A new field of studies has recently arisen in solid-state physics--quantum crystals. The currently existing theoretical and experimental data indicate--leaving no doubt--that a quantum crystal is a new state of matter, which had been predicted theoretically by A. F. Andreev and I. M. Lifshitz. At low temperatures one has not too large a number of quasiparticles to deal with. This permits one to treat a quantum crystal as a ``rarefied gas'' of quasiparticles in the discrete space of the crystal lattice. Starting with the founding ideas of I. M. Lifshitz, a method is developed of describing the process of pairwise scattering of quasiparticles in the discrete space. The energy spectrum of the quantum crystal is discussed. The features of behavior of vacancions in solution are elucidated.