Spectra of nonlocally bound quantum systems

Abstract

We discuss a class of nonlinear and nonlocal models for the dynamics of a composite quantum system. The models in question depend on the following constituents: on two subsystem Hamiltonians (denoted by H and Ĥ), an analytic function (f), and a real parameter (s). As demonstrated elsewhere before, the stationary states can be described in these models fairly explicitly. In this article, we build upon that result, and discuss the topological as well as statistical characteristics of the spectra. Here, we concentrate on the special case f = log. It turns out that an energy spectrum of the nonlocally bound system substantially differs from that of its components. Indeed, we show rigorously that, if H is the harmonic oscillator and Ĥ is completely degenerate with one energy level, then the energy spectrum of the composite system has the topology of the Cantor set (for s > 2). In addition, we show that, if H is replaced by the logarithm of the harmonic oscillator, then the spectrum consists of finitely many intervals separated by gaps (for s sufficiently large). In the last case, the key analytic object is the series Σn−s. In particular, as an interesting offshoot, this structure furnishes a nontautological immersion of fundamental number-theoretic functions into the quantum formalism.