Regularized trace of the Laplace-Beltrami operator perturbed by the   function on manifolds with specially perturbed sphere metric
  The paper is devoted to investigation of the spectrum of a perturbed Laplace-Beltrami operator on manifolds with closed geodesics of the same length and where metric is a perturbation of the metric of the unit sphere. As the result, we give the regularized trace formula for the eigenvalues ??of this operator when the metric of manifold is presented in abstract form. 
T.Zykova
math.SP