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 (- 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 .