Semigroups associated with analytic Schrödinger operators

Abstract

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 2 iφ . 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 2 iφ yields semigroups { U( t, ± iεe 2 iφ , φ)} ( t > 0 and t < 0). Under the assumption that the potential is local and belongs to suitable 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.