Lasing Eigenvalue Problem for a Circular Quantum Wire Partially Covered with Graphene

We consider the Lasing Eigenvalue Problem (LEP) for a circular quantum wire, made of a gain material, with a partial cover, made of a graphene strip. This boundary value problem is reduced to a dual series equation for the Fourier series coefficients of the magnetic field function. Using the Riemann-Hilbert Problem solution, we transform this equation to a Fredholm second-kind set of linear algebraic equations. To find the LEP eigenvalues, we apply the method of the residual inverse iteration. Some preliminary results of the lasing mode frequencies, threshold gains and field patterns are presented.