Transmission and scattering problems involving complex potentials are important in physics, in particular in describing nuclear collisions. We describe many pedagogical features of transmission in one dimension and scattering and absorption cross sections in three dimensions for a rectangular absorptive potential and compare the results with scattering from real barriers and a hard sphere. For a given energy in one dimension we show that the absorption reaches a maximum for a critical value of the absorption strength and then monotonically decreases with an increase in the absorption strength. An infinitely absorptive well becomes almost fully reflective and thus is similar to an infinitely high barrier. Similar results are found in three dimensions. We show that the absorption cross section in three dimensions diverges at a threshold energy, which is of critical importance for exothermic reactions.Transmission and scattering problems involving complex potentials are important in physics, in particular in describing nuclear collisions. We describe many pedagogical features of transmission in one dimension and scattering and absorption cross sections in three dimensions for a rectangular absorptive potential and compare the results with scattering from real barriers and a hard sphere. For a given energy in one dimension we show that the absorption reaches a maximum for a critical value of the absorption strength and then monotonically decreases with an increase in the absorption strength. An infinitely absorptive well becomes almost fully reflective and thus is similar to an infinitely high barrier. Similar results are found in three dimensions. We show that the absorption cross section in three dimensions diverges at a threshold energy, which is of critical importance for exothermic reactions.

American Journal of PhysicsFeb 12, 20072 citations

Scattering in One Dimension

S. Brandt, Hans Dieter Dahmen

Springer eBooksJan 1, 19910 citations

tmatrix in hard-core two-body scattering in three and two dimensions

Chih-Chun Chang, Richard M. Friedberg

The Beliaev-Galitskii relation between the two-body t matrix and the scattering amplitude is shown to be slightly erroneous in hard-core scattering because of an incorrect interchange of limits. The correction term is computed exactly at zero momentum both in three dimensions [to relative order O( ${\mathit{Ea}}^{2}$ )] and in two dimensions [to relative order O( ${\mathit{Ea}}^{2}$ ln ${\mathit{Ea}}^{2}$ )].The Beliaev-Galitskii relation between the two-body t matrix and the scattering amplitude is shown to be slightly erroneous in hard-core scattering because of an incorrect interchange of limits. The correction term is computed exactly at zero momentum both in three dimensions [to relative order O(${\mathit{Ea}}^{2}$)] and in two dimensions [to relative order O(${\mathit{Ea}}^{2}$ln${\mathit{Ea}}^{2}$)].

Physical Review AFeb 1, 19946 citations

Universal properties of Fermi gases in arbitrary dimensions

Manuel Valiente, Nikolaj Thomas Zinner, Klaus Mølmer

We consider spin- $1/2$ Fermi gases in arbitrary, integer, or noninteger spatial dimensions, interacting via a Dirac $\ensuremath{\delta}$ potential. We first generalize the method of Tan's distributions and implement short-range boundary conditions to arbitrary dimension and we obtain a set of universal relations for the Fermi gas. Three-dimensional scattering under very general conditions of transversal confinement is described by an effectively reduced-dimensional scattering length, which we show depends on the three-dimensional scattering length in a universal way. Our formula for noninteger dimensions interpolates between the known results in integer dimensions 1, 2, and 3. Without any need to solve the associated multichannel scattering problem, we find that confinement-induced resonances occur in all dimensions different from $D=2$ , while reduced-dimensional contacts, related to the tails of the momentum distributions, are connected to the three-dimensional contact by a correction factor of purely geometric origin.We consider spin-$1/2$ Fermi gases in arbitrary, integer, or noninteger spatial dimensions, interacting via a Dirac $\ensuremath{\delta}$ potential. We first generalize the method of Tan's distributions and implement short-range boundary conditions to arbitrary dimension and we obtain a set of universal relations for the Fermi gas. Three-dimensional scattering under very general conditions of transversal confinement is described by an effectively reduced-dimensional scattering length, which we show depends on the three-dimensional scattering length in a universal way. Our formula for noninteger dimensions interpolates between the known results in integer dimensions 1, 2, and 3. Without any need to solve the associated multichannel scattering problem, we find that confinement-induced resonances occur in all dimensions different from $D=2$, while reduced-dimensional contacts, related to the tails of the momentum distributions, are connected to the three-dimensional contact by a correction factor of purely geometric origin.

arXiv:1205.6388Physical Review AOct 15, 201268 citations

{\mathit{Ea}}^{2}

)] and in two dimensions [to relative order O(

{\mathit{Ea}}^{2}

{\mathit{Ea}}^{2}

)].

We consider spin-

1/2

Fermi gases in arbitrary, integer, or noninteger spatial dimensions, interacting via a Dirac $\ensuremath{\delta}$ potential. We first generalize the method of Tan's distributions and implement short-range boundary conditions to arbitrary dimension and we obtain a set of universal relations for the Fermi gas. Three-dimensional scattering under very general conditions of transversal confinement is described by an effectively reduced-dimensional scattering length, which we show depends on the three-dimensional scattering length in a universal way. Our formula for noninteger dimensions interpolates between the known results in integer dimensions 1, 2, and 3. Without any need to solve the associated multichannel scattering problem, we find that confinement-induced resonances occur in all dimensions different from

D=2

, while reduced-dimensional contacts, related to the tails of the momentum distributions, are connected to the three-dimensional contact by a correction factor of purely geometric origin.