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In this paper we exploit the technique used in \cite{A}-\cite{5b} to dealwith delta interactions in a rigorous way in a curved spacetime represented bya cosmic string along the $z$ axis. This mathematical machinery is applied inorder to study the discrete spectrum of a point-mass particle confined in aninfinitely long cylinder with a conical defect on the $z$ axis and perturbed bytwo identical attractive delta interactions symmetrically situated around theorigin. We derive a suitable approximate formula for the total energy. As aconsequence, we found the existence of a mixing of states with positive or zeroenergy with the ones with negative energy (bound states). This mixture dependson the radius $R$ of the trapping cylinder. The number of quantum bound statesis an increasing function of the radius $R$. It is also interesting to note thepresence of states with zero total energy (quasi free states). Apart from thegravitational background, the model presented in this paper is of interest inthe context of nanophysics and graphene modeling. In particular, the graphenewith double layer in this framework, with the double layer given by theaforementioned delta interactions and the string on the $z-$axis modelingtopological defects connecting the two layers. As a consequence of thesesetups, we obtain the usual mixture of positive and negative bound statespresent in the graphene literature.

international journal of geometric methods in modern physics

The spectrum of the Schr\"{o}dinger Hamiltonian for trapped particles in  a cylinder with a topological defect perturbed by two attractive delta  interactions