Some Exact Results for Quantum Lattice Problems

Physics is increasingly moving towards nano-scale technology, and with this comes the need to understand small lattices or quasi-lattices supporting a small number of quanta. Often this gives the quantum equivalent of a classical breather, though with some interesting differences. In this short paper we review some work on quantum lattice problems, concentrating mainly on the quantum discrete nonlinear Schrödinger (QDNLS) model. The corresponding classical DNLS lattice is discussed elsewhere in this book. Our eventual aim is to understand such systems as quantum dot arrays, BoseEinstein Condensate lattices, and even models for Quantum Computers. We adopt here a pedagogical approach with many details omitted– see for a fuller description of the background material in our approach, and 3,4 for some other interesting developments by other authors. Consider the classical DNLS model in one spatial dimension with nearest-neighbour interactions