The spectral shift function (SSF) is one of scattering theory's fundamental objects: it counts, in a regularized sense, how many eigenvalues shift past a given energy when a potential is turned on. For self-adjoint perturbations, the Lifshits–Krein trace formula connects the SSF to the trace of the difference of resolvents. The SSF is real-valued because the eigenvalues are real.
The paper on the SSF for non-self-adjoint perturbations (arXiv: 2603.21773) extends this to complex-valued potentials, where eigenvalues leave the real line and become complex.
The extension requires care: the trace formula involves regularization via functional calculus, and spectral singularities — points where the continuous spectrum behaves anomalously — create additional difficulties. For Schrödinger operators with complex short-range potentials in three dimensions, the SSF becomes complex-valued and reveals information about the complex eigenvalues: their location and accumulation pattern.
The toy models are suggestive: the imaginary part of the SSF detects the presence of complex eigenvalues (resonances), while the real part tracks the shift of the continuous spectrum. The SSF unifies two kinds of spectral information — bound states (discrete) and scattering (continuous) — into a single function.
The through-claim: the spectral shift function complexifies when the perturbation does, and the imaginary part is the new information. Self-adjoint perturbations have real SSF because eigenvalues stay real. Non-self-adjoint perturbations push eigenvalues off the real axis, and the imaginary part of the SSF detects this displacement. The complex extension is natural: the SSF tracks eigenvalues wherever they go.
2603.21773. Spectral theory / scattering theory / non-self-adjoint operators / spectral shift function / Lifshits–Krein formula.