Study on Physical Properties of Quantum Singularity

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The present study aims at providing a coherent framework for singularities in quantum mechanics and thereby seek their possible physical effects in various situations. To this end, we have investigated primarily on the three topics: (1) statistical properties of many particle systems in the presence of quantum singularity, (2) quantum singularity and quantum integrable systems, and (3) quantum singularity and quantum information.First, for (1) we have studied systems consisting of bosonic/fermionic particles confined in a quantum well in one dimension, where a quantum singularity is placed at the center of the well acting as a partition wall. Different quantum pressures emerge depending on the nature of the singularity, which is also dependent on the temperature and the number of the particles and their statistics. We have found a curious scaling law satisfied by the particle number and temperature, which suggests a novel method of detecting the nature of particles by observation of the scaling law in low temperature regimes. Concerning (2), we have considered the N = 3 Calogero model with generalized singularity in the potential, and found a new set of solutions there. This shows explicitly that physical properties do change according to the treatise of the singularity. Finally, for (3) we have proposed a simple model of quantum computation by means of quantum singularities (quantum abacus). We have also investigated the physical effects of quantum entanglement by using game theory as a testing ground. We have succeeded to provide a complete framework of two-player games in which all possible quantum strategies can be treated and yet the entanglement is transparent. The resultant Nash equilibrium can also be classified easily whereby one can observe the superiority of quantum strategies over the classical counterparts due to quantum entanglement.