Is this a general identity for the Resolvent? [solved: integral representation of the resolvent] And if so, what is it called?
$$i(H-\lambda - i\epsilon)^{-1}\phi = \int_0^\infty e^{-\epsilon t} e^{i\lambda t}e^{-iHt}\phi\,\text dt$$
as in Reed-Simon XIII.7 example 1.
It is stated there for $H=-i\frac{\text d}{\text dx}$ on $L^2(\mathbb R)$, but later (example 2) it seems to be used for any $H$ self-adjoint.
Looks like some complex-analysis-used-on-operators argument, but I don't recall. I would like the name of this identity, but I have difficulty coming up with the correct terms to search for and it is not referenced in the book (although I would assume it is in there somewhere).
Thanks
 A: Definition [$C_0$ Semigroup]: Let $X$ be a Banach space, and let $T : [0,\infty)\rightarrow\mathcal{B}(X)$ be a function into the bounded linear operators on $X$. Then $T$ is a semigroup if $T(0)=I$ and $T(t)T(t')=T(t+t')$ for all $t,t' \ge 0$. $T$ is a $C_0$ semigroup if $\lim_{t\downarrow 0}T(t)x=x$ for all $x\in X$.
Suppose $X$ is a Banach space, and suppose that $T(t)$ is a $C_0$ semigroup of bounded linear operators on $[0,\infty)$, then you can show that $e^{-\omega t}T(t)$ is a uniformly bounded $C_0$ semigroup on $[0,\infty)$ for some $\omega > 0$. That is, $\|e^{-\omega t}T(t)\| \le M$ for some constant $M$. This follows from the exponential property of $T$, from the $C_0$ property,
and from the Uniform Boundedness Principle.
Any $C_0$ semigroup $T$ has the property that
$$
            Ax=\lim_{h\downarrow 0} \frac{1}{h}\{T(h)x-x\}
$$
exists for all $x$ in a dense subspace $\mathcal{D}(A)$ of $X$. The resulting operator $A$ is the so-called generator of the $C_0$ semigroup. The generator is a closed densely-defined operator. You can show that $\{ \lambda : \Re\lambda > \omega \}$ is always contained in the resolvent set $\rho(A)$ of the operator $A$, and the resolvent is given by
$$
             (A-\lambda I)^{-1}x = \int_{0}^{\infty}e^{-\lambda t}T(t)xdt,
   \;\;\; x\in\mathcal{D}(A),\;\;\Re\lambda > \omega.
$$
These facts holds very generally for a $C_0$ semigroup. And, you can characterize the properties of $A$ that guarantee $A$ is the generator of a $C^0$ semigroup. All of this is classical, and well worked out.
Unitary groups $U(t)$ are examples where the generator $A$ is selfadjoint. Then $U(t)$ and $U(-t)$ are $C_0$ semigroups of contractions. So the resolvent formula for unitary semigroups is a special case of the more general $C_0$ semigroup.
History: The earliest notions of semigroups of operators as a function of $t \ge 0$ grew out of the work of Oliver Heaviside, the Father of modern Electrical Engineering. He noticed that a solution operator $T(t)$ that would take a state of a fixed electrical system from time $0$ to time $t$ would necessarily have the exponential property, which is amazing in itself. He reasoned as follows: If you evolve the state at time $0$ through $t$ seconds, and then you start with that as the initial state and evolve through $t'$ seconds, the end result must be the same as if you took the initial state at time $0$ and evolved through $t+t'$ seconds; thus, $T(t')T(t)x=T(t'+t)x$ would have to hold. All of Heaviside's early methods were operator methods (this was in the latter part of the 19th century.) He assumed a circuit working a linear regime that was not changing in time.
Through this reasoning Heaviside derived a functional Calculus that is now referred as the Laplace transform (people named it after Laplace because Heaviside was a jerk, and nobody wanted to give him credit for anything.) Later, people were able to use Heaviside's solution method to derive the Bromwich integral for the inverse Laplace transform. Eventually this morphed into what you see today. It took a couple of decades for Heaviside's work to lead people to what we now call the Laplace transform. Then, once everything was nicely understood, semigroup theory evolved, and it was known that the Laplace transform of the time evolution semigroup would be the resolvent. And that was applied to abstract operators. This brought it back full circle; Heaviside started with operators and derived the Functional Calculus, transform methods, and solution methods, before the Laplace transform. These methods were then applied to operators.
Reference: A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations.
