SCATTERING MATRIX FOR FINITE TIME IN INTERACTING PARTICLE WAVE FUNCTION SPACE

V. V. Alekseev

Rostislav V. Vedrinskii

V.V. Komarov, Anna M. Popova

Octavio Cano Corona

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.

Dimitri R. Yafaev

Manuel Jesús Maldonado Lozano

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.