I know that since differential operators are not bounded then generaly we tend to use their inverses (integral) to study their characteristics, does we do the same for the case of essential Spectrum?
Do you mean by the definition "the essential Spectrum is the spectrum minus the point Spectrum" ?
I assume that the operator is self-adjoint? You can then use what is called Weyl's criterion. This can be used to prove that essential spectrum of the Laplace operator on L^2(R) with domain H^2 is [0,\infty) (See Hislop & Sigal, Introduction to Spectral Theory page 73). The essential spectrum of A and B are the same if A-B is A-compact. This can be used if you study a perturbation of the Laplace operator.
If your operator is non-selfadjoint, then there are several definitions of the essential spectrum. For one of them the Weyl criterion mentioned by Christian holds. Also for all these definitions one has a stability of essential spectrum under relatively compact perturbations. These questions are discussed in several books, I would recommend the survey "Non-self-adjoint operators and their essential spectra" by W. D. Evans, R. T. Lewis and A. Zettl - http://link.springer.com/chapter/10.1007/BFb0076796 .
Thank you for your valuable answers I'll read the references you provied
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