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and then take its [[trace of an operator|trace]]. In a formal sense, ignoring all the difficult points of [[mathematical analysis]] involved, this will be Σ. Therefore the existence of such 'trace formulae' for ''T'' means that the explicit formulae essentially encode the nature of ''T'', from the point of view of [[spectral theory]], at least as far as its eigenvalues ([[spectrum of an operator|spectrum]]) is concerned.
For the case the ''Spectrum '' is just the one belonging to a Hamiltonian ''H'' , the semiclassical approach can give a definition of the sum by means of an integral of the form:
:<math> \Sigma= \int_{-\infty}^{\infty} \int_{-\infty}^{\infty} e^{-aH(x,p)}\,dx\,dp </math>
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