In this dissertation we develop a numerical approach that is applicable to the investigation of different few-body quantum systems in states close to a threshold of dissociation. These systems are extremely loosely bound and require advanced numerical techniques for accurate investigation. Our results are corroborated by analytical estimates.

For high-precision direct three- and four-particle calculations we use the stochastic variational method. This method has proven its effectiveness for a variety of few-body quantum problems. We use it to compute near-threshold states and calculate binding energies for different values of the system parameters.

With this approach we systematically study the energy stability thresholds for the general quantum three-particle problem where two particles are identical. We use both short range and long range interactions and investigate a wide spectrum of mass ratios.

Our main results were presented in a form of stability diagrams that show domains, in parameter space, where the system is stable in a particular state. We use the stability diagrams of different three-particle systems with short-range and Coulomb interactions to take a new look at such well-known near-threshold effects as the Efimov effect and the Thomas effect. We also reveal a number of new effects in the behavior of the three-particle stability thresholds.

We make a thorough analysis of the dependency of the stability thresholds on the masses of the particles and on the properties of the short-range interaction. We do so for different symmetries of the states. Analytical estimates are used to confirm the asymptotic behavior of threshold lines.

Our numerical approach is extended to three-particle systems with non-zero total angular momentum and to four-particle systems. Though a systematic investigation of their near-threshold behavior is a subject of future research, some results for such systems were presented in the context of this dissertation. Specifically, we are going to study the possible existence of bound multi-neutron systems, which are expected to be extremely loosely bound if they exist at all.