Mathematical Principles of Theoretical Physics

3.6 Spectral Theory of Differential Operators

Since the orbital electrons are bound in the interior of the atom, the following condition holds
true:
φ= 0 for|x|>r 0 ,

wherer 0 is the radius of an atom. Thus, if ignoring the electromagnetic interactions be-
tween orbital electrons, then the bound energy of an electron is a negative eigenvalue of the
following elliptic boundary problem

(3.6.4)

−

h ̄^2

2 m 0

∇^2 φ−

Ze^2

r

φ=λ φ forx∈Br 0 ,

φ= 0 forx∈∂Br 0 ,

whereBr 0 is a ball with the atom radiusr 0.
According to the spectral theory for elliptic operators, the number of negative eigenvalues
of (3.6.4) is finite. Hence, it is natural that the energy levels in (3.6.1) and (3.6.2) are finite
and discrete.

3.6.2 Classical spectral theory

Consider the eigenvalue problem of linear elliptic operators as follows

(3.6.5)

−D^2 ψ+Aψ=λ ψ forx∈Ω,

ψ= 0 forx∈∂Ω,

whereΩ⊂Rnis a bounded domain,ψ= (ψ 1 ,···,ψm)T: Ω→Cmis a complex-valued
function withmcomponents,

(3.6.6) D=∇+i~B, ~B= (B 1 ,···,Bn),

andA,Bk( 1 ≤k≤n)arem-th order Hermitian matrices:

(3.6.7) A= (Aij(x)), Bk= (Bkij(x)).

Letλ 0 be an eigenvalue of (3.6.5). The corresponding eigenspace atλ 0 is

Eλ 0 ={ψ∈L^2 (Ω,Cm)|ψsatisfy( 3. 6. 5 )withλ=λ 0 }

is finite dimensional, and its dimension

N=dimEλ 0

is called the multiplicity ofλ 0. Physically,Nis also called the degeneracy providedN>1.
Usually, we count the multiplicityNofλ 0 asNeigenvalues, i.e., we denote

λ 1 =···=λN=λ 0.

Based on the physical background, we mainly concern the negative eigenvalues. How-
ever, for our purpose the following classical spectral theorem is very important.