Study of Exactly Soluble One-Dimensional N-Body Problems

Abstract

In this paper it is shown that several cases of one-dimensional N-body problems are exactly soluble. The first case describes the motion of three one-dimensional particles of arbitrary mass which interact with one another via infinite-strength, repulsive delta-function potentials. It is found in this case that the stationary-state solution of the scattering of the three particles is analogous to an electro-magnetic diffraction problem which has already been solved. The solution to this analogous electro-magnetic problem is interpreted in terms of particles. Next it is shown that the problem of three particles of equal mass interacting with each other via finite- but equal-strength delta-function potentials is exactly soluble. This example exhibits rearrangement and bound-state effects, but no inelastic processes occur. Finally it is shown that the problem of N particles of equal mass all interacting with one another via finite- but equal-strength delta functions is exactly soluble. Again no inelastic processes occur, but various types of rearrangements and an N-particle bound state do occur. These rearrangements and the N-particle bound state are illustrated by means of a series of sample calculations.

The chiral Potts models revisited

“…Hence, at zero temperature, this not only signifies a wetting transition, but for N > 3 a more special phenomenon is taking place which we shall call "superwetting," with the maximal amount of interface degeneracy as each interface of type r is free to break up into two interfaces of types j and r − j, for any j between 1 and r − 1. 7 It is as if we have energy levels given by spin operators S z for 2S + 1 = N, similar to what happens in the superintegrable chiral Potts model. (26,32) It should be obvious that the horizontal couplings E have no role to play at zero temperature.…”

On higher spin $U_q(\operatorname {sl}_2)$-invariant $R$-matrices

“…(6) Here P j is the projection onto the spin j subspace V j in V ⊗2 s , and r j (λ) is a scalar function. Property (5) is equivalent to the requirement of U q (sl 2 )-invariance, i.e., (7) [R(λ), ∆(ξ)] = 0, ξ ∈ U q (sl 2 ).…”

“…Equations (8) and (10) are preserved under rescaling of the spectral parameter, λ → γλ, (11) by an arbitrary finite constant γ. R-matrices related by such a transformation with a finite nonzero γ will be regarded as equivalent. For q = 1, four different types of sl 2 -invariant R-matrices are known [7,8,9,10,11]:…”

Connections and effective S-matrix in triangle representation for quantum scattering

“…We consider the quantum system of three spinless particles interacting with deltafunction potentials of equal strength g E R in one dimension [22][23][24][25][26]301. We suppose the wave function of the system to be symmetric; i.e., consider the statistics of boson particles.…”