Investigations Of Quantum Three-body Systems Such As Halo Nuclei

Quantum few-body problems are attracting more and more interest in recent years. The newly found structure called quantum halos can be well described by few-body mod-els. A bizarre few-body phenomenon, which is theoretically predicted by Vitaly Efimov in1970s, is usually present in quantum halos. Three identical bosons with resonant two-body interactions may form infinitely many trimer states, while the two-body subsystems are very close to the zero-energy threshold. Any further increase of the strength of two-body interactions will decrease the three-body bound states to a finite number.The Efimov effect has turned into a hot topic as the tunable ultracold atomic gases experiments become realizable. The so-called Feshbach resonance provides us a tool for controlling the two-body interaction via an external magnetic field. This ability of control makes ultracold atomic gases a preferred experimental platform for investigating Efimov effect. Several experiments have been done, which have revealed the occurrence of the Efimov effect and its universality in few-body physics.In the beginning, we present a brief introduction to the quantum three-body problems. Related topics such as quantum halos and Efimov effect are then detailedly introduced, especially the universality of Efimov effect in few-body physics. As an experimental tool for observing Efimov effect, Feshbach resonance in ultracold atomic gases is discussed. After that, the evidences for the existence of the Efimov effect are given and discussed.We adopt two theoretical methods in our study, which are briefly introduced in Chap-ter2. Firstly, the original equivalent two-body method is proposed with its spirit and formalism. Then several improved and extended versions of this method are presented. respectively, by taking more and more improvements into account. Secondly, three-body hyperspherical formulation based on solving the Faddeev equations with hyperspherical coordinates expansion is simply given.Following the methodological chapter, we begin to investigate concrete physical sys-tems. Our study is divided into four parts, each of which focuses on a specific lype of two-body interaction in three-body systems. First of all. the Coulomb potential is considered in Chapter3. As a most common example, the bound state energies and wave functions of helium-like ions are calculated by the equivalent two-body method. The results are consistent with the experimental data much better, in contrast to those obtained by a one-parameter calculus of variations approach. A two-parameter calculus of variations approach is then adopted to verify the results. After that, the H+2ion system is investigated. It’s indicated that improvements and extensions on the original equivalent two-body method could produce much better results.In Chapter4, three-body systems with two-body potentials of a general power series form are discussed. By means of the equivalent two-body method, an effective potential of a power series form is obtained, as well as its asymptotic behavior. Based on the analyt-ical expression of the effective potential, the linear and the harmonic oscillator potentials are studied respectively. We come back to a general discussion on the power law of the effective potential at the end of the chapter.The attractive r-2two-body interactions, which are the division between short-and long-range potentials, are the central topic of Chapter5. An effective potential of the R-2form is derived. Infinitely many bound states are obtained in the system with two heavy particles and a light particle, even if all the two-body subsystems have no bound states. However, as we adjust the three particles towards three identical bosons, the infinitely many bound states disappear, while these bound states are considered to be present in such a system. In order to overcome this problem, we adopt two modified versions of the equivalent two-body method. While some improvements can be made, these modified methods still fail to explain this discrepancy, which is probably due to the accuracy of the methodological framework. It should be mentioned that these infinitely many bound states are actually different from the Efimov states.In the last part we focus on the short-range two-body interactions. First of all. two identified halo nuclei.6He and11Li. are quantitatively solved by the Faddeev equations. Good consistency between the calculational results and experimental data is obtained, which indicates the availability of few-body model in halo structure. Next, the0+bound states of three-body system with spherical square well potential are discussed. An asymp-totic behavior of1/R3for the effective potential is obtained, even if no specific forms of short-range two-body potentials are considered. Furthermore, specific forms of short-range two-body interactions are adopted by means of solving the Faddeev equations. Gen-era10+bound state properties are obtained despite of the specific forms of the potentials, too. On the one hand, a number of bound0+excited states appear in the system with two heavy particles and a light particle. On the other hand, a weakly bound0+excited state is obtained in the system with two light particles and a heavy particle.Finally, we give a brief summary of this dissertation, and some outlook is also pro-posed for possible investigations in the future.