  The Ba\~{n}ados-Teitelboim-Zanelli (BTZ) black hole model corresponds to a
solution of (2+1)-dimensional Einstein gravity with negative cosmological
constant, and by a conformal rescaling its metric can be mapped onto the
hyperbolic pseudosphere surface (Beltrami trumpet) with negative curvature.
Beltrami trumpet shaped graphene sheets have been predicted to emit Hawking
radiation that is experimentally detectable by a scanning tunnelling
microscope. Here, for the first time we present an analytical algorithm that
allows variational solutions to the Dirac Hamiltonian of graphene
pseudoparticles in BTZ black hole gravitational field by using an approach
based on the formalism of pseudo-Hermitian Hamiltonians within a
discrete-basis-set method. We show that our model not only reproduces the exact
results for the real part of quasinormal mode frequencies of (2+1)-dimensional
spinless BTZ black hole, but also provides analytical results for the real part
of quasinormal modes of spinning BTZ black hole, and also offers some
predictions for the observable effects with a view to gravity-like phenomena in
a curved graphene sheet.
