An n-dimensional pseudo-differential operator involving linear canonical transform and some applications in quantum mechanics

Abstract

In this work, an n-dimensional pseudo-differential operator involving the n-dimensional linear canonical transform associated with the symbol ?(x1,..., xn; y1,..., yn) ? C?(Rn ? Rn) is defined. We have introduced various properties of the n-dimensional pseudo-differential operator on the Schwartz space using linear canonical transform. It has been shown that the product of two n-dimensional pseudodifferential operators is an n-dimensional pseudo-differential operator. Further, we have investigated formal adjoint operators with a symbol ? ? Sm using the n-dimensional linear canonical transform, and the Lp(Rn) boundedness property of the n-dimensional pseudo-differential operator is provided. Furthermore, some applications of the n-dimensional linear canonical transform are given to solve generalized partial differential equations and their particular cases that reduce to well-known n-dimensional time-dependent Schr?dinger-type-I/Schr?dinger-type-II/Schr?dinger equations in quantum mechanics for one particle with a constant potential.