Two-point one-dimensionalδ-${\delta }^{\prime }$ interactions: non-abelian addition law and decoupling limit

Abstract

In this contribution to the study of one dimensional point potentials, we prove that if we take the limit q → 0 on a potential of the type v0δ(y) + 2v1δ (y) + w0δ(y − q) + 2w1δ (y − q), we obtain a new point potential of the type u0δ(y) + 2u1δ (y), when u0 and u1 are related to v0, v1, w0 and w1 by a law having the structure of a group. This is the Borel subgroup of SL2(R). We also obtain the non-abelian addition law from the scattering data. The spectra of the Hamiltonian in the decoupling cases emerging in the study are also described in full detail. It is shown that for the v1 = ±1, w1 = ±1 values of the δ couplings the singular Kurasov matrices become equivalent to Dirichlet at one side of the point interaction and Robin boundary conditions at the other side.

Vacuum Energy for Generalized Dirac Combs at T = 0

“…being {ω 2 n = k n } the eigenvalues characterising the one-particle states of the quantum field theory given by the equation (4). The ultraviolet divergences that appear naturally in this expression must be subtracted taking into account the self-energy of the individual potential that makes up the comb and the fluctuations of the field in the chosen background.…”

Breaking Waves and Spectral Analysis of the Two‐Dimensional KdV–Bogoyavlenskii Equation

“…It forms part of a more general three-parameter family of similarity solutions of KdV, see [43][44][45] for the relation with the second Painlevé's function P(x). Finally, note that if t = l 2 and X ≡ (x − cy) 3/4 we see that the corresponding initial data is See also [46] for other developements in connection with these singular point potentials.…”

“…Consider now the general situation C x < ∞, so we only have that bound (46) holds (stated differently, F x is bounded on L 2 (x, ∞) but needs not be a contraction). Let ∈ L 2 andˆ be its Fourier transform; for notational convenience, we introduce here…”

Double General Point Interactions: Symmetry and Tunneling Times

“…From the above expressions we observe that when x 2 → x 1 the double point interaction converges to a single point interaction, with the transfer matrix given by Ŵ = 2 1 , i.e., in this limit the composition law for the parameters has the U(1)×SL(2, R) group structure (see [55] for the group structure of a restricted class of double point interactions). The explicit form of the interaction term s[ψ] in Equation (1) is suitable to study its symmetry properties [39].…”