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 /dx 2 ) + q on (0, ∞) with purely discrete spectra. Roughly speaking, the class considered is generated by potentials q that, for some fixed C 0 > 0, ε > 0, x 0 ∈ (0, ∞), diverge at infinity in the manner that q(x) ≥ C 0 x (2/3)+ε 0 for all x ≥ x 0 . We treat all self-adjoint boundary conditions at the left endpoint 0.

On a Nonlocal Integral Operator Commuting with the Laplacian and the Sturm-Liouville Problem I: Low Rank Perturbations of the Operator

“…The proposed approach exploits in a fundamental way the Taylor and/or Laurent expansion of the resolvent function (or Neumann series) corresponding to linear operators. This approach aligns with the recent work of Gesztesy and Kirsten [23,24,25]. Our procedure explicitly calculates expressions of these operators for all GSARC problems.…”

“…The dimension of the range corresponding to P is the algebraic multiplicity, m(T ; 0) of z = 0 [23,24,25]:…”

“…We also investigate a couple of non-GSARC problems that appear in the literature, namely the periodic and anti-periodic cases treated in [22]. Our results in this article focus on perturbation questions not pursued in [22,23,24,25].…”

The two lowest eigenvalues of the harmonic oscillator in the presence of a Gaussian perturbation

“…Motivated to a certain extent by [11], it might be worth pointing out that the method can be extended to the 2D/3D analogues of our model, even though, given that the 2D/3D counterparts of our Birman-Schwinger operator are no longer nuclear but only Hilbert-Schmidt integral operators (see [3,19,24,25,26,27]), the Fredholm determinant will have to be replaced by its modified version used for such operators (see [29]). Work in this direction is in progress.…”

On Hermite Functions, Integral Kernels, and Quantum Wires

“…where for any Hilbert-Schmidt operator A, we define the modified Fredholm determinant as det 2 [I + A] := det[I + A] e −Tr A , where det[I + A] is the ordinary Fredholm determinant [5,19,20] and I the identity operator. Note that A either could be or could not be trace class.…”