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Additional resources for Exact Solutions of Relativistic Wave Equations
1) implied by Eq. 4) IL Here, as in the close-coupling Eq. 38), index s labels independent degenerate solutions of the continuum Schrodinger equation, one for each open channel at given total energy E. 5) which imply Eq. 35). Index q refers to open channels only, and H is the (N + I)-electron Hamiltonian operator. 6) for arbitrary fixed values of the coefficients (Xips. 7) When Eq. , which can be assumed to be real numbers, '::'st pq = '\"\' £.. £.. 8) where, as a consequence of Eq. 9) IL v The matrices combined in Eq.
3. Anomalies in the Kohn Formalism The anomalous singularities inherent in the Kohn formula were first considered by Schwartz (1961a,b), who encountered these anomalies in variational calculations of e - -H elastic scattering. Nevertheless, the s- wave phase shifts obtained by Schwartz, after empirically smoothing the variationally computed numbers, still stand as the most accurate work available. 1 (Schwartz, 1961b, Fig. 4ao\ evaluated with wave functions expanded in basis sets containing from three to 34 functions of rb r2, and r12.
Special problems occur when m (which is a real unsymmetric matrix) has complex eigenvalues. A transformation to reorder diagonal elements must be used to ensure that m~l remains nonsingular in the limit of an exact scattering solution. These properties of the method introduce some arbitrariness and lead in some applications to irregular energy dependence that appears to be a computational artifact. A special algorithm for reordering the eigenvalues of m' in the OAF method has been published (Nesbet, 197 8b) together with a modified method (OAF2) that obtains a symmetric K matrix.