Book contents
- Frontmatter
- Contents
- 1 Introduction
- 2 Unbounded operators, adjoints, and self-adjoint operators
- 3 Representation theorems
- 4 Semibounded operators and the Friedrichs extension
- 5 Compact operators: general properties and examples
- 6 Spectral theory for bounded operators
- 7 Applications to statistical mechanics and partial differential equations
- 8 Self-adjoint unbounded operators and spectral theory
- 9 Essentially self-adjoint operators
- 10 The discrete spectrum and essential spectrum
- 11 The max–min principle
- 12 Spectral questions about the Rayleigh equation
- 13 Non-self-adjoint operators and pseudospectra
- 14 Applications to non-self-adjoint one-dimensional models
- 15 Applications in kinetic theory: the Fokker–Planck operator
- 16 Problems
- Bibliography
- Index
2 - Unbounded operators, adjoints, and self-adjoint operators
Published online by Cambridge University Press: 05 March 2013
- Frontmatter
- Contents
- 1 Introduction
- 2 Unbounded operators, adjoints, and self-adjoint operators
- 3 Representation theorems
- 4 Semibounded operators and the Friedrichs extension
- 5 Compact operators: general properties and examples
- 6 Spectral theory for bounded operators
- 7 Applications to statistical mechanics and partial differential equations
- 8 Self-adjoint unbounded operators and spectral theory
- 9 Essentially self-adjoint operators
- 10 The discrete spectrum and essential spectrum
- 11 The max–min principle
- 12 Spectral questions about the Rayleigh equation
- 13 Non-self-adjoint operators and pseudospectra
- 14 Applications to non-self-adjoint one-dimensional models
- 15 Applications in kinetic theory: the Fokker–Planck operator
- 16 Problems
- Bibliography
- Index
Summary
The development of spectral theory is strongly related to quantum mechanics, and the main operators that immediately appear in the theory are the operators of multiplication by x (in, say, L2(ℝ)), the operator of differentiation d/dx, and the harmonic oscillator −d2/dx2 + x2. These operators are unbounded and, in fact, not defined for any element of L2(ℝ). Of course, we can start by simply restricting the operator by introducing a smaller domain in the Hilbert space L2(ℝ) of definition, but what is the right notion of continuity for the operator? How do we choose the “maximal” domain of definition? This is what we shall start to explain in this chapter.
Unbounded operators
We consider a Hilbert space ℋ. We assume that the reader has some basic knowledge of Hilbertian theory. The scalar product will be denoted by 〈u, v〉ℋ or, more simply, by 〈u, v〉 when no confusion is possible. We adopt the convention that the scalar product is antilinear with respect to the second argument.
- Type
- Chapter
- Information
- Spectral Theory and its Applications , pp. 11 - 21Publisher: Cambridge University PressPrint publication year: 2013