Higher-Order Spectral Shift Functions and Associated Trace Formulas for One-Dimensional Schrödinger Operators

Abstract

Spectral shift functions (SSFs) provide a powerful framework for understanding how the spectrum of a self-adjoint operator changes under perturbation, and they play a central role in trace formulas that generalize the classical results of Krein and Koplienko. While higher-order SSFs have been extensively developed in abstract settings—particularly under Schatten class assumptions or within noncommutative frameworks with τ-compact resolvents—their explicit computation remains challenging, especially for differential operators arising in quantum mechanics. In this paper, we define and compute the SSF of order k for a one-dimensional Schrödinger operator perturbed by a constant potential, and rigorously verify the associated trace formula. Our approach, grounded in classical Hilbert–Schmidt theory and Fourier analysis, bypasses abstract machinery while capturing physically meaningful scenarios. These results also serve as a foundation for future work on more singular perturbations, such as delta and square-well potentials.