Mathematical Principles of Theoretical Physics

3.6. SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 173

It is known thatφnis weakly differentiable, andφn∈H 01 (Ω). These functionsφn( 1 ≤n≤N)
are what we need. Let

φ=

N

∑

n= 1

αnφn, ||φ||L 2 = 1.

By Assertion (2) in Theorem3.36,φn( 1 ≤n≤N)are orthonormal:
∫

B 1

φiφjdx=

∫

Ω

eiejdx=δij.

Therefore we have

(3.6.33) ||φ||L 2 =

N

∑

n= 1

αn^2 = 1.

Thus the integral in (3.6.31) is

∫

B 1

[|∇φ|^2 +r^2 +αV 0 (x)φ^2 ]dx=

∫

Ω

−

(

N

∑

n= 1

αnen

)(

N

∑

n= 1

αn∆en

)

dx+r^2 +α

∫

Ω

V 0 (x)φ^2 dx

=

N

∑

n= 1

αn^2 λn+r^2 +α

∫

Ω

V 0 (x)φ^2 dx

≤

N

∑

n= 1

αn^2 λn−θr^2 +α by (3.6.24)

< 0 by (3.6.32) and (3.6.33).

It follows from Theorem3.36that there are at leastNnegative eigenvalues for (3.6.30). When
θr^2 +α≫1 is sufficiently large, the relation (3.6.32) implies that

(3.6.34) λN≃θr^2 +α.

On the other hand, by (3.6.27) in Lemma3.37,

(3.6.35) λN∼λ 1 N^2 /n (m= 1 ).

Hence the relation (3.6.28) follows from (3.6.34) and (3.6.35). The proof is complete.

Remark 3.39.In Section6.4.6, we shall see that for particles with massm, the parameters
in (3.6.28) are
α= 0 , r= 1 , n= 3 , θ= 4 mρ 12 Ag^2 /h ̄^2 ρ,

whereg=gworgsis the weak or strong interaction charge,ρis the particle radius,ρ 1 is the
weak or strong attracting radius, andAis the weak or strong interaction constant. Hence the
number of energy levels of massive particles is given by

N=

[

4

λ 1

ρ 12 A

ρ

mc

h ̄

g^2

̄hc

] 3 / 2

.

whereλ 1 is the first eigenvalue of−∆in the unit ballB 1.