A new approach for solving the Bethe ansatz (Gaudin-Richardson) equations of the standard pairing problem is established based on the Heine-Stieltjes correspondence. For $k$ pairs of valence nucleons on $n$ different single-particle levels, it is found that solutions of the Bethe ansatz equations can be obtained from one $(k+1)\times (k+1)$ and one $(n-1)\times (k+1)$ matrices, which are associated with the extended Heine-Stieltjes and Van Vleck polynomials, respectively. Since the coefficients in these polynomials are free from divergence with variations in contrast to the original Bethe ansatz equations, the approach provides an efficient and systematic way to solve the problem, which by extension, can also be used to solve a large class of Gaudin-type quantum many-body problems, including an efficient angular momentum projection method for multiparticle systems.

The role of pairing interactions on fission barriers and static fission paths in Th, U and Pu isotopes is investigated by using the deformed mean-field plus standard pairing model. It is found that pairing interaction plays different roles in different stages of the fission process. The first and second saddle point of the fission process are sensitive to the pairing interaction strength, while the ground-state and prescission points are mainly driven by nuclear deformation. It is also demonstrated that pairing interactions play a crucial role in the region of the inner and outer barriers. The barrier heights obtained by using the present model are found to reproduce the experimental data remarkably well and yield better results than those obtained by using the BCS approximation, thus indicating the importance of exact solutions of pairing interactions. Through numerical analysis, we provide possible microscopic pictures of spontaneous fission in the current pairing model.

The role of pairing interactions on the scission configurations, the total kinetic energy, and the mass distributions in U isotopes is investigated using the deformed mean-field plus standard pairing model. The total kinetic energy and the mass distributions of ${}^{232–238}\mathrm{U}$ isotopes obtained using the shape-dependent probability distribution expressed by the Wigner function reproduces the experimental data remarkably well. The model calculations show that the scission region is sensitive to the variation of the pairing interaction strength, particularly for the asymmetric and symmetric scission points. The apparent changes in the peak-to-valley ratio in mass distribution by varying the pairing interaction strength confirm that the pairing interaction plays an important role in achieving the scission process for ${}^{236}\mathrm{U}$ under the present model. These results also suggest that the pairing interaction strength in the current work should increase with the elongation of the nucleus to yield better fission products. Through numerical analysis, we provide possible microscopic pictures of spontaneous fission around the scission configurations in the exactly solvable pairing model.