1978/09/01 by S. M. Blinder · 1 voice · 1 citation
Mathematics · Physics and Astronomy · #Quantum Mechanics and Non-Hermitian Physics #Quantum chaos and dynamical systems #Spectral Theory in Mathematical Physics
paper · doi:10.1103/physreva.18.853
openalex publication_date 1978/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A modification of the Fermi contact interaction is proposed in which the nuclear moment is represented by a uniformly magnetized spherical shell of radius r0. In effect, the delta function \ensuremathδ(r) in the Fermi Hamiltonian is replaced by \ensuremathδ(r\ensuremath-r0). The Schr"odinger equation for a hydrogenlike system thus perturbed is exactly solvable in terms of the Coulomb Green's function. Negative energy eigenvalues have the form E_\ensuremathν=\ensuremath-\fracZ22\ensuremathν2, with \ensuremathν a nonintegral quantum number. An asymptotic formula is derived for the quantum defect \ensuremathδ=\ensuremathν\ensuremath-n. The l=0 eigenfunctions are multiples of the Whittaker functions: M_\ensuremathν,(1)/(2)(\frac2Zr\ensuremathν) for r<r0 and W_\ensuremathν,(1)/(2)(\frac2Zr\ensuremathν) for r>r0. Explicit forms are given by expansion of the Whittaker functions to first order in quantum defect. In the limit r0\ensuremath→0 results pertaining to the original Fermi Hamiltonian are approached. It is shown that a repulsive delta function maintains the unperturbed Coulomb energy while an attractive delta function pulls all bound state energies to \ensuremath-\ensuremath∞. Perturbation expansions are discussed and comparisons made with earlier calculations. It is shown that second-order and higher perturbation energies diverge as r0\ensuremath→0.