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

In this work, an n-dimensional pseudo-differential operator involving the n-dimensional linear canonical transform associated with the symbol sigma(x1, ... , xn; y1, . . . , yn) is an element of C infinity(Rn x 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 pseudo-differential operators is an n-dimensional pseudo-differential operator. Further, we have investigated formal adjoint operators with a symbol sigma is an element of 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 Schro center dot dinger-type-I/Schro center dot dinger-type-II/Schro center dot dinger equations in quantum mechanics for one particle with a constant potential.