The regularized trace of the perturbed Laplace-Beltrami operator on a certain family of manifolds

Abstract

225 This paper is devoted to calculating the regularized trace of the perturbed Laplace–Beltrami operator on a certain family of manifolds with closed geodesics. After the development of the theory of integral Fourier operators, an overwhelming progress in the study of the spectrum of differential operators on com� pact manifolds with periodic bicharacteristic flow has been achieved. In this theory, the Laplace–Beltrami operator perturbed by the operator of multiplication by a smooth function on manifolds with closed geode� sic flow occupies a special place as the basic model and as the physically most interesting case. Consider the Laplace–Beltrami operator Δ 0 on the sphere S n in R n+1 . By Δ = Δ0 +