 Mixed-boundary drift-diffusion eigenmodes on a rectangle

For $u_t=u_{xx}+u_{yy}+\beta u_x$ on $(0,L_x)\times (0,L_y)$, impose zero Dirichlet boundary conditions on the two vertical sides and the top, and a zero Neumann boundary condition at the bottom. The Dirichlet gauge transform for constant drift $u=e^{-\beta x/2}v$ gives $v_t=\Delta v-{\beta }^{2}v/4$. Separation of variables produces the displayed modes, with $m,n\ge 1$ and ${\lambda }_{mn}=(m\pi /L_x{)}^2+((n-1/2)\pi /L_y{)}^2+{\beta }^2/4$. Their spatial parts are orthogonal with weight $e^{\beta x}$; equivalently, expand $e^{\beta x/2}{u}_0$ in the ordinary sine-cosine basis.