Without this condition, the discontinuity would be, in general, a hyperfunction, which is inconvenient for some necessary limiting operations. Therefore, we assume that the discontinuity can be represented as a tempered distribution on $[-\infty ,0]$, namely, $D(-{\sigma }^2-i\epsilon )-D(-{\sigma }^2+i\epsilon )\stackrel{\epsilon \to +0}{\to }\mathrm{Disc}D(-{\sigma }^2)\in {𝒮}^{\prime }([0,\infty ])$, where ${𝒮}^{\prime }([0,\infty ])$ is the dual space of $𝒮([0,\infty ])≔\{f(\lambda )=g(-(1+\lambda {)}^{-1})$; $g$ is a $C^{\infty }$ function on $[-1,0]\}$. For details, see (Sec. A.3. in [40]). Note that a tempered distribution with positive support can be extended to a tempered distribution on $[0,\infty ]$.

The assumptions (1) boundedness of complex singularities in $|k^2|$ and (2) holomorphy of $D(k^2)$ in a neighborhood of the real axis except for the timelike singularities are required to justify this statement.

Even in a finite dimensional vector space, the completeness of eigenstates of a hermitian operator does not always hold. Instead, by the Jordan decomposition, “generalized eigenstates” defined by a sequence $\{|E^0⟩,|E^1⟩,\dots ,|E^{n-1}⟩\}$: $(H-E)|E^0⟩=E|E^1⟩$, $(H-E)|E^1⟩=E|E^2⟩,\dots$, $(H-E)|E^{n-1}⟩=0$ spans the vector space. Nevertheless, allowing generalized eigenstates does not change the conclusion.