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After recalling a fundamental identity relating traces and modified Fredholmdeterminants, 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 mannerthat $q(x) \geq C_0 x^{(2/3) + \varepsilon_0}$ for all $x \geq x_0$. We treatall self-adjoint boundary conditions at the left endpoint $0$.

On traces and modified Fredholm determinants for half-line Schr\"odinger operators with purely discrete spectra