rgoerke said:
I am led to believe (because it is in a paper I am reading) that
\frac{1}{H - z} \left|\phi\rangle = \frac{1}{E - z}\left|\phi\rangle
where H is the hamiltonian, \left|\phi\rangle is an energy eigenstate with energy E, and z is a complex variable.
In attempting to understand this expression, I have realized I do not know what is meant by
\frac{1}{A}\left|\phi\rangle
for some operator A.
Welcome to the wonderful world of the spectral theorem(s)! :-)
The basic idea is this: let A is a self-adjoint operator on a Hilbert space,
and suppose it has a (continuous) spectrum from 0 to infinity.
Then it is possible to find a basis in the Hilbert space where each basis
state corresponds to a particular value in the spectrum, and in fact
a complex-analytic function of A, written "f(A)", can be expressed as
<br />
f(A) ~\leftrightarrow~ \int_0^\infty\! dk\; f(k) \; |k\rangle\langle k|<br />
where each |k> is one of the eigenstates of A, with eigenvalue k.
(The set of all such k is called the "spectrum" of A.)
In your case, we're dealing with the Hamiltonian operator H, and your
particular complex-analytic function is f(H) := 1/(H-z), so we can
express it as
<br />
\frac{1}{H - z} ~\leftrightarrow~ \int_0^\infty\! dk\; \frac{ |k\rangle\langle k|}{k - z}<br />
where now each |k>is an eigenstate of H with eigenvalue k, and these states are
"normalized" according to \langle k|k'\rangle = \delta(k - k').
Now apply this to your |\phi\rangle, but first let's rewrite |\phi\rangle
as |E\rangle, since this is a more helpful notation. This is ok because
|\phi\rangle is (by definition) the eigenstate of H with eigenvalue E. We get:
<br />
\int_0^\infty\! dk\; \frac{ |k\rangle\langle k|}{k - z} ~ |E\rangle<br />
~=~ \int_0^\infty\! dk\; \frac{ |k\rangle}{k - z} ~ \delta(k - E)<br />
~=~ \frac{1}{E - z} \; |E\rangle<br />
To understand the spectral theorem(s) in more detail, it's probably best
to start with the finite-dimensional matrix versions in linear algebra
textbooks, and then progress to the functional-analytic versions,
then to the versions for rigged Hilbert space which is what I've been
using above. Or, for a more QM-flavored overview, try Ballentine ch1,
in particular sections 1.3 and 1.4.
HTH.