
416 Commutation Relations, Normal Ordering, and Stirling Numbers
U(t + s)=U(t)U (s) is very natural to assume as well as the property of being strongly
continuous (by a theorem of John von Neumann it is enough to assume the measurability of
the matrix elements), so that Stone’s theorem implies that U (t)=e
itH
for some self-adjoint
operator H. For all f ∈ D(H) one has with ψ(t)=U(t)f that
1
i
d
dt
ψ(t)=Hψ(t)whichis
an abstract version of Schr¨odinger’s equation.Thus,theHamiltonian operator H has to be
self-adjoint.
F.3 Basic Facts on Spectral Theory
Let us turn to some points concerning spectral theory. Recall that in C
n
a symmetric
linear map T can be diagonalized ...