In the compact version, there is many application of the spectral decomposition of a bounded self-adjoint operator: Sturm-Liouville, spectre of the laplacian,...
But for the general version which insure that up to decompose the Hilbert space into a countable sum of subspace, your operator $T$ is equivalent to a multiplication on a certain $L^2(Sp(T),\mu)$ on each factor. In every books i know, they give as application the functional calculus for Borel functions. But does anyone know nice application to PDE?