Commun. Math. Phys. 188, 233 – 249 (1997) Communications in Mathematical Physics c  Springer-Verlag 1997 On-Diagonal Estimates on Schr¨ odinger Semigroup Kernels and Reduced Heat Kernels Adam Sikora Centre for Mathematics and its Applications, School of Mathematical Sciences, Australian National University, Canberra, ACT 0020, Australia Received: 25 May 1996 / Accepted: 29 January 1997 Abstract: We prove various estimates for the kernels of semigroups generated by Schr¨ odinger operators with magnetic field and potential of polynomial growth. We also investigate the reduced heat kernels. 1. Introduction Let M be a connected and complete Riemannian manifold with Riemannian metric 〈·, ·〉. By d we denote the Riemannian distance on M and by H we denote the operator (Hψ,ψ)=  M dx ( |grad ψ(x)+ iψ(x)Y | 2 + V (x)|ψ(x)| 2 ) , (1) where dx is a Riemannian measure on M , ψ ∈ C ∞ c (M ), Y is a real vector field such that 〈Y,Y 〉∈ L 1 loc (M ), V : M → R, V ∈ L 1 loc (M ) and V ≥ 0. With some abuse of notation, we will also denote by H the Friedrichs extension of this operator. For any bounded Borel function F : [0, ∞) → C we define the operator F (H) by the spectral decomposition and we denote its kernel by K F (H) , i.e. F (H)(ψ)(x)=  M dy K F (H) (x, y)ψ(y) . The operator H is called a Schr ¨ odinger operator with magnetic field. Various properties of such operators were studied in many papers, see e.g. [2, 9, 12, 14]. In the sequel we will always assume that the following Nash inequality holds: ‖ψ‖ L 2 ≤ ε ( ‖grad ψ‖ 2 L 2 + γ 2 ‖ψ‖ 2 L 2 ) 1/2 + c (γε) −k/2 ‖ψ‖ L 1 (2)