Topology in Molecular Biology

11 One-Forms and Deformed de Rham Complex 197

It can be calculated that

Ht=dtd∗t+d∗tdt=dd∗+d∗d+t^2 (df)^2 +t

∑

i,j

∇^2 (i,j)(f)[ ̃ai, ̃aj∗], (11.13)

where (df)^2 =(df,df)x=gij∂x∂fi∂x∂fjand

̃ai(ξ)=dxi∧ξ, ∇^2 (i,j)=∇i∇j−Γijk∇k.

As the “potential energy”t^2 (df)^2 of the HamiltonianHtbecomes very
large fort→+∞the eigenfunctions ofHtare concentrated near the critical
points df= 0 and the low-lying eigenvalues ofHtcan be calculated by ex-
panding about the critical points. Taking the Morse coordinatesxiin some
neighbourhoodWof a critical pointP

f(x)=

1

2

∑

λi(xi)^2 ,λ 1 =···=λq=− 1 ,λq+1=···=λn=1,

whereqis the index of the critical pointP, and introducing a Riemmanian
metricgijonMnsuch thatxiare Euclidean coordinates forgijinWone can
locally evaluate the HamiltonianHt:

Ht=

∑

i

(

−

∂^2

∂xi^2

+t^2 xi

2

+tλi[ ̃ai, ̃ai∗]

)

. (11.14)

The operator

Hi=−

∂^2

∂xi^2

+t^2 xi

2

is the Hamiltonian of the simple harmonic oscillator and it has the following
set of eigenvalues

t(1 + 2Ni),Ni=0, 1 , 2 ,...

with simple multiplicities. The operatorHicommutes with [ ̃ai,a ̃i∗] and the
eigenvalues of the last operator are equal to±1:

[ ̃ai, ̃ai∗](ψ(x)dxi^1 ∧...∧dxip)=

{

ψ(x)dxi^1 ∧···∧dxip,i∈(i 1 ,...,ip),

−ψ(x)dxi^1 ∧···∧dxip,i/∈(i 1 ,...,ip).

Hence the eigenvalues of the restrictionHt|Ware equal to

t

∑

i

(1 + 2Ni+λili),Ni=0, 1 , 2 ,..., li=± 1. (11.15)

The corresponding eigenfunctionsΨt=ψ(x, t)dxi^1 ∧···∧dxipare defined
inWand not on the whole manifoldMn. Using the partition of unit one can