Continuity and Asymptotic Behavior of Integral Kernels Related to Schrodinger Operators on Manifolds

The continuity of integral kernels related to Schrödinger operators (the kernel of the heat equation, Green’s function) plays an important role in the study of different properties of quantum systems. In the Euclidean case, it was shown in [1] that sufficiently general functions of the Schrödinger operator have continuous integral kernels for scalar potentials belonging to the Kato class. In [2], this result was generalized to operators with nontrivial vector potential of the magnetic field; in this case, arbitrary domains in the Euclidean space were admitted as configuration spaces. Simultaneously, in view of several problems in mesoscopy and gravitation quantization [3, 4], it seems to be of interest to study the integral kernels related to the Schrödinger operators on Riemann manifolds. In this paper, we presents several results concerning the existence and continuity of integral kernels for different operators generated by the Schrödinger operator on a manifold. In addition, we give several estimates for Green’s function for the cases in which its arguments are far from one another or, conversely, close to one another. Let X be a ν-dimensional manifold of bounded geometry. By d(x, y) we denote the geodesic distance between points x, y ∈ X ; by B(x, r) we denote an open ball of radius r centered at x ; and by D we denote the set {(x, y) ∈ X × X, x = y} ⊂ X × X . We consider a 1-form A on X with smooth coefficients; this form determines the connection in a trivial linear bundle over X ; by ∆A we denote the corresponding Bochner Laplacian (for A = 0, we obtain ∆A = ∆, which is the Beltrami–Laplace operator). The method used to prove our results requires several (rather weak) restrictions on the scalar potentials under study; in what follows, we assume that this potential belongs to the class P(X) of real functions U on X with the properties