By using the formulation of the reconstruction, we explicitly construct
models of k-essence, which unify the inflation in the early universe and the
late accelerating expansion of the present universe by a single scalar field.
Due to the higher derivative terms, the solution describing the unification can
be stable in the space of solutions, which makes the restriction for the
initial condition relaxed. The higher derivative terms also eliminate tachyon.
Therefore we can construct a model describing the time development, which
cannot be realized by a usual inflaton or quintessence models of the canonical
scalar field due to the instability or the existence of tachyon. We also
propose a mechanism of the reheating by the quantum effects coming from the
variation of the energy density of the scalar field.Comment: LaTeX, 13 pages, 10 figure
