Absence of embedded eigenvalues in semi-Dirac semi-metals

Abstract Semi-Dirac semi-metals are recently constructed nanostructures, which behave like both conventional zero-gap semiconductors and graphene-type materials in a sense. These nano-structures have been extensively studied in the physics literature, however, the rigorous mathematical investigation is missing. In the pioneering work [32], Krejčiřík and Antunes analyze the spectral stability properties under small perturbations, notably sufficient conditions of absence/existence of discrete eigenvalues. The main objective of this paper is to formulate sufficient conditions of absence of eigenvalues embedded in the essential spectrum. The first approach is in the spirit of the virial theorem, the other approach is the method of multipliers. The main idea of both methods is the choice of an anisotropic dilation operator corresponding to the anisotropy of semi-Dirac semi-metals.