Action billiards

Abstract

Truncation of the Hamiltonian matrix in the harmonic oscillator representation of quantum mechanics corresponds to cutting off the action dependence of the classical Hamiltonian. The conjugate angle variable suffers a discontinuous jump, when the orbit collides with the action boundary in a manner analogous to the specular reflection on the border of a common billiard. The authors derive the connection rule for the angles by analysing the limit of smooth cut-offs in the classical Hamiltonian, for which examples are given. It is found that different definitions of the classical cut-off may lead to diverse orbit structures, though the corresponding finite Hamiltonian matrices are identical.