Short Course: Pairing in the continuum

R. M. ID BETAN

Ciudad de México

Conferencia; Quantum Fest; 2014

Cinvestav

There are two characteristic ingredients in any many-body calculation: (i) the presence of a lot of degrees of freedom, and (ii) the interaction. Both of these features depend on the specific physical system and because of that there is a whole universe of techniques for dealing with many-body systems. Besides, one must distinguish between finite or infinite systems. Nuclei are finite many-body systems and some approximation must be done to study their properties. Typically, one comes across with the evaluation of the expectation value $\langle \Psi | V | \Psi \rangle$, where $\Psi$ is the wave function which describes the many-body system and $V$ is the interaction between their constituents. Then, in practical calculations, one may approximate the many-body wave function $\Psi$, the interaction $V$ or both. In the nuclear shell model, for example, one approximates the wave function $\Psi$ by a shell-model wave function $\Psi_{SM}$, and uses the ``exact'' form of $V$. Alternatively, one may approximate the interaction $V$ instead of the wave function and try to find the ``exact'' solutions $\Psi$ for an approximated interaction of $V$. The pairing interaction is of this second kind of approach. The importance of the pairing Hamiltonian is that it contrives to get the most important part of the  particle-particle interaction and as a consequence it constitutes an important approximation in many-body systems. The pairing is important in many physical phenomena, for example, for understanding the structure of the low-lying states of nuclei, the properties of neutron stars, exotic decay, alpha decay and fission among others. Since only for the special case of constant pairing (it is quantum integrable) one is able to find the exact wave function, one also appeals to the approximation of the wave function. As a consequence the approximate wave function does not conserve the number of particles of the system. In this short course we will show how to obtain the exact solution (Richardson) of the pure pairing interaction. We will introduce the Bardeen-Cooper-Schrieffer (BCS) and Lipkin-Nogami BCS approximate solutions. We will compare the approximate and the exact solutions. Because the dynamics of unstable systems occur in the continuous part of the energy spectrum, we will introduce the complex energy basis (Berggren representation) as a representation of the continuum and we will study its influence in the pairing interaction.