  In (J. Funct. Anal. 257, 1092-1132 (2009)), Dykema and Skripka showed the
existence of higher order spectral shift functions when the unperturbed
self-adjoint operator is bounded and the perturbations is Hilbert-Schmidt. In
this article, we give a different proof for the existence of spectral shift
function for the third order when the unperturbed operator is self-adjoint
(bounded or unbounded, but bounded below).
