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Two-point one-dimensional - interactions: non-abelian addition law and decoupling limit
In this contribution to the study of one dimensional point potentials, we
prove that if we take the limit on a potential of the type
, we
obtain a new point potential of the type , when and are related to , , and
by a law having the structure of a group. This is the Borel subgroup of
. We also obtain the non-abelian addition law from the
scattering data. The spectra of the Hamiltonian in the exceptional cases
emerging in the study are also described in full detail. It is shown that for
the , values of the couplings the
singular Kurasov matrices become equivalent to Dirichlet at one side of the
point interaction and Robin boundary conditions at the other side
All the Groups of Signal Analysis from the (1+1)-affine Galilei Group
We study the relationship between the (1+1)-affine Galilei group and four
groups of interest in signal analysis and image processing, viz., the wavelet
or the affine group of the line, the Weyl-Heisenberg, the shearlet and the
Stockwell groups. We show how all these groups can be obtained either directly
as subgroups, or as subgroups of central extensions of the affine Galilei
group. We also study this at the level of unitary representations of the groups
on Hilbert spaces.Comment: 28 pages, 1 figur
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