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In this work, we present a complete spectral study of a family of non-normaloperators arising in Reggeon field theory. This family of operators is anoriginal example who permit us to discover the recent theory of physicalrequirement of space-time reflection symmetry (PT symmetry) without losing anyof the essential physical features of quantum mechanics [Bender]. Early studiesof Reggeon field theory, in the late 1970s led a number of investigators toobserve that model cubic quantum-mechanical Hamiltonians might have realeigenvalues [Bower et al]. The study of this family of operators, permit us todiscover some fine results of Spectral Theory and Functional Analysis inparticular the results connected with completeness of elementary solutions ofmathematical physics problems. We use knowledge basics of holomorphic functionsof one complex variable and the properties of Hilbert spaces. In this work theknowledge of spectral theory and the functional analysis of standard level isrequired (see [Kato], [Ghohberg2 et al] and [Markus]). It is also requested toknow the basic properties of semigroup theory see for example ([Pazy]).

Spectral analysis of some non-normal operators arising in Reggeon field theory