Semigroups associated with analytic Schrödinger operators

If the potential in a two-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has an analytic continuation H(φ). The continuous spectrum of H(φ) consists of the half-line Y(0, φ) which runs from 0 to ∞e^2iφ. Integrating along lines parallel to Y(0, φ), this paper examines the Fourier transform of the resolvent R(λ, φ). The integration path passing through ±iεe^2iφ yields semigroups {U(t, ±iεe^2iφ, φ)} (t > 0 and t < 0). Under the assumption that the potential is local and belongs to suitable $\text{L}$^p-spaces, it is shown that the semigroups tend to norm limits as ε tends to 0. The proof is based on the Paley-Wiener theorem for functions in a strip. It generalizes to multiparticle systems under conditions on R(λ, φ) that are to be verified with the help of the theory of smooth operators.