  We consider an arbitrary selfadjoint operator on a separable Hilbert space.
To this operator we construct an expansion in generalized eigenfunctions in
which the original Hilbert space is decomposed as a direct integral of Hilbert
spaces consisting of general eigenfunctions. This automatically gives a
Plancherel type formula. For suitable operators on metric measure spaces we
discuss some growth restrictions on the generalized eigenfunctions. For
Laplacians on locally finite graphs the generalized eigenfunctions are exactly
the solutions of the corresponding difference equation.
