Mathematical Principles of Theoretical Physics

170 CHAPTER 3. MATHEMATICAL FOUNDATIONS

Proof.Assertion (1) follows directly from the following classical formula for the first eigen-
valueλ 1 of (3.6.5):

λ 1 = min

ψ∈H 01 (Ω,Cm)

1

||ψ||L 2

∫

Ω

[

(Dψ)†(Dψ)+ψ†Aψ

]

dx.

We now prove Assertion (2) by contradiction. Assume that it is not true, thenK>N. By
Theorem3.34, theKfunctionsψjin (3.6.15) can be expended as

(3.6.16) ψj=

N

∑

i= 1

αjiei+

∞

∑

l= 1

βjlφl for 1≤j≤K,

whereei( 1 ≤i≤N)andφlare eigenfunctions corresponding to negative and nonnegative
eigenvalues. SinceK>N, there exists aK-th order matrixPsuch that

(3.6.17) Pα=

(

0 ··· 0

∗

)

,

where

α=







α 11 ··· α 1 N

..

.

..

.

αK 1 ··· αKN





 withαijas in(^3.^6.^16 ).

Thus, under the transformationP,

(3.6.18) ψ ̃=P

(

ψ

0

)

∈EK, ψ= (ψ 1 ,···,ψN)T,

whereEKis as in (3.6.15).
However, by (3.6.14) and (3.6.15), the first termψ ̃ 1 in (3.6.18) can be expressed in the
form

(3.6.19) ψ ̃ 1 =

∞

∑

l= 1

θlφl∈EK.

Note thatφlare the eigenfunctions corresponding to the nonnegative eigenvalues of (3.6.5).
Hence we have

∫

Ω

[(Dψ ̃ 1 )†(Dψ ̃ 1 )+ψ ̃† 1 Aψ ̃ 1 ]dx=

∫

Ω

ψ ̃† 1 (−D^2 ψ ̃ 1 +Aψ ̃ 1 )dx=

∞

∑

l= 1

(3.6.20) |θl|^2 λl> 0.

Hereλl≥0 are the nonnegative eigenvalues of (3.6.5). Hence we derive, from (3.6.19) and
(3.6.20), a contradiction with the assumption in Assertion (2). Theproof of the theorem is
complete.