Topology in Molecular Biology

12 The Spectral Geometry of Riemann Surfaces 213

Both worries will be taken care of by showing that “not much is happening
far out in the cusps,” by comparing what happens there to what happens closer
in. What we will show is:

Theorem 2.LetCyjbe the part of the cusp wherey>yj.
Letfbe anL^2 function onCwith eigenvalueλ=1/ 4 −s^2 ,andy 1 >y 0.
Then
∫

Cy 1

f^2 ≤

(

y 0

y 1

) 2 s∫

Cy 0

f^2.

The idea of the proof is the following: we may fix coordinates (x, y)inC,
where 0≤x≤1andysufficiently large. The Laplacian offis then given by

∆(f)=−y^2

(

∂^2

∂x^2

f+

∂^2

∂y^2

f

)

.

Sincef is periodic inx, we may write out its Fourier series inxas a
function ofy:

f(x, y)=

∑

n

an(y) cos(2πnx)+bn(y)sin(nx).

The equation

∆(f)=λf

then translates to the differential equations

a

′′

n(y)=

(

4 π^2 n^2 −

λ

y^2

)

an,

where differentiation is understood with respect toy, similarly forbn.
Putting aside the nuisance that these functions are of length 1/2 and not
1, we have that

∫

Cyi

f^2 =

∫∞

y=yi

∑

na

2

n(y)+b

2

n(y)

y^2

dy.

Let us first examine the casen= 0. We are then looking at the equation

a

′′

0 =

−λ

y^2

a 0.

This has solutions

a 0 =c 1 y(1/2)−s+c 2 y(1/2)+s.

In order for this term to beL^2 ,wemusthavec 2 = 0. Thus, we must have

a 0 =c 1 y(1/2)−s,