Semiclassical principal symbols and Gutzwiller's trace formula
The paper studies the semiclassical trace formula of Gutzwiller [proved by \textit{M. Gutzwiller}, J. Math. Phys. 12, 343-358 (1971)] for the asymptotic expansion of the spectral density for the Schrödinger operator. This formula is closely related to a theorem of \textit{J. J. Duistermaat} and \textit{V. W. Guillemin} [Invent. Math. 29, 39-79 (1975; Zbl 0307.35071)] on the spectral asymptotics of positive, first order, elliptic pseudo-differential operators on compact manifolds, but a rigorous proof has apparently been given only recently by \textit{T. Paul} and \textit{A. Uribe} [C. R. Acad. Sci., Paris, Sér. I 313, No. 5, 217-222 (1991; Zbl 0738.58046)] after preliminary work of Guillemin-Uribe and Brummelhuis-Uribe. The present paper gives a new proof of the result by a direct adaptation of the Fourier integral calculus to the semiclassical context.
scientific article; zbMATH DE number 459126
scientific article