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.

A new angular momentum projection for systems of particles with arbitrary spins is formulated based on the Heine-Stieltjes correspondence, which can be regarded as the solutions of the mean-field-plus -pairing model in the strong-pairing interaction $G\to \infty$ limit. Properties of the Stieltjes zeros of the extended Heine-Stieltjes polynomials, whose roots determine the projected states, and the related Van Vleck zeros are discussed. An electrostatic interpretation of these zeros is presented. As examples, applications to $n$ nonidentical particles of spin $1/2$ and to identical bosons or fermions are made to elucidate the procedure and properties of the Stieltjes zeros and the related Van Vleck zeros. It is shown that the new angular momentum projection for $n$ identical bosons or fermions can be simplified with the branching multiplicity formula of U$\left(N\right)\downarrow \mathrm{O}\left(3\right)$ and the special choices of the parameters used in the projection. Especially, it is shown that the solutions for identical bosons can always be expressed in terms of zeros of Jacobi polynomials. However, unlike nonidentical particle systems, the $n$-coupled states of identical particles are nonorthogonal with respect to the multiplicity label after the projection.

The exact eigenenergies of the $T_{4c}=[cc][\overline{c}\overline{c}]$, $T_{4b}=[bb][\overline{b}\overline{b}]$, and $T_{2[bc]}=[bc][\overline{b}\overline{c}]$ tetraquarks are calculated within the extended transitional Hamiltonian approach, in which the so-called Bethe ansatz within an infinite-dimensional Lie algebra is used. We fit the parameters appearing in the transitional region from phenomenology associated with potential candidates of tetraquarks. The rotation and vibration transitional theory seems to provide a better description of heavy tetraquarks than other attempts within the same formalism. Our results indicate that the pairing strengths are large enough to provide binding; an extended comparison with the current literature is also performed.

The nucleon pair shell model (NPSM) is cast into the so-called $M$ scheme for the cases with isospin symmetry and without isospin symmetry. The odd system and even system are treated on the same footing. The uncoupled commutators for nucleon pairs, which are suitable for the $M$ scheme, are given. Explicit formula of matrix elements in $M$ scheme for overlap, one-body operators, and two-body operators are obtained. It is found that the CPU time used in calculating the matrix elements in $M$ scheme is much shorter than that in the $J$ scheme of NPSM.

We propose an algebraic approach to elucidate the dynamic characteristics of triaxial rotor modes in nuclei by mapping a triaxial rotor Hamiltonian to the interacting boson model one within a finite-$N$ framework. Our method unveils striking features not observed in conventional modes, exemplified by the $B\left(E2\right)$ anomaly, characterized by $B(E2;4_1^{+}\to 2_1^{+})/B(E2;2_1^{+}\to 0_1^{+})<1$. Using specific examples, we demonstrate that the peculiar properties of low-lying states in both neutron-rich and neutron-deficient Os nuclei can be comprehensively understood through the proposed Hamiltonian, which incorporates both rigid and soft triaxial rotor modes. This algebraic method not only offers fresh insights into triaxial dynamics but also showcases its capability in uncovering emergent exotic collective modes in nuclear structure.