Spectral structure and wavefunction behavior of the degenerate tensor-based periodization operator

Characterize the spectral structure and corresponding wavefunction behavior of the extended Schrödinger operator for the tensor-based periodization model of a bilayer incommensurate system, which is degenerate elliptic and whose properties remain largely unknown.

Background

The tensor-based periodization model embeds the original Schrödinger operator for a bilayer incommensurate system into a higher-dimensional space in order to restore periodicity. Its extended Schrödinger operator is degenerate elliptic, which prevents the direct application of the standard analytic and numerical theory available for uniformly elliptic periodic operators.

The paper introduces a regularization term that makes the extended operator uniformly elliptic and thereby enables characterization of observables such as the density of states and electron density. However, the underlying spectral structure and wavefunction behavior of the unregularized extended operator are identified as unresolved, motivating further theoretical analysis beyond the regularized framework.

References

However, the associated extended Schrödinger operator is not well behaved: it is degenerate elliptic, and both its spectral structure and the behavior of corresponding wavefunctions are highly intricate and remain largely unknown.

— A Regularization Based Computational Method for Quantum Incommensurate Problems  (2609.29357 - Ding et al., 24 Sep 2026) in Section 2.3, “Regularized model”