Consistent Parametrization of Multiband Hamiltonians: Mathematical Foundations and Data-Driven Applications in Nanoscience

Bandstructure methods occupy a central place in the physics of nanostructures, and the multiband k·p theory of Luttinger, Kohn, and Kane has served as one of the most widely used computational frameworks for modelling electronic states and energies in low-dimensional semiconductor systems for several decades. Despite its broad success, the theory harbours a fundamental mathematical difficulty that has been largely overlooked: the multiband Luttinger–Kohn Hamiltonians are non-elliptic partial differential operators for the overwhelming majority of common III–V and III-nitride crystalline materials, a fact that violates the axiomatic requirements of quantum mechanics and is the root cause of the long-standing problem of spurious solutions. In this paper, we derive ellipticity conditions rigorously for the 6×6, 8×8, and 14×14 zinc-blende Hamiltonians, demonstrating that non-ellipticity affects a substantially larger class of materials than previously reported. We develop and justify a systematic parameter rescaling procedure for the 8×8 Kane Hamiltonian and obtain admissible parameter sets for GaAs, AlAs, InAs, GaP, AlP, InP, GaSb, AlSb, InSb, GaN, AlN, and InN. The inversion-asymmetry parameter B is shown to play an essential and previously unrecognized role in maintaining ellipticity, and it is used to optimize the bandstructure fit of the rescaled parameter sets. Analysis of several known 14×14 models reveals structural sources of non-ellipticity, pointing to the need for a revision of perturbative assumptions regarding out-of-basis band contributions. The consistent parametrization framework developed here provides the rigorous mathematical foundation required by inverse design methodologies, AI-enhanced electronic structure calculations, and data-driven multifidelity approaches in nanoscience and nanotechnology.