second step for lambda complex (Real(lambda)>-1/2). Finally, we justify the Watson resummation of the partial wave amplitudes in an angular sector of the lambda-plane in terms of the various components of the polar manifold of T with equation sigma(lambda,k)=0. While integrating the basic Regge notion of interpolation of resonances in the upper half-plane of lambda, this unified representation of the singularities of T also provides an attractive possible description of antiresonances in the lower half-plane of lambda. Such a possibility, which is forbidden in the usual theory of local potentials, represents an enriching alternative to the standard Breit-Wigner hard-sphere picture of antiresonances.
