ON LIMITS OF SEPARABLE POTENTIALS AND OPERATOR EXTENSIONS

A family of self-adjoint Hamiltonians with a separable potential leading towards a contact potential (zero range) is analyzed by tools of functional analysis. It is shown that the family of time evolution operators e-iHt converges strongly (for all t) though the family of Hamiltonians does not converge even weakly. In the case of three dimensions a renormalization procedure is discussed and a correspondence between the renormalized coupling constant and the self-adjoint extensions of the free Hamiltonian is established. © 1969 Springer-Verlag.