potentials, which describes an atom in a magnetic field. Announcements of some of our results appear in [4,5].Basic to many of our results is the following: If V is fixed and H() H0() + V---t)+ Vthen le-"<)4,1 <_ e-'')14,(1.1) pointwise. The history of (1.1) and references for proofs can be found in 2

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Using dilation invariance and dilation analytic techniques, and with the help of a new virial theorem, we give a detailed description of the spectral properties of the operator (p 2 + m 2 ) 112 -Ze 2 /r. In the process the norm of the operator |*|~α|/>|~α is calculated explicitly in L P (1R N ).

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We analyze the spectral properties and discuss the scattering theory of operators of the form H = H 0 +V> H 0 =-A+Ex.Among our results are the existence of wave operators, Ω ± (H,H 0 ), the nonexistence of bound states, and a (speculative) description of resonances for certain classes of potentials.