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The Lorentz Group Symmetry of the Hydrogen Atom and the Coulomb *T* Matrix

### Abstract

The existence of solutions for the Lippmann‐Schwinger (LS) equation for the Coulomb problem is studied by investigating whether its kernel belongs to the Hilbert‐Schmidt (HS) class. The kernel for each partial wave is shown to belong to the HS class for complex energies whereas for real energies it becomes unbounded. Unlike the case of short range potentials, e.g., the Yukawa potential, even the suitably symmetrized kernel does not belong to the HS class for real, positive energies. These formal properties strongly indicate that a unique limit for the partial wave off‐shell Coulomb *T* matrix as it approaches the unitarity axis may not exist. It is found that exploiting the *O*(4) symmetry of the Coulomb Hamiltonian (*H*) in the subspace of the negative spectrum of *H* and the *O*(3, 1) symmetry in the subspace of the positive spectrum of *H*, one can construct the off‐shell Coulomb *T* matrix in terms of the eigen‐solutions (Sturmians) of the kernel of the Lippmann‐Schwinger equation. These follow from the work of Perelemov and Popov, and Schwinger on the Coulomb Green's function. On the basis of the generating functions for the Sturmians, various integral, contour integral, and discrete sum representations for the complete off‐shell Coulomb *T* matrix are derived. In this way, we explicitly demonstrate that indeed the *T* matrix has a nonunique limit as one approaches the unitarity axis. It is also shown that when the asymptotic Coulomb distortion is taken into account, the physical Coulomb amplitude can be deduced from this Coulomb *T* matrix. These results incidentally rectify some errors in the earlier works. Since for both negative and positive energies the Coulomb *T* matrix is obtained as the explicit solution of the LS equation, the validity of the generalized unitarity—the Low equation—in a certain sense is guaranteed. This is proved in a general way, by showing that a generalized Low equation follows when the energy is complex only from the LS equations and some defining relations; in the Coulomb case, the generalized unitarity relationship for real energies must be interpreted as the limit when the imaginary part becomes zero.

© 1971 The American Institute of Physics

Received 17 May 1971
Revised 25 June 1971
Published online 28 October 2003

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/content/aip/journal/jmp/12/11/10.1063/1.1665548

2003-10-28

2016-10-24

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