Topology in Molecular Biology

198 D.V. Millionschikov

define a new smoothq-formΨ ̃tonMnsuch thatΨ ̃tcoincides withΨtin some
W ̃⊂WandΨ ̃t≡0 outside ofW.Theq-formΨ ̃tis called a quasi-mode:

HtΨ ̃t=t

(

∑

i

(1 + 2Ni+λili)+

B

t

+

C

t^2

+···

)

Ψ ̃t,t→+∞. (11.16)

The numberst

∑

i

(1 + 2Ni+λili) are calledasymptotic eigenvaluesand their

minimal valueEas 0 approximates the minimal eigenvalue ofHtast→+∞.
In order to findEas 0 , we must setNi= 0 for alli. The sum

∑q

i=1

(1−li)+

∑n

i=q+1

(1 +li).

is non-negative and it is equal to zero if and only if

l 1 =···=lq=1,lq+1=···=ln=− 1.

This means thatHthas precisely one zero asymptotic eigenvalue for each
critical point of indexq. Hence we have preciselymq(f) asymptotic zero eigen-
values (forq-forms). Vanishing of the first term of the asymptotical expansion
(11.16) for a minimal eigenvalue ofHtis only a necessary condition to have
zero energy level; hence the numberbq(Mn) of zero eigenvalues does not exceed
the number of zero asymptotic eigenvalues. In other words we have established
the Morse inequalities
mq(f)≥bq(Mn).
It was Pajitnov who remarked that it is possible to apply Witten’s ap-
proach to the Morse–Novikov theory [4]. Letωbe a closed 1-form onMn
andta real parameter. As in the construction earlier one can define a new
deformed differential dtωinΛ∗(M)

dtω=d+tω∧, dtω(ξ)=da+tω∧ξ.

If the 1-formωis not exact, the cohomologyHtω∗(M,R) of the de Rham com-
plex with the deformed differential dtωgenerally speaking is not isomorphic
to the standard oneH∗(M,R). ButHtω∗(M,R) depends only on the cohomol-
ogy class ofω: for any pairω, ω′of 1-forms such thatω−ω′=dφ, where
φis a smooth function onMn; the cohomologyHtω∗(M,R)andH∗tω′(M,R)
is isomorphic to each other. This isomorphism can be given by the gauge
transformation

ξ→etφξ;d→etφde−tφ=d+tdφ∧.

It is convenient also to consider a complex parameterλinstead oft.Itwas
remarked in [3, 4] that the cohomologyHλω∗(M,C)ofΛ∗(M) with respect
to the deformed differentialdλωcoincides with the cohomologyHρ∗λω(M,C)