Random renormalization in the semiclassical long-time limit of a precessing spin
Several authors have studied the (complicated) behavior of the sum \(S_ j(\tau)=\sum^{j}_{m=1}\exp (i\pi \tau^ 2_ m)\) [see for instance \textit{F. M. Dekking} and \textit{M. Mendès France}, J. Reine Angew. Math. 329, 143-153 (1981; Zbl 0459.10025); for related results see also \textit{J. M. Deshouillers}, Elementary and analytic theory of numbers, Banach Cent. Publ. 17, 75-82 (1985; Zbl 0616.10031)]. In the paper under review the author gives a physical interpretation for this sum: the trace of the propagator for a particle of spin \(J^ 2=\hslash^ 2j(j+1)\) with Hamiltonian \(J^ 2_ z\times c^ r\) can be expressed in terms of \(S_ j(t/j)\) where t is a dimensionless propagation time. The long-limit corresponds to \(t\to \infty\) (j fixed), and the semiclassical limit is \(j\to \infty\) (t fixed). Taking the combined limit \(j\to \infty\), \(t\to \infty\) \((\tau =t/j\) fixed), one obtains complex patterns for \(S_ j(\tau)\) which consist of an infinite hierarchy of scaling spirals: these pattern strongly depend on the arithmetic properties of \(\tau\).
scientific article; zbMATH DE number 4091606