Gelfand Triplets, Ladder Operators and Coherent States

Inspired by a similar construction on Hermite functions, we construct two series of Gelfand triplets, each one spanned by Laguerre–Gauss functions with a fixed positive value of one parameter, considered as the fundamental one. We prove the continuity of different types of ladder operators on these triplets. Laguerre–Gauss functions with negative values of the fundamental parameter are proven to be continuous functionals on one of these triplets. Different sorts of coherent states are considered and proven to be in some spaces of test functions corresponding to Gelfand triplets.