### 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 $\sigma(x_{1},\dots,x_{n};y_{1},\dots,y_{n})\in{C^{\infty}(\mathbb{R}^{n}\times\mathbb{R}^{n})}$ 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 to be an n-dimensional pseudo-differential operator. Further, we have investigated formal adjoint operators with a symbol $\sigma\in{\mathcal{S}^{m}}$ using the n-dimensional linear canonical transform, and the $L^{p}(\mathbb{R}^{n})$ boundedness property of 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\"{o}dinger-type-I/Schr\"{o}dinger-type-II/Schr\"{o}dinger equations in quantum mechanics for one particle with a constant potential.

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