Eigenvalues of the Yakubovskii equation kernel for a four-nucleon system

Abstract

Low-lying states of $^{4}\mathrm{He}$ have been studied by calculating the trajectories of the first two eigen-values of the kernels of the Yakubovskii four-body equation as the total energy $E$ increases from $\ensuremath{-}\ensuremath{\infty} \mathrm{to} \ensuremath{\infty}+i\ensuremath{\epsilon}$. The two-particle interactions used are of the separable Yamaguchi type and include spin-dependent forces. The integral equations are derived for each state with values of spin $S$, isospin $T$, and total angular momentum $L$. To obtain a set of single variable integral equations, the Schmidt expansion is applied. The deformation of the integration contour is performed for the complex eigenvalues, and the eigenvalue problems for these equations are solved to determine the binding energy or the resonance energy including $l=1$ subamplitudes for 3 + 1 subsystems. The binding energies for the ground and the first excited state are -45.009 MeV and -11.529 MeV, respectively. A resonance state is found to be about -4.889 MeV in the state with $ST=10$, $L=1$ corresponding to the degenerate state of the second, third, and ninth excited states of the $^{4}\mathrm{He}$ nucleus.[NUCLEAR STRUCTURE $^{4}\mathrm{He}$; calculated levels. Four-body, separable potential model.]