Explicit characterization of periodic potentials yielding spectral Schrödinger (Hill) operators

Determine an explicit characterization of complex-valued, 1-periodic potentials q integrable on [0,1] for which the one-dimensional Schrödinger (Hill) operator L(q) acting in L2(ℝ) and generated by the differential expression l(y) = −y'' + q(x) y is a spectral operator of scalar type in the sense of Dunford.

Background

The paper studies non-self-adjoint periodic differential operators, focusing on the Hill/Schrödinger operator L(q) with complex-valued periodic potentials. The authors emphasize that spectrality is rare for non-self-adjoint cases; in several key examples (Mathieu–Schrödinger and PT-symmetric optical potentials), L(q) is a spectral operator only when it is self-adjoint.

Although analytic and geometric criteria for spectrality (in the sense of Dunford) have been established by Gesztesy and Tkachenko, these criteria do not provide an explicit description in terms of the potential q itself. The author’s own asymptotic results yield conditions for asymptotic spectrality but do not resolve spectrality at small spectral singularities. Consequently, a long-standing challenge remains: to explicitly characterize the class of periodic potentials that render L(q) a spectral operator.