 Compact massive-Laplacian resolvent

The solution operator $(-\Delta +m^2{)}^{-1}:L^2(U)\to L^2(U)$ is compact on every bounded open set because it maps boundedly into $H_0^1(U)$ and the Rellich-Kondrashov compactness theorem for H01 embeds that space compactly into $L^2(U)$.