Potential scattering with confined channels

Journal Article · Mon Nov 01 04:00:00 EST 1976 · Ann. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1016/0003-4916(76)90255-4· OSTI ID:7333711

Dashen, R F; Healy, J B; Muzinich, I J

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: 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 and permanently confined channels, the Hamiltonian for which has only a point spectrum with an accumulation point at E = +infinity. These two types of channels are connected in the full Hamiltonian H for the multichannel system by off-diagonal local potentials. The scattering theory of such systems is developed and the following properties are rigorously established: (1) The generalized wave operators are complete, and the S-matrix is a unitary operator. The S-matrix has nonzero matrix elements only in the scattering channels. (2) The spectrum of the Hamiltonian H consists of a finite number of negative energy eigenvalues, a discrete set of positive energy eigenvalues, and an absolutely continuous spectrum on the remainder of the positive real energy axis. To each eigenvalue, 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 Schroedinger 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. 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. (AIP)

• Research Organization: Institute for Advanced Study, Princeton, New Jersey 08540
• OSTI ID: 7333711
• Journal Information: Ann. Phys. (N.Y.); (United States), Journal Name: Ann. Phys. (N.Y.); (United States) Vol. 102:1; ISSN APNYA
• Country of Publication: United States
• Language: English

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72 PHYSICS OF ELEMENTARY PARTICLES AND FIELDS
AMPLITUDES
BOUND STATE
COMPOSITE MODELS
COUPLED CHANNEL THEORY
EIGENFUNCTIONS
EIGENSTATES
EIGENVALUES
ELASTIC SCATTERING
FUNCTIONS
HADRON-HADRON INTERACTIONS
HAMILTONIAN FUNCTION
INTERACTIONS
MATHEMATICAL MODELS
MATRICES
PARTICLE INTERACTIONS
PARTICLE MODELS
POTENTIAL SCATTERING
QUARK MODEL
S MATRIX
SCATTERING
SCATTERING AMPLITUDES