Semiclassical theory of long time propagation in quantum chaos. First part

Abstract

We develop a semiclassical theory of wave propagation based on invariant Lagrangian manifolds existing in conservative Hamiltonian systems with chaotic dynamics. They are stable and unstable manifolds of unstable periodic orbits, and their intersections consist of homoclinic and heteroclinic orbits. For arbitrary long times, we find matrix elements of the evolution operator between wave functions constructed in the neighbourhood of short unstable periodic orbits, in terms of canonical invariants of homoclinic and heteroclinic orbits. We verify the accuracy of these expressions by computing millions of homoclinic orbits and thousands of heteroclinic ones in the hyperbola billiard. Here we describe diagonal matrix elements while in another article named second part we will describe off-diagonal matrix elements.

Semiclassical theory of long time propagation in quantum chaos. Second part

“…This parametrization evolves according to equation (5) in ref. [11] g t z u (τ , u) = z u (τ + t, u e λ γ t ), (1) with λ γ the stability exponent of γ. Moreover the transverse velocity vector field ξ u (τ , u) evolves following equation (7) in ref.…”

“…Moreover the transverse velocity vector field ξ u (τ , u) evolves following equation (7) in ref. [11] Dg t [z u (τ , u)]ξ u (τ , u) = e λ γ t ξ u (τ + t, u e λ γ t ).…”

“…This parametrization evolves according to equation (12) in ref. [11] g t z s (τ , s) = z s (τ + t, s e −λ δ t ), (3) with λ δ the stability exponent of δ, while the transverse velocity vector field ξ s (τ , s) evolves by equation (13) in ref. [11] Dg t [z s (τ , s)]ξ s (τ , s) = e −λ δ t ξ s (τ + t, s e −λ δ t ).…”

“…In order to understand this point we remember from equation (2) in ref. [11] that there is, in principle, an ambiguity in the definition of the vectors used for the parametrization of manifolds. Nevertheless, this ambiguity does not affect the value of D. Let us assume, for instance, that the vector ξ u at z γ is replaced with αξ u (α = 0).…”

“…In a previous article named first part we have presented the semiclassical evaluation of diagonal matrix elements [11]. This second part is a continuation of the first one and we will use formulae, figures and concepts derived there.…”