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

The regularized trace of the perturbed Laplace-Beltrami operator on a certain family of manifolds with closed geodesics is calculated. A Laplace-Beltrami operator on the sphere and a normalized Laplace-Beltrami operator is considered. The regularized trace of the perturbed Laplace-Beltrami operator on a family of manifolds is calculated. Sphero-conical coordinates and formulas expressing the Cartesian coordinates in terms of the sphero-conical coordinates are considered. The form of the standard metric on the two-dimensional sphere in sphero-conical coordinates are obtained. The initial space is divided into the four domains that depend on smooth functions not changing the asymptotics of the function. A manifold determined by some functional family of smooth almost Liouville metrics on the sphere and specified by formulas is considered and conditions satisfied by eigenvalues of the operator are found.