Mathematical Principles of Theoretical Physics

176 CHAPTER 3. MATHEMATICAL FOUNDATIONS

1) The eigenvalues of (3.6.40) are real and discrete, with finite multiplicities, and satisfy

−∞<λ 1 ≤ ··· ≤λk≤ ···, λk→∞ask→∞.

2) The eigenfunctions are orthogonal in the sense that

(3.6.44)

∫

Ω

ψk†

[

i(~σ·~D)ψj

]

dx= 0 ∀k 6 =j.

3) The number of negative eigenvalues is finite

−∞<λ 1 ≤ ··· ≤λN< 0.

4) Equations (3.6.40) have negative eigenvalues if and only if there exists a functionφ∈

H 01 (Ω,C^2 )satisfying (3.6.43) such that

∫

Ω

[

̄hc|(~σ·~D)φ|^2 +

ig

2

φ†{(~σ·~D),A 0 }φ

]

< 0.

Proof.It is easy to see that the operator

L=i(~σ·~D):H 01 (Ω,C^2 )→L^2 (Ω,C^2 )

is a Hermitian operator. Consider a functionalF:H^1 (Ω,C)→R^1 :

F(ψ) =

∫

Ω

[

̄hc|Lψ|^2 +

g

2

ψ†{L,A 0 }ψ

]

dx.

By (3.6.42), the operatorL^2 =−(~σ·~D)^2 is elliptic. HenceFhas the following lower bound
onS:
S={ψ∈H 01 (Ω,C^2 )|

∫

Ω

ψ†Lψdx= 1 },

namely
min
ψ∈S

F(ψ)>−∞.

Based on the Lagrange multiplier theorem of constraint minimization, the first eigenvalueλ 1
and the first eigenfunctionφ 1 ∈Ssatisfy

(3.6.45) λ 1 =F(ψ 1 ) =min
ψ∈S
F(ψ).

In addition, if
λ 1 ≤ ··· ≤λm

are the firstmeigenvalues with eigenfunctionsψk, 1 ≤k≤m, then the(m+ 1 )-th eigenvalue
λm+ 1 and eigenfunctionψm+ 1 satisfy

(3.6.46) λm+ 1 =F(ψm+ 1 ) = min
ψ∈S,ψ∈H⊥m

F(ψ),

whereHm=span{ψ 1 ,···,ψm}, andHm⊥is the orthogonal complement ofHmin the sense of
(3.6.44).
It is clear that Assertions (1)-(4) of the theorem follow from (3.4.45) and (3.4.26). The
proof is complete.