New Algebraic Approach to Scattering Problems

Abstract

Quantum scattering systems described by Hamiltonians which are constructed from the Casimir operators of certain noncompact groups G are considered. We obtain the following result: If U x and Ux are the Weyl-equivalent representations of the symmetry group G of the dynamical system, the corresponding S matrices are constrained to satisfy SU x ͑g͒ Ux ͑g͒S, for all g [ G. This relation enables one to derive S. As applications, the S matrices corresponding to the dynamical groups SO 0 ͑p, q͒ are derived. [S0031-9007(98)05542-2]

On solvable potentials related to SO(2,2). II. Natanzon potentials

“…In a recent paper by one of the present authors 13 was proposed a way which allows pure algebraic calculation of S-matrices for the systems whose Hamiltonians are related to the Casimir operator C of some noncompact group G. In that paper the S-matrices for the systems under consideration are associated with intertwining operators A between Weyl equivalent principal series representations of G. At this stage we note that the operator A is said to intertwine the representations U and U of the group G ͑Ref. 14͒ if relation…”

Intertwining operator in thermal CFTd

“…Orbits of the actions of these two subgroups coincide with the worldlines for an observer in semi-eternal uniform motion and a uniformly accelerating observer with finite lifetime, each of whose causally connected regions are the future (past) light-cone V ± and diamond D, respectively. 6 As discussed first by Buchholz for V ± [24] and later by Hislop and Longo for D and V ± [25] in the context of modular theory in operator algebra, CFTs restricted on these regions are shown to be thermal as well under the identifications of temporal coordinates with these SO(1, 1) group parameters. In this way, CFT in any spacetime dimension d easily gets thermalized by just putting it on the Rindler wedge W R/L , light-cone V ± , or diamond D with suitably-chosen temporal coordinates.…”

“…There is, however, a crucial difference. The difference is the basis for the representation of so (2, d): the Kerimov's method [6] is based on the basis that diagonalizes the maximal compact subgroup SO(2) × SO(d) ⊂ SO (2, d), whereas our method is based on the basis that diagonalizes the noncompact subgroup SO(1, 1)×SO(1, d −1) ⊂ SO (2, d). 4 For the case of two-dimensional thermal CFT, the momentum-space two-and three-point Wightman functions for a scalar primary operator were calculated in [12][13][14] and [15].…”

so(3) algebraic approach to the Morse potential

“…Among above mentioned approaches, algebraic techniques have been proven rather useful and valid in the descriptions of bound states in nuclear and molecular physics, and have been extended to quantum mechanics. [8][9][10] A major advantage of the algebraic approach is to obtain energy eigenvalues and eigenfunctions of some solvable physical systems without dealing with the Schrödinger equations. [11][12][13][14][15][16] In particular, there are two successful examples that the algebraic method is applied to Pöschl-Teller potential 13,14 and Coulomb potential.…”