After recalling a fundamental identity relating traces and modified Fredholm determinants, we apply it to a class of half-line Schrödinger operators
(
−
d
2
/
d
x
2
)
+
q
(- d^2/dx^2) + q
on
(
0
,
∞
)
(0,\infty )
with purely discrete spectra. Roughly speaking, the class considered is generated by potentials
q
q
that, for some fixed
C
0
>
0
C_0 > 0
,
ε
>
0
\varepsilon > 0
,
x
0
∈
(
0
,
∞
)
x_0 \in (0, \infty )
, diverge at infinity in the manner that
q
(
x
)
≥
C
0
x
(
2
/
3
)
+
ε
0
q(x) \geq C_0 x^{(2/3) + \varepsilon _0}
for all
x
≥
x
0
x \geq x_0
. We treat all self-adjoint boundary conditions at the left endpoint
0
0
.