Three impenetrable particles on a ring and the quantum billiard in a triangle

Remarkable similarities are found in the problem of three impenetrable quantum mechanical particles on a ring and the problem of the quantum billiard in a triangle. If the energy contributed by the motion of the center of mass in the ring problem is subtracted from the total energy, the energy eigenvalues of the particles on the ring are proportional to the energy eigenvalues of the wave function in the corresponding triangle. The eigenvalues derived from the ring solution are unique to that triangle and that energy level. A mathematical relationship is derived, which connects the masses of the particles on the ring to the angles of the triangle. There are three quantum billiard triangles that have previously been solved by the method of separation of variables. The three quantum particles on a ring problem, however, has now been solved for many cases. By correlating the three known triangle solutions to the masses on the ring problem we derive and verify a relationship between the two problems. For the three known triangle solutions, the eigenvalues found in the ring problem are proportional to those found in the triangle. The correlation between the masses on the ring and the triangle is then used to find solutions to other triangles, which do not yield to solution by separation of variables.