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We present an exact solution to the Lippmann-Schwinger equation for a two-dimensional circular billiard. After diagonalizing an integral operator whose kernel is a zeroth order Hankel function of first kind, we use its eigenfunctions and eigenvalues to obtain in a straightforward way the exact wavefunctions of the referred Lippmann-Schwinger equation.

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We solve analytically the Lippmann-Schwinger equation for a linear potential. As an application of this result we investigate the two-dimensional scattering of a scalar particle by a linear potential and by an arbitrary barrier modeled as a boundary-wall.

A solution to the Lippmann-Schwinger equation for the scattering of a plane wave by a spherical barrier is presented. The spherical barrier is described by a boundary-wall with coupling constant γ = γ(θ, f). A general method is presented, along with several possible examples for the coupling constant such as q sin and q cos dependencies, Janus particle and a ring-shaped surface. The exact scattering amplitude for the general case is also presented. RECEIVED