Mathematical Principles of Theoretical Physics

3.6. SPECTRAL THEORY OF DIFFERENTIAL OPERATORS 177

In the following, we consider the estimates of the number of negative eigenvalues for the
Weyl operators. If the interaction potentialAμtakes approximatively the following

Aμ= (K, 0 , 0 , 0 ) withK>0 being a constant.

Then (3.6.40) becomes

(3.6.47)

−∆φ=i(λ+K)(~σ·~∂)φ forx∈Ω⊂Rn,

φ= 0 forx∈∂Ω.

Obviously, the numberNof negative eigenvalues of (3.6.47) satisfies

(3.6.48) βN<K<βN+ 1 ,

whereβkis thek-th eigenvalue of the equation

(3.6.49)

−∆φk=iβk(~σ·~∂)φk,

φk|∂Ω= 0.

For (3.6.49) we have the Weyl asymptotical relation

(3.6.50) βN∼β 1 N^1 /n.

Here the exponent is 1/n, since (3.6.47) is an 2m-th order elliptic equation withm= 1 /2.
Hence we deduce, from (3.6.48) and (3.6.50), the estimates of the numberNof negative
eigenvalues of (3.6.47) as

(3.6.51) N≃

(

K

β 1

)n

,

whereβ 1 is the first eigenvalue of (3.6.49).

Remark 3.44.For the mediators such as the photon and gluons, the number ofenergy levels
is given byNin (3.6.51), which can be estimated as

N=

(

A

β 1

ρ 1

ρ

g^2 w

hc ̄

) 3

,

whereρ 1 ,ρ,A,gware as in Remark3.39.