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Mathematical Methods in Quantum Mechanics
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In this paper, a self-contained introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators, is presented, which is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books.Abstract:
Quantum mechanics and the theory of operators on Hilbert space have been deeply linked since their beginnings in the early twentieth century States of a quantum system correspond to certain elements of the configuration space and observables correspond to certain operators on the space This book is a brief, but self-contained, introduction to the mathematical methods of quantum mechanics, with a view towards applications to Schrodinger operators Part 1 of the book is a concise introduction to the spectral theory of unbounded operators Only those topics that will be needed for later applications are covered The spectral theorem is a central topic in this approach and is introduced at an early stage Part 2 starts with the free Schrodinger equation and computes the free resolvent and time evolution Position, momentum, and angular momentum are discussed via algebraic methods Various mathematical methods are developed, which are then used to compute the spectrum of the hydrogen atom Further topics include the nondegeneracy of the ground state, spectra of atoms, and scattering theory This book serves as a self-contained introduction to spectral theory of unbounded operators in Hilbert space with full proofs and minimal prerequisites: Only a solid knowledge of advanced calculus and a one-semester introduction to complex analysis are required In particular, no functional analysis and no Lebesgue integration theory are assumed It develops the mathematical tools necessary to prove some key results in nonrelativistic quantum mechanics Mathematical Methods in Quantum Mechanics is intended for beginning graduate students in both mathematics and physics and provides a solid foundation for reading more advanced books and current research literature This new edition has additions and improvements throughout the book to make the presentation more student friendlyread more
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Real and Complex Analysis. By W. Rudin. Pp. 412. 84s. 1966. (McGraw-Hill, New York.)
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
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Light–matter interaction in the long-wavelength limit: no ground-state without dipole self-energy
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Introduction to Model Spaces and their Operators
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References
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Journal ArticleDOI
Handbook of Mathematical Functions
Book
Perturbation theory for linear operators
TL;DR: The monograph by T Kato as discussed by the authors is an excellent reference work in the theory of linear operators in Banach and Hilbert spaces and is a thoroughly worthwhile reference work both for graduate students in functional analysis as well as for researchers in perturbation, spectral, and scattering theory.
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Real and complex analysis
TL;DR: In this paper, the Riesz representation theorem is used to describe the regularity properties of Borel measures and their relation to the Radon-Nikodym theorem of continuous functions.
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A treatise on the theory of Bessel functions
TL;DR: The tabulation of Bessel functions can be found in this paper, where the authors present a comprehensive survey of the Bessel coefficients before and after 1826, as well as their extensions.