Existence and exponential decay of bound state solutions for the Brown-Ravenhall operator with a critical potential for non-confining systems in genuinely two-dimensional spaces

We study the Brown-Ravenhall operator (the suitably projected Dirac operator) in dimension 2 using the Foldy-Wouthuysen unitary transformation. This allows us to write the operator in diagonalized form, so that the kinetic energy is equal to $〈\psi ,{(-\Delta +m^2)}^{1/2}\psi 〉$. This suggests that we use some interesting results found in recent literature for equations driven by the fractional operator ${(-\Delta +m^2)}^s$ with $s\in (0,1)$ and $m>0$. Here, we are interested in the Brown-Ravenhall operator perturbed by a short-range attractive potential given by a Bessel-Macdonald function (also known as $K_0$-potential) to model relativistic effects in graphene. The existence of bound states for the Brown-Ravenhall operator with the $K_0$-potential is proven using a variant of the Caffarelli-Silvestre extension method, which permits to characterize ${(-\Delta +m^2)}^{1/2}$ as an operator that maps a Dirichlet boundary condition to a Neumann-type condition via an extension problem in the upper-half space $R_{+}^3$. In this process, the minimization occurs for an auxiliary energy functional associated with the weak solutions of the Neumann problem, defined on $H^1(R_{+}^3;C)$. In addition, the lower bound for the smallest eigenvalue is established via Herbst operator. Exponential decay, in an $L_2$ sense, of the bound states of the Brown-Ravenhall operator with the $K_0$-potential is also investigated.