On 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 R_z − R⁰_z 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 R_z − R⁰_z 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 R_z − R⁰_z is not nuclear for any of the (suitably regular) potential V(Q_N.W.) where Q_N.W. denotes the usual Newton‐Wigner position operator of the relativistic particle. We also investigate in Sec. 3 the necessary and sufficient conditions for $R_z^{\beta }{\mathrm{V\; R}}_z^{\text{0β}}\text{(β>0;β≠1)}$ 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.