  This paper is devoted to giving definitions of Besov spaces on an arbitrary
open set of $\mathbb R^n$ via the spectral theorem for the Schr\"odinger
operator with the Dirichlet boundary condition. The crucial point is to
introduce some test function spaces on $\Omega$. The fundamental properties of
Besov spaces are also shown, such as embedding relations and duality, etc.
Furthermore, the isomorphism relations are established among the Besov spaces
in which regularity of functions is measured by the Dirichlet Laplacian and the
Schr\"odinger operators.
