Groups generated by analytic Schrödinger operators

AbstractIf 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) which is not normal. In case the potential is local and belongs to suitable Lp-spaces, there is a bounded operator P(0, φ) projecting onto the continuous subspace of H(φ). This paper shows that P(0, φ) H(φ) e−2iφ generates a strongly differentiable group. It is proved that P(0, φ) H(φ) is spectral, and details of the spectral projection operators are presented. The reasoning is based on the Paley-Wiener theorem for functions in a strip. It applies to larger systems provided the resolvent of the multiparticle operator H(φ) satisfies certain regularity conditions that come from the theory of smooth operators. There are no smallness conditions on the potential

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

Space-time rotations and isobaric spin

After some preparatory work on rotations and generalized Eulerian angles, the transformations of special relativity are discussed in terms of six Eulerian angles in Minkowski space. Families of functions Z are constructed which transform Iinearly among themselves under rotations in space-time as well as under spatial reflection. It is found that for any finite-dimensional representation of the full Lorentz group there exist several families of functions Z, which are distinguished from each other by two family-indices. ... Zie: Summary.