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In Galperin billiards, two balls colliding with a hard wall form an analog calculator for the digits of the number &pi, This classical, one-dimensional three-body system (counting the hard wall) calculates the digits of &pi, 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 &pi, itself. This article reviews previous results for Galperin billiards and then pushes these results farther. We provide a complete explicit solution for the balls&rsquo, 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 &pi, with Galperin billiards, including curious cases with irrational number bases.

We present a new class of electrostatics problems that are exactly solvable by adding finitely many image charges. Given a charge at some location inside a cavity bounded by up to four conducting grounded segments of spheres: if the spheres have a symmetry derived via a stereographic projection from a 4D finite reflection group, then this is a solvable generalization of the familiar problem of a charge inside a spherical cavity. There are 19 three-parametric families of finite groups formed by inversions relative to at most four spheres, each member of each family giving a solvable problem. We solve a sample problem which derives from the reflection group $\mathbf{D}_{4}$ and requires 191 image charges., 26 pages, 6 figures Submitted to Foundations of Physics

We show that the dynamics of particles in a one-dimensional harmonic trap with hard-core interactions can be solvable for certain arrangements of unequal masses. For any number of particles, there exist two families of unequal mass particles that have integrable dynamics, and there are additional exceptional cases for three, four and five particles. The integrable mass families are classified by Coxeter reflection groups and the corresponding solutions are Bethe ansatz-like superpositions of hyperspherical harmonics in the relative hyperangular coordinates that are then restricted to sectors of fixed particle order. We also provide evidence for superintegrability of these Coxeter mass families and conjecture maximal superintegrability., 9.5+4.5 pages, 5 figures, 2 tables; v3: a few corrections and additions

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., Comment: 9 pages, 3 figures

Let ( G , K ) be the complex symmetric pair associated with a real reductive Lie group G 0 . We discuss an algorithmic approach to computing generators for the centralizer of K in the universal enveloping algebra of g . In particular, we compute explicit generators for the cases G 0 = SU ( 2 , 2 ) , SL ( 3 , R ) , SL ( 4 , R ) , Sp ( 4 , R ) , and the exceptional group G 2 ( 2 ) .

A quantum Galilean cannon is a 1D sequence of $N$ hard-core particles with special mass ratios, and a hard wall; conservation laws due to the reflection group $A_{N}$ prevent both classical stochastization and quantum diffraction. It is realizable through specie-alternating mutually repulsive bosonic soliton trains. We show that an initial disentangled state can evolve into one where the heavy and light particles are entangled, and propose a sensor, containing $N_{\text{total}}$ atoms, with a $\sqrt{N_{\text{total}}}$ times higher sensitivity than in a one-atom sensor with $N_{\text{total}}$ repetitions., Comment: 5 pages, 3 figures