Mathematical Principles of Theoretical Physics

3.6. SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 169

and the constraint functionalGis given by

(3.6.11) G(ψ) =

∫

Ω

|ψ|^2 dx.

It is easy to see that the equation of (3.6.5) is of the form:

δF(ψ) =λ δG(ψ),

andF,Gare as in (3.6.10) and (3.6.11), which is as the variational equation (3.6.9) with the
constraint onΓ.
Hence, the eigenvaluesλk(k= 1 , 2 ,···)of (3.6.5) can be expressed in the following forms

(3.6.12)

λ 1 =min

ψ∈Γ

F(ψ),

λk= min

ψ∈Γ,ψ∈Hk⊥− 1

F(ψ),

whereFis as in (3.6.10),ΓandHk⊥− 1 are the sets:

(3.6.13)

Γ={ψ∈H 01 (Ω,Cm)| ||ψ||L 2 = 1 },

Hk⊥− 1 =

{

ψ∈H 01 (Ω,Cm)

∣

∣

∣

∫

Ω

ψ†ψjdx= 0 , 1 ≤j≤k− 1

}

,

andψj( 1 ≤j≤k− 1 )are the eigenfunctions corresponding to the first(k− 1 )eigenvalues
λ 1 ,···,λk− 1. NamelyHk⊥− 1 is the orthogonal complement ofHk− 1 =span{ψ 1 ,···,ψk− 1 }in
H 01 (Ω,Cm).
Based on (3.6.12)-(3.6.13), we readily deduce the spectral theorem, Theorem3.34, for
the elliptic eigenvalue problem (3.6.5).

3.6.3 Negative eigenvalues of elliptic operators

The following theorem provides a necessary and sufficient condition for the existence of
negative eigenvalues of (3.6.5), and a criterion to estimate the number of negative eigenvalues.

Theorem 3.36.For the eigenvalue problem (3.6.5), the following assertions hold true:

1) Equations (3.6.5) have negative eigenvalues if and only if there is a functionψ∈

H 01 (Ω,Cm), such that

(3.6.14)

∫

Ω

[(Dψ)†(Dψ)+ψ†Aψ]dx< 0 ,

where D is as in (3.6.6).

2) If there are K linear independent functionsψ 1 ,···,ψK∈H 01 (Ω,Cm),such that

(3.6.15) ψsatisfies (3.6.14) for anyψ∈EK=span{ψ 1 ,···,ψK},

then the number N of negative eigenvalues is larger than K, i.e., N≥K.