Speaker: Dr. Horst R. Beyer.
Date and Time
October 3 – December 8, 2017.
Sala de Usos Múltiples de la Facultad de
Ciencias en Física y Matemáticas (UNACH),
Ciudad Universitaria Carretera Emiliano Zapata Km. 8,
Real del Bosque (Terán).
Tuxtla Gutiérrez, Chiapas, México.
MCTP, CONACYT, UNACH y FCFM-UNACH.
Horst R Beyer.
Sendic Estrada Jiménez.
In addition to abundant experimental evidence, quantum theory has a
firm mathematical foundation, provided by the spectral theorems for
(densely-defined, linear and) self-adjoint operators in Hilbert Spaces, called “observables” in quantum theory. In particular, these theorems provide the
existence of spectral measures associated to self-adjoint operators and
elements of the Hilbert space, the latter called “states” in quantum theory.
Such spectral measures are the primary observables in quantum theory. One
well-known example in one space dimension is the absolute square of the
wave-function in quantum mechanics, which generates the spectral measure
corresponding to the position operator and that wave function.
Unfortunately, these mathematical facts are not widely known neither inside
the mathematics nor the physics community and deserve a lot more
distribution, since providing an additional firm foundation of quantum
theory, turning it into a “mathematical theory.”
This short course introduces into basic aspects of applications of
operator theory to quantum mechanics. It follows to a large extent
Martin Schechter's book “Operator Methods in Quantum Mechanics,"
Dover 2003. “This advanced undergraduate and graduate-level text
introduces the power of operator theory as a tool in the study of
quantum mechanics, assuming only a working knowledge of advanced
- A brief Sketch of the Mathematical Structure of Quantum Theory
- Introduction to Banach and Hilbert Spaces
- Representation of Formal Differential Operators
- Linear Operators in Banach Spaces
- Definition and Properties of the Fourier Transform
- Spectra of Operators
- The Essential Spectrum
- One-Dimensional Motion
- The Negative Eigenvalues
- Estimating the Spectrum
- Scattering Theory
- Long-Range Potentials
- Time-Independent Theory
- Strong Completeness
- Oscillating Potentials
- Eigenfunction Expansions
- Restricted Particles
- Hard-Core Potentials
- The Invariance Principle
- Trace Class Operators