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The objective of this monograph is to present the most advanced topics in analysis of mathematical structure in Schrödinger equations. Quantum science becomes important in recent years because we need industrial and electronic applications of microscopic phenomena in quantum computation, quantum communication, and quantum cryptography etc. It is also expected in future that computer scientists apply the result in fundamental theory to the development of quantum devices like quantum computers. The exciting results will be of interest to researchers, graduate and undergraduate students in mathematics, physics and computer science, who are interested in linear Schrödinger type equations and understand the basic methods for analysis of partial differential equations. The author hopes to develop the further research in the relevant fields of mathematical analysis on the Cauchy problems for Schrödinger equations.