Groups generated by 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) which is not normal. In case the potential is local and belongs to suitable L p -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 −2 iφ 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.