Band invariants and closed trajectories on \(S^ n\)
The author considers the spectral properties of the Schrödinger operator \(\Delta +g\) where \(\Delta\) is the Laplace-Beltrami operator on \(S^ n\) and \(g\in C^{\infty}(S^ n)\) is a given potential. The spectrum of \(\Delta +g\) forms bands of fixed width around the spectrum of the Laplace-Beltrami operator. It is shown that, under certain conditions on g, there is a correspondence between clustering in the bands and the existence of closed classical orbits, i.e. integral curves of the Hamiltonian \(| p|^ 2+g\) on \(T^*S^ n\). The author also relates the asymptotic behavior of the periods of the trajectories to the spectrum of \(\Delta +g\).
scientific article; zbMATH DE number 3942574
scientific article