New Estimates of the Potential Schrödinger Equation

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The Schrödinger equation, a fundamental construct in quantum mechanics, plays a pivotal role in understanding the properties and behaviors of quantum systems across various scientific fields, including but not limited to physics, economics, and geophysics. This research paper delves into the exploration of novel invariants associated with the Schrödinger equation, uncovering previously unrecognized symmetries and properties that open new pathways for theoretical and applied scientific exploration. The investigation reveals that the energy of bound states remains invariant under transformations of the coordinate system, highlighting a universal symmetry embedded within the equation. This discovery, coupled with an analysis of the variation in scattering amplitudes’ phase with different coordinate systems, not only enriches the theoretical framework of quantum mechanics but also has significant practical implications across various domains such as fluid dynamics, material science, and medical imaging. Furthermore, by applying the principles of the Poincaré-Riemann-Hilbert boundary-value problem, the study offers a novel approach to estimate potentials within the Schrödinger equation, advancing the three-dimensional inverse problem of quantum scattering theory. This methodological innovation allows for a more profound understanding of the unitary scattering operator, facilitating enhanced modeling of complex systems and phenomena. The implications of these findings extend beyond theoretical physics, offering transformative insights and applications in areas ranging from fluid dynamics, where it aids in refining models for the Navier-Stokes equations, to seismic exploration, tomography, and ultrasound imaging, where it enhances phase selection and interpretation of measurement results. In essence, this research not only contributes to a deeper understanding of the Schrödinger equation’s fundamental properties but also paves the way for interdisciplinary advancements, demonstrating the profound impact of theoretical discoveries on practical applications in diverse scientific and technological domains.