  After recalling a fundamental identity relating traces and modified Fredholm
determinants, we apply it to a class of half-line Schr\"odinger operators $(-
d^2/dx^2) + q$ on $(0,\infty)$ with purely discrete spectra. Roughly speaking,
the class considered is generated by potentials $q$ that, for some fixed $C_0 >
0$, $\varepsilon > 0$, $x_0 \in (0, \infty)$, diverge at infinity in the manner
that $q(x) \geq C_0 x^{(2/3) + \varepsilon_0}$ for all $x \geq x_0$. We treat
all self-adjoint boundary conditions at the left endpoint $0$.
