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Quantization in quantum mechanics deals with the problem of correct defining various classical structures, for example, quantum-mechanical observables such as Hamiltonian, momentum, self-adjoint operators in some Hilbert space and so on. Though there exists a naive treatment, based on experience in finite-dimensional algebra or even infinite-dimensional algebra with bounded operators, it results in paradoxes and inaccuracies. This exposition is devoted to a consistent treatment of such problems, based on appealing to some nontrivial items of functional analysis concerning the theory of linear operators in Hilbert spaces.It begins by considering quantization problems in general, emphasizing the non-triviality of consistent operator construction by presenting paradoxes to the naive treatment. It then builds the necessary mathematical background following it by the theory of self-adjoint extensions. By considering several problems such as the one-dimensional Calogero problem, the Aharonov-Bohm problem, the problem of delta-like potentials and relativistic Coulomb problem it then shows how quantization problems associated with correct definition of observables can be treated consistently for comparatively simple quantum-mechanical systems. In the end, related problems in quantum field theory are briefly introduced.This well organized text is most suitable for students and post graduates interested in deepening their understanding of mathematical problems in quantum mechanics. However, scientists in mathematical and theoretical physics and mathematicians will also find it useful.
