$$\mathscr {D}{-}$$ D - Deformed and SUSY-Deformed Graphene: First Results

We discuss some mathematical aspects of two particular deformed versions of the Dirac Hamiltonian for graphene close to the Dirac points, one involving \(\mathscr {D}\)-pseudo bosons and the other supersymmetric quantum mechanics. In particular, in connection with \(\mathscr {D}\)-pseudo bosons, we show how biorthogonal sets arise, and we discuss when these sets are bases for the Hilbert space where the model is defined, and when they are not. For the SUSY extension of the model we show how this can be achieved and which results can be obtained.