The use of exactly solvable Richardson-Gaudin models to describe the physics of systems with strong pair correlations is reviewed. The article begins with a brief discussion of Richardson's early work, which demonstrated the exact solvability of the pure pairing model, and then shows how that work has evolved recently into a much richer class of exactly solvable models. The Richardson solution leads naturally to an exact analogy between these quantum models and classical electrostatic problems in two dimensions. This analogy is then used to demonstrate formally how BCS theory emerges as the large-
N limit of the pure pairing Hamiltonian. Several applications to problems of relevance to condensed-matter physics, nuclear physics, and the physics of confined systems are considered. Some of the interesting effects that are discussed in the context of these exactly solvable models include (i) the crossover from superconductivity to a fluctuation-dominated regime in small metallic grains; (ii) the role of the nucleon Pauli principle in suppressing the effects of high-spin bosons in interacting boson models of nuclei, and (iii) the possibility of fragmentation in confined boson systems. Interesting insight is also provided into the origin of the superconducting phase transition both in two-dimensional electronic systems and in atomic nuclei, based on the electrostatic image of the corresponding exactly solvable quantum pairing models.
A simple, "exactly solvable" atomic model will be presented that helps to discuss some of the interesting features of multiphoton ionization of atoms. The electromagnetic field is treated non-perturbatively, it can be strong. The atomic electrons are assumed to be non-interacting (Hartree model). They are bound in a harmonic oscillator potential (one- or three-dimensional). At the ionization energy, the potential is cut off and the free electron has no Coulomb attraction. The Schrodinger equation has exact analytic solutions valid in the inner or outer regions. If the field is turned on adiabatically, the exact solution can be found by matching these solutions at the boundary; both of their value and the normal derivative at all times. Thus the machine calculation reduces to repeated application of fast Fourier transforms and to the solution of linear equations (unfortunately a lot of them).
In both molecular physics and condensed matter theory, deeper understanding of the correlation energy density
ϵc(r) remains a high priority. By adopting L\"owdin's definition of correlation energy as the difference between the exact and the Hartree-Fock values, here we propose two alternative routes to define this. One of these involves both exact and Hartree-Fock (HF) wave functions, while the second requires a coupling constant integration. As an exact analytical example of the first route, we treat the two-electron model atom of Moshinsky, for which both confinement potential and interactions are harmonic. Though the correlation energy density
ϵc(r) is known analytically, we also investigate numerically its relation to the exact ground-state density in this example.