arXiv subjects

S. G. Jackson

Publications and source records attributed to S. G. Jackson.

3 recordsLinked to original sources

arXiv2026

Diffraction induced quantum chaos in a one-dimensional Bose gas

We investigate the Lieb--Liniger model of interacting one-dimensional bosons coupled to a localized impurity, modeled by a delta barrier. While the Lieb--Liniger gas is integrable, the impurity breaks integrability and induces a transition towards quantum chaos. We show that the low-energy spectrum exhibits random-matrix statistics, in striking contrast to the Bohigas--Giannoni--Schmit conjecture, where chaotic behavior typically emerges at high energy. For two bosons, the odd-parity sector remains integrable, whereas the even-parity sector displays clear signatures of chaos at low energy and a crossover back to quasi-integrable behavior at higher energies. For three bosons, both parity sectors exhibit spectral statistics close to chaos at low energy. We argue that this unconventional form of few-body quantum chaos originates from diffractive processes induced by the impurity.

cond-mat.quant-gas↗

arXiv2020

The dynamics of digits: Calculating pi with Galperin's billiards

In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number $π$. This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of $π$ in a base determined by the ratio of the masses of the two particles. This base can be any integer, but it can also be an irrational number, or even the base can be $π$ itself. This article reviews previous results for Galperin billiards and then pushes these results farther. We provide a complete explicit solution for the balls' positions and velocities as a function of the collision number and time. We demonstrate that Galperin billiard can be mapped onto a two-particle Calogero-type model. We identify a second dynamical invariant for any mass ratio that provides integrability for the system, and for a sequence of specific mass ratios we identify a third dynamical invariant that establishes superintegrability. Integrability allows us to derive some new exact results for trajectories, and we apply these solutions to analyze the systematic errors that occur in calculating the digits of $π$ with Galperin billiards, including curious cases with irrational number bases.

math.DS↗

arXiv2016

Exactly solvable quantum few-body systems associated with the symmetries of the three-dimensional and four-dimensional icosahedra

The purpose of this article is to demonstrate that non-crystallographic reflection groups can be used to build new solvable quantum particle systems. We explicitly construct a one-parametric family of solvable four-body systems on a line, related to the symmetry of a regular icosahedron: in two distinct limiting cases the system is constrained to a half-line. We repeat the program for a 600-cell, a four-dimensional generalization of the regular three-dimensional icosahedron.

math-ph↗