Mathematical Principles of Theoretical Physics

168 CHAPTER 3. MATHEMATICAL FOUNDATIONS

Theorem 3.34(Spectral Theorem of Elliptic Operators).Let the matrices in (3.6.7) are Her-
mitian, and the functions Aij,Bkij∈L∞(Ω). The the following assertions hold true:

1) All eigenvalues of (3.6.5) are real with finite multiplicities, and form an infinite conse-

quence as follows:

−∞<λ 1 ≤λ 2 ≤ ··· ≤λk≤ ···, λk→∞ask→∞.

whereλkis counting the multiplicity.

2) The eigenfunctionsψkcorresponding toλkare orthogonal to each other, i.e.

∫

Ω

ψk†ψjdx= 0 ∀k 6 =j.

In particular,{ψk}is an orthogonal basis of L^2 (Ω,Cm).

3) There are only finite number of negative eigenvalues in{λk},

(3.6.8) −∞<λ 1 ≤ ··· ≤λN< 0 ,

and the number N of negative eigenvalues depends on the matrices A,Bjin (3.6.7) and

the domainΩ.

Remark 3.35.For the energy levels of subatomic particles introduced in Chapter 5, we are
mainly interested in the negative eigenvalues of (3.6.5) and in the estimates of the numberN
in (3.6.8).

Theorem3.34is a corollary of the classical Lagrange multiplier theorem. We recall the
variational principle with constraint. LetHbe a linear normed space, andFandGare two
functionals onH:
F,G:H→R^1.

LetΓ⊂Hbe the set
Γ={u∈H|G(u) = 1 }.

Ifu 0 ∈Γis a minimum point ofFwith constraint onΓ:

F(u 0 ) =min

u∈Γ

F(u),

thenu 0 satisfies the equation

(3.6.9) δF(u 0 ) =λ δG(u 0 ),

whereλis a real number.
For the eigenvalue equation of (3.6.5), the corresponding functional is

(3.6.10) F(ψ) =

∫

Ω

[

|Dψ|^2 +ψ†Aψ

]

dx,