Eigenvalue Statistics and Eigenstate Wigner Functions for the Discrete Self-Trapping Equation

Abstract

In some recent papers [1, 2] the discrete self-trapping (DST) equation [3] was shown to be an effective working model to analyse the semiclassical limit of the quantum version of classically non-integrable dynamics [4-]. Canonical quantisation of the DST eq. with m freedoms leads in fact to the Hamiltonian

$$ \hat H = \Gamma \hbar ^2 \sum\limits_{j = 1}^m {\left( {\hat A_j ^ + \hat A_j + 1/2} \right)^2 + \hbar \varepsilon \sum\limits_{j,k = 1}^m {M_{jk} } \left( {\hat A_j ^ + \hat A_K + \hat A_K \hat A_j ^ + } \right)} /2 $$

((1))

which commutes with the norm operator

$$ \hat N = \hbar \left( {m/2 + \sum\limits_{j = 1}^m {\hat N_j } } \right) \equiv \hbar \left( {m/2 + \sum\limits_{j = 1}^m {\hat A_j ^ + \hat A_j } } \right) $$

((2))

whose eigenspaces are finite-dimensional.