Convolutions of semi-classical spectral distributions and periodic-orbit theory (Q859633)

In terms of the spectral distribution concept attached to an energy level \(E\Upsilon^P(E,h,\varphi)\) and to a Schrödinger operator \(P_h=-h^2\Delta +V\), with \(V\in C^\infty(\mathbb{R}^n,\mathbb{R})\), the author introduces a new spectral quantity by means of a convolution \(\Upsilon^{P,Q}(u,h,\varphi)=(\Upsilon^P*\Upsilon^Q) (u,h,\varphi)\). Under suitable hypotheses, he investigates the asymptotics of the convolution spectral distribution as \(h\to 0\). Very accurate estimates are obtained and several applications of the main results are provided, among them, the validation (in the author's opinion) of the theory of orbits pairs used in physics.