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Revista Mexicana de Física

Condiciones a la frontera y formalismo S de la dispersión nuclear

Manuel Jesús Maldonado Lozano, Nicolau F. Medina

Published May 10, 2009• 0 citations
OpenAlex

Abstract

THE PURPOSE OF THE PRESENT IS TO FIND THE RELATION BETWEEN THE S-MATRIX FORMALISM AND THE DESCRIPTION OF INTERACTIONS BY MEANS OF BOUNDARY CONDITIONS. IT IS SHOWN THAT IN THE ABSENCE OF AN ESSENTIAL SINGULARITY AT -- FOR ----, THE DYNAMICAL DESCRIPTION SUGESTED IN A NATURAL WAY BY THE S-FORMALISM, GIVES THE SAME TIME-DEPENDENT WAVE FUNCTIONS THAT ARE OBTAINED WHEN THE INTERACTIONS ARE REPRESENTED BY BOUNDARY CONDITIONS. THE DYNAMICAL DESCRIPTION OF THE S-FORMALISM IS EXTENDED TO THE CASE WHERE ----- MAY HAVE AN ESSENTIAL SINGULARITY AT ---.

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THE PURPOSE OF THE PRESENT IS TO FIND THE RELATION BETWEEN THE S-MATRIX FORMALISM AND THE DESCRIPTION OF INTERACTIONS BY MEANS OF BOUNDARY CONDITIONS. IT IS SHOWN THAT IN THE ABSENCE OF AN ESSENTIAL SINGULARITY AT -- FOR ----, THE DYNAMICAL DESCRIPTION SUGESTED IN A NATURAL WAY BY THE S-FORMALISM, GIVES THE SAME TIME-DEPENDENT WAVE FUNCTIONS THAT ARE OBTAINED WHEN THE INTERACTIONS ARE REPRESENTED BY BOUNDARY CONDITIONS. THE DYNAMICAL DESCRIPTION OF THE S-FORMALISM IS EXTENDED TO THE CASE WHERE ----- MAY HAVE AN ESSENTIAL SINGULARITY AT ---.
THE PRESENT NOTE ANALYSES THE DYNAMICAL DESCRIPTION OF THE SCATTERING DUE TO A SHORT RANGE POTENTIAL. IN PARTICULAR, IT DEVELOPS A CLASS OF TRANSFORMS THAT ARE USEFUL FOR THIS DYNAMICAL DESCRIPTION. A TIME-DEPENDENT GREEN FUNCTION FOR THE SCATTERING PROCESS IS DISCUSSED, THAT TAKES US FROM THE INITIAL STATE (PLANE WAVE) TO THE FINAL STATE (PLANE PLUS SCATTERED WAVES). THIS GREEN FUNCTION CAN BE USED TO GIVER THE MOTION OF ANY WAVE PACKET THAT INITIALLY WAS OUTSIDE THE POTENTIAL. THE PRESENT PAPER COMPLEMENTS A PREVIOUS DYNAMICAL DESCRIPTION OF A SCATTERING PROCESS, IN WHICH THE INTERACTION WAS REPRESENTED BY APPROPRIATE BOUNDARY.
THE PRESENT NOTE ANALYSES THE DYNAMICAL DESCRIPTION OF THE SCATTERING DUE TO A SHORT RANGE POTENTIAL. IN PARTICULAR, IT DEVELOPS A CLASS OF TRANSFORMS THAT ARE USEFUL FOR THIS DYNAMICAL DESCRIPTION. A TIME-DEPENDENT GREEN FUNCTION FOR THE SCATTERING PROCESS IS DISCUSSED, THAT TAKES US FROM THE INITIAL STATE (PLANE WAVE) TO THE FINAL STATE (PLANE PLUS SCATTERED WAVES). THIS GREEN FUNCTION CAN BE USED TO GIVER THE MOTION OF ANY WAVE PACKET THAT INITIALLY WAS OUTSIDE THE POTENTIAL. THE PRESENT PAPER COMPLEMENTS A PREVIOUS DYNAMICAL DESCRIPTION OF A SCATTERING PROCESS, IN WHICH THE INTERACTION WAS REPRESENTED BY APPROPRIATE BOUNDARY.
IN THIS PAPER WE GIVE THE TIME DEPENDENT DESCRIPTION OF THE SCATTERING OF A WAVE PACKET BY A SHORT RANGE POTENTIAL. WE SHOW THAT WHEN THE WAVE PACKET IS INITIALLY OUTSIDE THE SCATTERER, IT IS POSIBLE TO INTRODUCE AN APPROPIATE TIME DEPENDENT GREEN FUNCTION TO DESCRIBE THE SCATTERING PROCESS. THIS GREEN FUNCTION WILL BE EVALUATED EXPLICITLY IN TERM OF ERROR INTEGRAL FUNCTIONS, AND THE ONLY PARAMETERS THAT APPEAR IN THE GREEN FUNCTION WILL BE THE POLES OF THE S MATRIX. FROM THE ASYMPTOTIC BEHAVIOR OF THE GREEN FUNCTIOR WHEN T, WE OBTAIN RESTRICIONS ON THE POSITION OF THE POLES OF THE S MATRIX WHICH AGREE WITH THOSE DUE TO THE DEFINITION OF THE S IN TERMS OF THE DERIVATIVE MATRIX R. THE PRESENT ANALYSIS IS CARRIED OUT FOR ARBITRARY ANGULAR MOMENTUM.
IN APREVIOUS PAPER, WE HAVE STUDIED THE TIME DEPENDENT DESCRIPTION OF THE SCATTERING BY A GENERAL POTENTIAL, AND SHOWN THAT THE LAPLACE TRANSFORM OF THE TIME DEPENDENT GREEN FUNCTION OF THE PROBLEM SATISFIES DISPERSION RELATIONS. IN THIS PAPER WE OBTAIN THE EXPLICIT FORM OF THE TIME DEPENDENT GREEN FUNCTION FOR A POTENTIAL OF THE BARGMANN TYPE, AND DISCUSS THE RELATION BETWEEN THE TIME DPENDENT BEHAVOIR OF THE GREEN FUNCTION AND THE POSITION OF THE POLES AND ZEROS OF S(K) IN THE COMPLEX WAVE NUMBER PLANE.