The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 42

G(z) is the Green operator

where

−1

G(z) =

z1−H . (2.40)

H, it follows
G ∗ (z) =

For a Hermitean Hamiltonian

G(z ∗ ).

(2.41)

The existence of the inverse operator G(z) is guaranteed if z is not an eigen
energy E k from the spectrum of H.

This will be the case for a Hermitean

Hamiltonian H provided that

z + = E + i0 + z = E + iǫ (2.42)

where ǫ is an infinitesimally small positive number. On the other hand, the
resolvent G + (z) can be viewed as the standard Fourier integral of the time
evolution operator

∞

G(E) =−i

dtθ(t) U(t)e iEt

(2.43)

−∞

U(t) = e −i Ht

(2.44)

where θ(t) is the usual Heaviside step function and t is the real time variable.
As it stands, G(z) is the operator Pade approximant, i.e., the OPA, because
this resolvent is defined by the product of the unity operator and (z1−H) −1 ,
and formally, G(z) = 1(z1−H) −1 = (z1−H) −1 1. Therefore, diagonal or off
diagonal matrix elements of the Green function G are the customary scalar
Pade approximant. This follows from the corresponding Schrodinger eigen
value problem

H|Υ k = E k |Υ k

(2.45)

where|Υ k is the eigenstate corresponding to the eigenenergy E k . Employ

ing completeness of the set{|Υ k

}, in the form of the spectral decomposition

K
k=1

of the unity operator, i.e., 1 = |Υ k Υ k |, we have from (2.40)

K

|Υ k Υ k |
z−E k .

G(z) =

(2.46)

k=1

Nonnegative integer K is finite or infinite, depending on the number of eigen
states E k in the spectrum of H. The associated diagonal (r = s) and off
diagonal (r = s) matrix elements of G(z) taken over the states|Φ s andΦ r |

can be extracted from (2.46) as

K

d (r,s)
k
z−E k

≡ A K−1 (z)
B K (z)

G r,s (z)≡Φ r |G|Φ s = (2.47)

k=1