Potential scattering with confined channels

Abstract

We introduce a class of nonrelativistic multichannel potential scattering models which are characterized by the simultaneous presence of and communication between two distinct types of channels: First, ordinary two-particle scattering channels, the Hamiltonian for which has an absolutely continuous spectrum on the positive real energy axis and perhaps a finite number of negative energy bound states; second, permanently confined channels (like those of the quark model), the Hamiltonian for which has only a point spectrum with an accumulation point at E = + ∞. These two types of channels are connected in the full Hamiltonian H for the multichannel system by off-diagonal local potentials which satisfy suitable smoothness and integrability conditions. The scattering theory of such systems is developed and, under certain general restrictions on the potentials, the following properties are rigorously established: (1) Asymptotic completeness; i.e., the generalized wave operators exist and are complete, and the S-matrix is a unitary operator in the scattering channels. The S-matrix has nonzero matrix elements only in the scattering channels. (2) The spectrum of the Hamiltonian H consists of 3 parts, namely, a finite number of negative energy eigenvalues, a discrete set of positive energy eigenvalues with only possible accumulation point at E = + ∞, and an absolutely continuous spectrum on the remainder of the positive real energy axis. To each eigenvalue, positive or negative, corresponds a finite number of orthonormal eigenvectors, the bound states of H ; and to each positive energy there corresponds a unique bounded solution, the distorted plane wave, to the time-independent Schrödinger equation. (3) There is an eigenfunction expansion associated with H in which enter only the bound-state eigenvectors and the distorted plane wave eigenfunctions. (4) The subspaces of discontinuity and absolute continuity of H are orthogonal complements to each other. In addition the scattering amplitude for these models is constructed and shown to be related to the S-matrix in the usual way. Both the S-matrix and the scattering amplitude are well defined and continuous at the positive boundstate energies.