PHYSICS

Exact solutions of the two-dimensional Logunov–Tavkhelidze equation with the “delta-circle” potential for scattering states

A. V. Paulenka, V. N. Kapshai, Yu. A. Grishechkin

Francisk Skorina Gomel State University

Abstract: The exact solutions of the two-dimensional partial integral equations of Logunov-Tavkhelidze for the scattering states of a system of two scalar particles of equal mass were obtained. The particle interaction was modeled by a “delta-circle” quasipotential defined in the relativistic configurational representation and by a superposition of two such quasipotentials. The analysis of the partial scattering cross-sections and the full two-dimensional scattering amplitude revealed their resonant behavior. It was established that a peculiarity of two-dimensional scattering, unlike its three-dimensional counterpart, is the unlimited growth of the scattering cross-section corresponding to states with a zero azimuthal quantum number as the rapidity tends to zero (energy tends to the rest mass). This feature is caused by the logarithmic behavior of the partial Green’s function at small rapidity values. Using the found exact solutions as an example, the fulfillment of the unitarity condition for the two-dimensional partial scattering amplitudes is demonstrated.

Keywords: two-dimensional Logunov–Tavkhelidze equation, two-dimensional Green’s function, wave function, two-particle system, “delta-circle” potential, partial integral equation, exact solution, unitarity condition, two-dimensional scattering amplitude, two-dimensional scattering cross-section.

UDC: 539.12.01

Received: 12.06.2025

DOI: 10.54341/20778708_2025_3_64_67