The role of quantum mechanics in signal processing - Signal Processing in MRS with Biomedical Applications - page 51

For a given physical state|Φ n , which could be an eigenvector of a part of

H, or an orbital from a basis set, two state vectors|Ψ n

and |ψ (s) can be

n

introduced as

= G(z)|Φ n = Q n ( H)|Φ s |ψ (s)

n

|Ψ n . (2.99)

Given a fixed s, the vector|ψ (s)

can be obtained by a repeated application
of (2.98). This procedure yields the Lanczos algorithm for physical states

n

β n+1 |ψ (s) = ( H−α n 1)|ψ (s) −β n |ψ (s) . (2.100)

n+1 n n−1

Both diagonal and offdiagonal matrix elements of the Green operator can be
computed in this manner

∞

Φ r |G(z)|Φ s = I r,n,s =Φ r |ψ (s)

γ n (z)I r,n,s . (2.101)

n

n=0

We could also calculate the matrix elements of any number of Green operators
that might be multiplied by some other operators. This gives the expression

∞

= G(z)|Φ s γ n (z)|ψ (s)

|Ψ s = . (2.102)

n

n=0

In particular, it can be deduced from here that

|G(z) V G(z)|Φ s |V|Ψ s Φ r =Ψ r . (2.103)

By reliance upon the quantummechanical trace, Tr, of the Green function,
the density of states (DOS), as denoted by ρ(E), can be introduced via
ρ(E) =− 1

π Im(Tr){G(z + )}.

(2.104)

On the other hand, by definition, the trace represents the sum of the diagonal
elements of δ(E1−H)

L

Φ s |δ(E1−H)|Φ s . ρ(E) = (2.105)

s=0

Substituting (2.37) into (2.104) gives the following expression for ρ(E)

∞ L

ρ(E) = W n (E)Q n (E)I n I n = I s,n,s (2.106)

n=0 s=0

where L is the number of the retained vectors from the set{|Φ s }. Another

important quantity is the integrated density of state (IDOS), as denoted by
N(E), which can be obtained through integration of ρ(x)

∞

E E

N(E) = dxρ(x) = I n W n (x)Q n (x)dσ(x). (2.107)

−∞ −∞

n=0