Mathematical Principles of Theoretical Physics

172 CHAPTER 3. MATHEMATICAL FOUNDATIONS

Theorem 3.38.Under the assumptions of (3.6.22) and (3.6.23), the number N of the negative
eigenvalues of (3.6.21) satisfies the following approximative relation

(3.6.28) N≃

(

θr^2 +α

λ 1

)n/ 2

,

provided thatθr^2 +α/λ 1 ≫ 1 is sufficiently large, where r andθ are as in (3.6.21) and
(3.6.24), andλ 1 is the first eigenvalue of (3.6.25).

Proof.The ballBrcan be written as

Br={y=rx|x∈B 1 }.

Note that∂/∂y=r−^1 ∂/∂x, (3.6.21) can be equivalently expressed as

(3.6.29)

−∆φ+r^2 V(rx)φ=β φ forx∈B 1 ,

φ= 0 forx∈∂B 1 ,

and the eigenvalueλof (3.6.21) is

λ=

1

r^2

β, whereβis the eigenvalue of (3.6.29).

Hence the number of negative eigenvalues of (3.6.21) is the same as that of (3.6.29), and we
only need to prove (3.6.28) for (3.6.29).
By (3.6.22), the equation (3.6.29) is approximatively in the form

(3.6.30)

−∆φ+r^2 +αV 0 (x)φ=β φ forx∈B 1 ,

φ= 0 forx∈∂B 1.

Based on Assertion (2) in Theorem3.36, we need to findNlinear independent functions
φn∈H 01 (B 1 ) ( 1 ≤n≤N)satisfying

(3.6.31)

∫

B 1

[

|∇φ|^2 +r^2 +αV 0 (x)φ^2

]

dx< 0 ,

for anyφ∈span{φ 1 ,···,φN}with||φ||L 2 =1.
To this end, we take the eigenvalues{λn}and eigenfunctions{en}of (3.6.25) such that

0 <λ 1 ≤ ··· ≤λN<λN+ 1 ,

and

(3.6.32) λN<θr^2 +α≤λN+ 1.

For the eigenfunctionsen, we make the extension

φn=

{

en forx∈Ω,

0 forx∈B 1 /Ω.