Abstract
We introduce the quadratic analogue of the Bogolyubov endomorphisms of the canonical commutation relations (CCR) associated with the re-normalized square of white noise algebra (RSWN-algebra). We focus on the structure of a subclass of these endomorphisms: each of them is uniquely determined by a quadruple (C, S, X, alpha), where C, X are linear transformations from a test-function space T subset of L-2(R-d, lambda) into itself, while S is anti-linear on T and a is real. Precisely, we prove that C, S and X are uniquely determined by two arbitrary complex-valued Borel functions of modulus 1 and two maps of R-d, into itself. Under some additional analytic conditions on C, S and X, we discover that we have only two equivalent classes of Bogolyubov endomorphisms, one of them corresponds to the case alpha > 0 and the other corresponds to the case alpha < 0. Finally, we close the paper by building some examples in one and multi-dimensional cases.