The purpose of this paper is to investigate the spectra of the Dirac operator
H
=
H
0
+
V
=
−
i
c
α
⋅
∇
+
β
m
c
2
+
V
H=H_0+V=-ic\alpha \cdot \nabla +\beta mc^2+V
. The local compactness of
H
H
is shown under some assumption on
V
V
. This method enables us to prove that if
|
V
(
x
)
−
a
β
|
→
0
|V(x)-a\beta |\to 0
as
|
x
|
→
∞
|x|\to \infty
, then
σ
ess
(
H
)
=
(
−
∞
,
−
m
c
2
−
a
]
∪
[
m
c
2
+
a
,
∞
)
\sigma _{\operatorname {ess}}(H)=(-\infty ,-mc^2-a]\cup [mc^2+a,\infty )
and to give a significant sufficient condition that
H
+
H^{+}
or
H
−
H^{-}
has a purely discrete spectrum.