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Title: Propagator for finite range potentials

Abstract

The Schroedinger equation in integral form is applied to the one-dimensional scattering problem in the case of a general finite range, nonsingular potential. A simple expression for the Laplace transform of the transmission propagator is obtained in terms of the associated Fredholm determinant, by means of matrix methods; the particular form of the kernel and the peculiar aspects of the transmission problem play an important role. The application to an array of delta potentials is shown.

Authors:
;  [1];  [2]
  1. Istituto di Fisica Applicata 'Nello Carrara', C.N.R., via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence (Italy)
  2. (Italy)
Publication Date:
OSTI Identifier:
20861555
Resource Type:
Journal Article
Resource Relation:
Journal Name: Journal of Mathematical Physics; Journal Volume: 47; Journal Issue: 12; Other Information: DOI: 10.1063/1.2401728; (c) 2006 American Institute of Physics; Country of input: International Atomic Energy Agency (IAEA)
Country of Publication:
United States
Language:
English
Subject:
71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ALGEBRA; INTEGRAL EQUATIONS; LAPLACE TRANSFORMATION; ONE-DIMENSIONAL CALCULATIONS; POTENTIAL SCATTERING; POTENTIALS; PROPAGATOR; SCHROEDINGER EQUATION

Citation Formats

Cacciari, Ilaria, Moretti, Paolo, and Istituto dei Sistemi Complessi, C.N.R., Sezione di Firenze, via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence. Propagator for finite range potentials. United States: N. p., 2006. Web. doi:10.1063/1.2401728.
Cacciari, Ilaria, Moretti, Paolo, & Istituto dei Sistemi Complessi, C.N.R., Sezione di Firenze, via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence. Propagator for finite range potentials. United States. doi:10.1063/1.2401728.
Cacciari, Ilaria, Moretti, Paolo, and Istituto dei Sistemi Complessi, C.N.R., Sezione di Firenze, via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence. Fri . "Propagator for finite range potentials". United States. doi:10.1063/1.2401728.
@article{osti_20861555,
title = {Propagator for finite range potentials},
author = {Cacciari, Ilaria and Moretti, Paolo and Istituto dei Sistemi Complessi, C.N.R., Sezione di Firenze, via Madonna del Piano 10, 50019 Sesto Fiorentino, Florence},
abstractNote = {The Schroedinger equation in integral form is applied to the one-dimensional scattering problem in the case of a general finite range, nonsingular potential. A simple expression for the Laplace transform of the transmission propagator is obtained in terms of the associated Fredholm determinant, by means of matrix methods; the particular form of the kernel and the peculiar aspects of the transmission problem play an important role. The application to an array of delta potentials is shown.},
doi = {10.1063/1.2401728},
journal = {Journal of Mathematical Physics},
number = 12,
volume = 47,
place = {United States},
year = {Fri Dec 15 00:00:00 EST 2006},
month = {Fri Dec 15 00:00:00 EST 2006}
}
  • Following a previous study on the transmission propagator for a finite range potential, the problem of reflection is considered. It is found that the Laplace transform of the reflection propagator can be expressed in terms of the usual Fredholm determinant {delta} and of a new similar determinant {gamma}, containing the peculiar characteristics of reflection. As an example, an array of delta potentials is considered. Moreover, a possible application to the calculation of quantum traversal time is shown.
  • We consider a system of three identical bosons near a Feshbach resonance in the universal regime with large scattering length usually described by model-independent zero-range potentials. We employ the adiabatic hyperspherical approximation and derive the rigorous large-distance equation for the adiabatic potential for finite-range interactions. The effective range correction to the zero-range approximation must be supplemented by a new term of the same order. The nonadiabatic term can be decisive. Efimov physics is always confined to the range between effective range and scattering length. The analytical results agree with numerical calculations for realistic potentials.
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