Direct and inverse scalar scattering problems for the Helmholtz equation in ℝm

The boundary value problem for the Helmholtz equation in the m-dimensional free space is considered. The problem is reduced to the Lippmann–Schwinger integral equation over the inhomogeneity domain. The operator of the integral equation is shown to be an invertible Fredholm operator. The inverse coefficient problem is considered. An application of the two-step method reduces the inverse problem to the source-type integral equation with a smooth kernel. Special classes of solutions to this equation are introduced. The uniqueness of a solution to the integral equation of the first kind is proved in the defined function classes.

Scalar scattering amplitude in the Gaussian wave-packet formalism

We compute an $s$-channel $2\to2$ scalar scattering $\phi\phi\to\Phi\to\phi\phi$ in the Gaussian wave-packet formalism at the tree level. We find that wave-packet effects, including shifts of the pole and the width of the propagator of $\Phi$, persist even when we do not take into account the time boundary effect for $2\to2$ proposed earlier. An interpretation of the result is that a heavy scalar $1\to2$ decay $\Phi\to\phi\phi$, taking into account the production of $\Phi$, does not exhibit the in-state time boundary effect unless we further take into account in-boundary effects for the $2\to2$ scattering. We also show various plane-wave limits.

On non-perturbative unitarity in gravitational scattering

We argue that the tree-level graviton-scalar scattering in the Regge limit is unitarized by non-perturbative effects within General Relativity alone, that is without resorting to any extension thereof. At Planckian energy the back reaction of the incoming graviton on the background geometry produces a non-perturbative plane wave which softens the UV-behavior in turn. Our amplitude interpolates between the perturbative graviton-scalar scattering at low energy and scattering on a classical plane wave in the Regge limit that is bounded for all values of s.

The phase-functions method and scalar amplitude of nucleon–nucleon scattering

A known phase-functions method (PFM) has been considered for calculation of a single-channel nucleon–nucleon scattering. The following partial waves of a nucleon–nucleon scattering have been considered using the phase shifts by PFM: 1S0-, 3P0-, 3P1-, 1D2-, 3F3-states for nn-scattering, 1S0-, 3P0-, 3P1-, 1D2-states for pp-scattering and 1S0-, 1P1-, 3P0-, 3P1-, 1D2-, 3D2-states for np-scattering. The calculations have been carried out using phenomenological nucleon–nucleon Nijmegen group potentials (NijmI, NijmII, Nijm93 and Reid93) and Argonne v18 potential. The scalar scattering amplitude has been calculated using the obtained phase shifts. Our results are not much different from those obtained by using the known phase shifts published in other papers. The difference between calculations depending on a computational method of phase shifts makes: for real (imaginary) parts 0.14–4.36% (0.16–4.05%) for NijmI. 0.02–4.79% (0.08–3.88%) for NijmII. 0.01–5.49% (0.01–4.14%) for Reid93 and 0.01–5.11% (0.01–2.40%) for Argonne v18 potentials.