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

Abstract

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

16 pages. Slight improvements were made in the file. This is a follow up to the submission arXiv:1712.00928