The spectral theory of linear operators plays a key role in the mathematical formulation of quantum theory. Furthermore, such a rigorous mathematical foundation leads to a more profound insight into the nature of quantum mechanics. This textbook provides a concise and comprehensible introduction to the spectral theory of (unbounded) self-adjoint operators and its application in quantum dynamics.
The book places emphasis on the symbiotic relationship of these two domains by (1) presenting the basic mathematics of nonrelativistic quantum mechanics of one particle, i.e., developing the spectral theory of self-adjoint operators in infinite-dimensional Hilbert spaces from the beginning, and (2) giving an overview of many of the basic functional aspects of quantum theory, from its physical principles to the mathematical models.
The book is intended for graduate (or advanced undergraduate) students and researchers interested in mathematical physics. It starts with linear operator theory, spectral questions and self-adjointness, and ends with the effect of spectral type on the large time behaviour of quantum systems. Many examples and exercises are included that focus on quantum mechanics. TOC:1 Linear Operators and Spectrum.- 2 Adjoint Operator.- 3 Fourier Transform and Free Hamiltonian.- 4 Operators via Sesquilinear Forms.- 5 Unitary Evolution Groups.- 6 Kato-Rellich Theorem.- 7 Boundary Triples and Self-Adjointness.- 8 Spectral Theorem.- 9 Applications of the Spectral Theorem.- 10 Convergence of Self-Adjoint Operators.- 11 Spectral Decomposition I.- 12 Spectral Decomposition II.- 13 Spectrum and Quantum Dynamics.- 14 Some Quantum Relations.
