The resolvent of a dilation-analytic three-particle system

AbstractIf the potential in a three-particle system is the boundary value of an analytic function, the physical Hamiltonian H(0) has a dilation-analytic continuation H(φ). The continuous spectrum of H(φ) consists of half-lines Y(λp, φ) starting at the thresholds λp of scattering channels and making angles 2φ with the positive real axis. If the interaction is the sum of local two-body potentials in suitable Lp-spaces, each half-line Y(λp, φ) is associated with an operator P(λp, φ) that projects onto an invariant subspace of H(φ). Suppose Y(λp, φ) does not pass through any two- or three-particle eigenvalues λ ≠ λp when φ runs through some interval 0 < α ⩽ φ ⩽ β < π2. For φ in [α, β], this paper shows that the resolvent R(λ, φ) has smoothness properties near Y(λp, φ) that are sufficient for P(λp, φ)[H(φ) − λp] e−2iφ to be spectral and to generate a strongly differentiable group. The projection, the group, and the spectral resolution operators are norm continuous in φ. These results are not affected by any spurious poles of the resolvent equation. At a spurious pole λ = λp + ze2iφ, the resolvent R(λp + ze2iφ,φ) is examined by a method that uses two resolvent equations in succession and shows that there is norm continuity in z, φ. The case of spurious poles on Y(λp, φ) is included