Moyal star product approach to the Bohr-Sommerfeld approximation

The Bohr-Sommerfeld approximation to the eigenvalues of a one-dimensional quantum Hamiltonian is derived through order $\hbar^2$ (i.e., including the first correction term beyond the usual result) by means of the Moyal star product. The Hamiltonian need only have a Weyl transform (or symbol) that is a power series in $\hbar$, starting with $\hbar^0$, with a generic fixed point in phase space. The Hamiltonian is not restricted to the kinetic-plus-potential form. The method involves transforming the Hamiltonian to a normal form, in which it becomes a function of the harmonic oscillator Hamiltonian. Diagrammatic and other techniques with potential applications to other normal form problems are presented for manipulating higher order terms in the Moyal series.Comment: 27 pages, no figure

The Berry-Tabor Conjecture

Abstract. One of the central observations of quantum chaology is that sta-tistical properties of quantum spectra exhibit surprisingly universal features, which seem to mirror the chaotic or regular dynamical properties of the under-lying classical limit. I will report on recent studies of simple regular systems, where some of the observed phenomena can be established rigorously. The results discussed are intimately related to the distribution of values of qua-dratic forms, and in particular to a quantitative version of the Oppenheim conjecture. Quantum chaos One of the main objectives of quantum chaology is to identify characteristic prop-erties of quantum systems which, in the semiclassical limit, reflect the regular or chaotic features of the underlying classical dynamics. Take for example the geodesic flow on the unit tangent bundle of a compact two-dimensional Riemannian surface M. The corresponding quantum system is described by the stationary Schrödinger equation −∆ϕj = λjϕj, (1) where ∆ is the Laplacian of M, λj represent the quantum energy eigenvalues and ϕj the corresponding eigenfunctions. The spectrum of the negative Laplacian is a discrete ordered subset of the real line, 0 ≤ λ1 < λ2 ≤ λ3 ≤ · · · → ∞. (2) According to Weyl’s law, the number of eigenvalues below λ is asymptotically #{j: λj ≤ λ} ∼ area(M)4pi λ (3) as λ → ∞. Hence the mean spacing between adjacent levels is asymptotically 4pi / area(M). For simplicity, we may assume in what follows that area(M) = 4pi