On traces and modified Fredholm determinants for half-line Schrödinger operators with purely discrete spectra

After recalling a fundamental identity relating traces and modified Fredholm determinants, we apply it to a class of half-line Schrödinger 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 $0$.