Solutions of simple systems of the deformed quantum mechanics based on a new extended uncertainty principle

Integral transforms serve as potent mathematical tools for converting functions between different domains, offering a versatile way to analyze systems from varied perspectives due to their linearity and invertibility. Here, we introduce a novel kernel for an integral transform, which plays a pivotal role in defining an extended uncertainty principle. Our proposed integral transform kernel significantly expands the applicability of generalized exponential functions within non-additive statistical mechanics, enabling these functions to encompass negative or complex values. Furthermore, it demonstrates analyticity by being mappable to the standard Fourier transform within the complex plane. The functions derived from this novel kernel, termed "deformed functions," exhibit properties reminiscent of hyperbolic and trigonometric functions. These deformed functions stand as solutions to straightforward systems within the framework of deformed quantum algebra, aligning with the extended uncertainty principle we introduced. Our work focuses on extending the formalism of quantum mechanics through an innovative uncertainty principle, leveraging integral transforms and specialized kernels to generate deformed functions that adhere to this extended principle and display characteristics akin to well-known mathematical functions.