In this paper, we propose a novel \( n -\) Dimensional Pseudo-Differential Operator ( \( n -\) DPDO) constructed using the robust framework of the \( n -\) Dimensional Quadratic Phase Fourier Transform ( \(n-\) DQPFT). This operator is characterized by a smooth symbol \( \delta (\nu _1, \dots , \nu _n; \omega _1, \dots , \omega _n) \in {\mathcal {C}}^\infty ({\mathbb {R}}^n \times {\mathbb {R}}^n) \) , paving the way for new directions in mathematical analysis. We provide a rigorous study of its fundamental properties within the context of Schwartz space, proving that the composition of two \( n -\) DPDOs yields another \( n -\) DPDO, thereby establishing its algebraic consistency. Furthermore, we analyze formal adjoint operators with symbols in \( {\mathcal {S}}^r \) , demonstrating their essential boundedness in \( L^p({\mathbb {R}}^n) \) spaces under the \( n -\) DQPFT framework. Additionally, the \( n -\) DQPFT is applied to solve generalized partial differential equations. Specific cases of these equations include well-known n-dimensional time-dependent generalized Schrödinger-type equations (Types I, II, and III) and the general Schrödinger equation in quantum mechanics for a single particle with a constant potential.
