  In this paper we study Spectral Decomposition Theorem [1] and translate it to
quantum language by means of the Wigner transform. We obtain a quantum version
of Spectral Decomposition Theorem (QSDT) which enables us to achieve three
distinct goals: First, to rank Quantum Ergodic Hierarchy levels [2,3]. Second,
to analyze the classical limit in quantum ergodic systems and quantum mixing
systems. And third, and maybe most important feature, to find a relevant and
simple connection between the first three levels of quantum ergodic hierarchy
(ergodic, exact and mixing) and quantum spectrum. Finally, we illustrate the
physical relevance of QSDT applying it to two examples: Microwave billiards
[4,5] and a phenomenological Gamow model type [6,7].
