Miscellaneous results on the theory of evolution operators and generalized transforms

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 by suitably manipulating identities involving exponential operators. Among the other things, we exploit Laplace transform method to define fractional order differential operators, particularly useful in the treatment of Schrodinger-type relativistic equations. The developed methods are used to treat various problems of physical interest, as, e.g., the solution of a Schrodinger equation with potentials linearly dependent on space coordinates as in the case of a charged particle moving in a time-dependent homogeneous electric field.