187,284 research outputs found

    Two-point one-dimensional δ\delta-δ′\delta^\prime interactions: non-abelian addition law and decoupling limit

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    In this contribution to the study of one dimensional point potentials, we prove that if we take the limit q→0q\to 0 on a potential of the type v0δ(y)+2v1δ′(y)+w0δ(y−q)+2w1δ′(y−q)v_0\delta({y})+{2}v_1\delta'({y})+w_0\delta({y}-q)+ {2} w_1\delta'({y}-q), we obtain a new point potential of the type u0δ(y)+2u1δ′(y){u_0} \delta({y})+{2 u_1} \delta'({y}), when u0 u_0 and u1 u_1 are related to v0v_0, v1v_1, w0w_0 and w1w_1 by a law having the structure of a group. This is the Borel subgroup of SL2(R)SL_2({\mathbb R}). 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 v1=±1v_1=\pm 1, w1=±1w_1=\pm 1 values of the δ′\delta^\prime 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

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    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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