Exactly solvable models have played an important role in the development of the nuclear shell model, in the advancement of our understanding of pairing properties in nuclei and in the description of collective nuclear phenomena in the context of geometric and algebraic models. An overview of exactly solvable models of nuclei is given with emphasis on conceptual rather than technical issues.

Table of Contents Introduction An algebraic formulation of the quantal n-body problem The nuclear shell model Racah’s seniority model Identical nucleons. Neutrons and protons. Wigner’s supermultiplet model Elliott’s rotation model The Lipkin model Geometric collective models Exactly solvable collective models The five-dimensional harmonic oscillator. The infinite square-well potential. The Davidson potential. Other analytic solutions. Triaxial models Rigid rotor models. The Meyer-ter-Vehn model. Approximate solutions for soft potentials. Partial solutions. Geometric collective models: an algebraic approach The interacting boson model Neutrons and protons: \(F\) spin Neutrons and protons: Isospin Supersymmetry Beyond exact solvability Further reading Acknowledgement References

The atomic nucleus is a many-body system predominantly governed by a complex and effective in-medium nuclear interaction and as such exhibits a rich spectrum of properties. These range from independent nucleon motion in nuclei near closed shells, to correlated two-nucleon pair formation as well as collective effects characterized by vibrations and rotations resulting from the cooperative motion of many nucleons.

The present-day theoretical description of the observed variety of nuclear excited states has two possible microscopic approaches as its starting point. Self-consistent mean-field methods start from a given nucleon--nucleon effective force or energy functional to construct the average nuclear field; this leads to a description of collective modes starting from the correlations between all neutrons and protons constituting a given nucleus (Bender, Heenen, & Reinhardt, 2003). The spherical nuclear shell model, on the other hand, includes all possible interactions between neutrons and protons outside a certain closed-shell configuration (Caurier, Martínez-Pinedo, Nowacki, Poves, & Zuker, 2005). Both approaches make use of numerical algorithms and are therefore computer intensive.

A review is given of a class of sub-models of both approaches, characterized by the fact that they can be solved exactly, highlighting in the process a number of generic results related to both the nature of pair-correlated systems as well as collective modes of motion in the atomic nucleus. Exactly solvable models necessarily are of a schematic character, valid for specific nuclei only. But they can be used as a reference or `bench mark' in the study of data over large regions of the nuclear chart (series of isotopes or isotones) with more realistic models using numerical approaches. The emphasis here is on the exactly solvable models themselves rather than on the comparison with data. The latter aspect of exactly solvable models is treated in several of the books mentioned at the end of this review.

Symmetry techniques and algebraic methods are not confined to certain models in nuclear physics but can be applied generally to find particular solutions of the quantal <math>n</math>-body problem. How that comes about is explained in this section.

To describe the stationary properties of an <math>n</math>-body system in non-relativistic quantum mechanics, one needs to solve the time-independent Schrödinger equation which reads

<math>\hat H\Psi(\xi_1,\dots,\xi_n)=

E\Psi(\xi_1,\dots,\xi_n), \label{e_schroed}</math>

where <math>H</math> is the many-body Hamiltonian

<math></math>\hat H=

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