Construction of Eigenfunctions of the Tricomi–Neumann Problem for Equations of Mixed Type with Characteristic Degeneration and Their Application

For the equation of mixed type
$$L_\alpha u\equiv u_{xx}+yu_{yy}+\alpha u_y+\lambda u=0,$$
where $0<\alpha<1$ and $\lambda$ is a complex parameter, we obtain eigenvalues in a special domain by the method of separation of variables and construct the system of corresponding eigenfunctions of the Tricomi–Neumann spectral problem. We construct the solution of the Tricomi–Neumann problem as a sum of biorthogonal series.