Trace-class operators of scattering theory and the asymptotic behavior of scattering cross section at high energy

We investigate necessary and sufficient conditions under which the difference of the resolvents Rz − R0z of two self-adjoint operators of the form H01 ≡ F(P) and H ≡ F(P) + V(Q) is nuclear. Here F(P) denotes a positive and continuous function of the usual momentum observables P and V(Q) a function of the conjugate coordinate observables. Roughly speaking, we prove that Rz − R0z is nuclear if and only if F(P) increases faster than |P|3/2 at large |P| and V(Q) falls off to zero faster than 1/|Q|3 at large |Q|. (For a precise statement of this result Sec. 3.) In particular, it is noticed that if H0 = (|P|2 + m2)1/2 the relativistic free Hamiltonian, then Rz − R0z is not nuclear for any of the (suitably regular) potential V(QN.W.) where QN.W. denotes the usual Newton-Wigner position operator of the relativistic particle. We also investigate in Sec. 3 the necessary and sufficient conditions for Rzβ V Rz0β(β>0;β≠1) to be nuclear. The implications of these results for the asymptotic behavior of the total scattering cross sections at high energy is discussed in Sec. 4.