Colston Chandler

Orthogonal polynomial expansion of the spectral density operator and the calculation of bound state energies and eigenfunctions

Abstract An orthogonal polynomial expansion method is presented, and illustrated with calculations, for calculating δ( E – H ), the spectral density operator (SDO), the projection operator that projects out of any L 2 wavepacket the eigenstate (s) of H having energy E . If applied to an L 2 wavepacket which overlaps the interaction, it yields either scattering-type (improper) eigenstates or proper bound eigenstates. For negative energies, the exact SDO yields zero away from an eigenvalue, and yields the energy eigenstate (times a constant) when E equals an eigenvalue. The finite orthogonal polynomial expansion of the SDO, acting on an L 2 wavepacket, yields approximately zero for E not equal to an eigenvalue, and becomes nonzero in the neighborhood of an eigenvalue.

Transition from time‐dependent to time‐independent multichannel quantum scattering theory

Rigorous proofs are given of several theorems establishing the connection between time‐dependent and time‐independent multichannel scattering theory. The method of proof involves a two‐Hilbert‐space formulation of time‐dependent multichannel theory and the theory of spectral integrals. In particular, the time‐independent theory in the form proposed by Alt, Grassberger, and Sandhas is derived.

The Coulomb problem a selective review

Abstract Selected topics in the nonrelativistic quantum theory of charged particle systems are reviewed. Recent results in the theory of bound states and resonances are briefly discussed. A more extended discussion of the scattering theory is presented. Emphasis is on the fundamental mathematical issues, but reference is made to numerical calculations.

N‐body quantum scattering theory in two Hilbert spaces. I. The basic equations

Derivations are given for some transition and resolvent operator equations for multichannel quantum scattering with short‐range potentials. The basic difference between these and previous equations is that the unknown operators act only on the channel subspaces. This is made possible by utilizing, and extending, the two‐Hilbert‐space formulation previously given by the authors [in J. Math. Phys. 14, 1328 (1973)]. The equations in abstract form are of the Lippmann–Schwinger type, differing only in the appearance of certain injection operators from one Hilbert space to the other. When applied to multichannel quantum scattering, the abstract theory yields a new system of equations for the transition and resolvent operators. Uniqueness of the solution to the equations is proved.

Time‐independent multichannel scattering theory for charged particles

Rigorous derivations are given of two time‐independent formulas for the multichannel scattering operator for nonrelativistic charged particle systems. The derivations are based on Dollards time‐dependent theory and use techniques of spectral integration. The formulas involve a complex power of the resolvent operator, in contrast to short‐range formulas. Bilateral Laplace transforms are used to derive a generalized multichannel resolvent equation and to prove existence and uniqueness of the solution. The formulas are applied to recover the well‐known two‐body Coulomb scattering amplitude.

N‐body quantum scattering theory in two Hilbert spaces. II. Some asymptotic limits

Within the framework of two‐Hilbert space scattering theory the existence of the strong Abel limit of a certain operator is proved, leading to the following results. A generalized Lippmann identity is derived that is valid for all channels, rather than only two‐body channels. On shell equivalence of the prior, post and AGS transition operators is rigorously proved, thus closing a gap in previous proofs. Results concerning the existence of the scattering operator as a strong, rather than weak, Abel limit are presented, and their implications with respect to the problem of unitarity are discussed. Finally, the possibility of exploiting operator limits of the Obermann–Wollenberg type is studied, with negative results.

Spurious solutions to N-particle scattering equations

Abstract The problem of spurious solutions to the integral equations of N -particle quantum scattering theory is treated. In particular, the Federbush discussion of the Weinberg-van Winter equation is enlarged. It is shown within this framework that the Narodetzkii-Yakubovskii equations, the Bencze-Redish equations, and the Kouri-Levin-Tobocman equations can all have spurious solutions. Certain model independent results are also presented.

Time‐dependent multichannel Coulomb scattering theory

The formulation by Mulherin and Zinnes of two‐particle Coulomb scattering theory is extended to the multichannel case. The wave operators so obtained are proved by a direct method to be identical with those of Dollard.

Spline interpolation and smoothing on hyperspheres

The authors generalize Wahba’s theory [SIAM J. Sci. Statist. Comput., 2 (1981), pp. 5–16] on spline interpolation and smoothing on the surface of the two-dimensional unit sphere to arbitrary dimensional hyperspheres. As a consequence, practical solutions to minimum norm interpolation and smoothing problems on hyperspheres are provided in terms of certain hyperspherical splines. In addition, Wahbas results for powers of the Laplace–Beltrami operator are extended to more general operators, and Wahba’s Hilbert space of constant functions is expanded to allow more than just constant functions. Extensive curve fitting calculations are made for some two- and higher-dimensional test problems using hyperspherical harmonics and hyperspherical splines. It is found that hyperspherical splines yield better fits than hyperspherical harmonics for test functions that possess no symmetry and are not infinitely differentiable. Tests have been run using several continuous functions; satisfactory absolute errors can be obt...

Multiparticle scattering theory and the method of coupled reaction channels

Abstract The two-Hilbert-space formulation of multiparticle scattering theory is used to investigate the validity and limitations of the conventional form of the coupled reaction channels (CRC) method. Also a new set of coupled dynamical equations is rigorously derived which includes the conventional CRC equations as a special case.