Notes on spectral clusters for semiclassical Schrodinger operators (Spectral and Scattering Theory and Related Topics)

Abstract

In this short note, we discuss the relationships between eigenvalues of Schrödinger operators and periodic trajectories of classical mechanics. For a Hamiltonian function H(x, p) : T^{*}mathrm{R}^{n}rightarrow mathrm{R}cup{pminfty}, let hat{H}equiv Op_{h}^{W}(H(x, p)) Ue a self-adjoint Weyl type pseudo-differential operator and Spec(H) be the spectrum of hat{H}. If $Lambda$_{h}(E, c)=Spec(hat{H})cap[E-c, E+mathrm{c}] consists of only eigenvalues, we define the (semiclassical) essential difference spectrum by D $sigma$(hat{H})equivoverline{{frac{E_{i}(h)-E_{J}(h)}{h}|E_{mathrm{t}}(h), E_{j}(h)in$Lambda$_{h}(E, c)}}subset mathrm{R} where displaystyle frac{wedge}{{cdot}^{ve}mathrm{s}mathrm{s} mems the set of accumulating points as hrightarrow 0. We prove the srcalled Helton type theorem including Hamiltonians with singular potentials, that is, either every classical Hamiltonian flow is periodic near E or D $sigma$(hat{H})=mathrm{R}.