Miscellaneous results on the theory of evolution operators and generalizedtransforms

We prove that most of existing transforms can be framed within the context of the theory of evolution operators. We show that Gauss, Fresnel, Mellin, Hankel, fractional Fourier and other kinds of transforms cap be expressed b y suitably manipulating identities involving exponential operators. Among t he other things, we exploit Laplace transform method to define fractional o rder differential operators, particularly useful in the treatment of Schrod inger-type relativistic equations. The developed methods are used to treat various problems of physical interest, as, e.g., the solution of a Schrodin ger equation with potentials linearly dependent on space coordinates as in the case of a charged particle moving in a time-dependent homogeneous elect ric field.