A Formula for the First Regularized Trace of a Perturbed Laplace–Beltrami Operator

Abstract

The theory of regularized traces is well developed for problems induced by ordinary differential expressions. It was shown in the fundamental paper [1] that the derivation of trace formulas can be reduced to studying zeros of entire functions possessing a specific asymptotic structure related to the particular form of the fundamental solution system of the differential equation. The case of problems induced by partial differential operators is much harder. This is primarily due to the complicated structure of the spectrum. The asymptotic spectrum distribution was obtained in [2] for the operator −∆+V with smooth odd potential V on the sphere S. However, the results of [2] do not allow finding regularized trace formulas. Further substantial progress in this problem was made in [3], where a formula for the first regularized trace was obtained for the first time for the operator −∆ + V with odd smooth potential V . Later, the method of [4] was used in [5] for finding the first regularized trace for an odd potential with n = 2 and in [6] for deriving a regularized trace formula for the operator −∆+V on compact symmetric spaces of rank 1. However, the condition that the potential V is C∞ was substantially used in all these papers. In the present paper, we derive a formula for the first trace of the operator −∆ +V on S in the case of a finite smoothness of the potential V and without the assumption that V is odd. If the potential is odd, then formula (1) coincides with the corresponding formula in [5]. Let L0 = −∆, and let V be the operator of multiplication by a real smooth function in L2 (S), where S is the two-dimensional sphere. By L we denote the operator L0 +V . Further, let R0(z) = (L0 − zE) and R(z) = (L− zE)−1; let λn = n(n+ 1) and μ n , n = 0, 1, . . . , −n ≤ i ≤ n, be the eigenvalues of the operators L0 and L, respectively, and let f (i) n = Y (i) n (φ, θ) be the orthonormal spherical functions, that is, the eigenfunctions of the operator L0. The following assertion is the main result of the present paper.