Interpolation Hilbert spaces between Sobolev spaces

We explicitly describe all Hilbert function spaces that are interpolation spaces with respect to a given couple of Sobolev inner product spaces considered over ${\mathbb{R}}^n$ or a half-space in ${\mathbb{R}}^n$ or a bounded Euclidean domain with Lipschitz boundary. We prove that these interpolation spaces form a subclass of isotropic H"ormander spaces. They are parametrized with a radial function parameter which is OR-varying at $+\mathrm{\infty }$ and satisfies some additional conditions. We give explicit examples of intermediate but not interpolation spaces.