Let
X
t
X_t
be the relativistic
α
\alpha
-stable process in
R
d
\mathbf {R}^d
,
α
∈
(
0
,
2
)
\alpha \in (0,2)
,
d
>
α
d > \alpha
, with infinitesimal generator
H
0
(
α
)
=
−
(
(
−
Δ
+
m
2
/
α
)
α
/
2
−
m
)
H_0^{(\alpha )}= - ((-\Delta +m^{2/\alpha })^{\alpha /2}-m)
. We study intrinsic ultracontractivity (IU) for the Feynman-Kac semigroup
T
t
T_t
for this process with generator
H
0
(
α
)
−
V
H_0^{(\alpha )} - V
,
V
≥
0
V \ge 0
,
V
V
locally bounded. We prove that if
lim
|
x
|
→
∞
V
(
x
)
=
∞
\lim _{|x| \to \infty } V(x) = \infty
, then for every
t
>
0
t >0
the operator
T
t
T_t
is compact. We consider the class
V
\mathcal {V}
of potentials
V
V
such that
V
≥
0
V \ge 0
,
lim
|
x
|
→
∞
V
(
x
)
=
∞
\lim _{|x| \to \infty } V(x) = \infty
and
V
V
is comparable to the function which is radial, radially nondecreasing and comparable on unit balls. For
V
V
in the class
V
\mathcal {V}
we show that the semigroup
T
t
T_t
is IU if and only if
lim
|
x
|
→
∞
V
(
x
)
/
|
x
|
=
∞
\lim _{|x| \to \infty } V(x)/|x| = \infty
. If this condition is satisfied we also obtain sharp estimates of the first eigenfunction
ϕ
1
\phi _1
for
T
t
T_t
. In particular, when
V
(
x
)
=
|
x
|
β
V(x) = |x|^{\beta }
,
β
>
0
\beta > 0
, then the semigroup
T
t
T_t
is IU if and only if
β
>
1
\beta >1
. For
β
>
1
\beta >1
the first eigenfunction
ϕ
1
(
x
)
\phi _1(x)
is comparable to
\[
exp
(
−
m
1
/
α
|
x
|
)
(
|
x
|
+
1
)
(
−
d
−
α
−
2
β
−
1
)
/
2
.
\exp (-m^{1/{\alpha }}|x|) \, (|x| + 1)^{(-d - \alpha - 2 \beta -1 )/2}.
\]