Unbounded Self-Adjoint Operator

Definition

An unbounded self-adjoint operator is a linear operator defined on a Hilbert space that is not bounded, meaning it does not have a finite operator norm, but is still equal to its adjoint. These operators are crucial in quantum mechanics and functional analysis, where they often represent physical observables. Understanding these operators is key to applying the spectral theorem and developing a functional calculus for them.

5 Must Know Facts For Your Next Test

1. Unbounded self-adjoint operators can have unbounded spectra, which means their eigenvalues may extend infinitely.
2. The domain of an unbounded self-adjoint operator is dense in the Hilbert space, ensuring that these operators can approximate any vector in the space.
3. The existence of a unique continuous functional calculus is established for unbounded self-adjoint operators, allowing for the evaluation of functions on these operators.
4. Self-adjoint operators possess real spectra, which implies that all eigenvalues are real numbers, a property essential in physical applications.
5. The resolvent set of an unbounded self-adjoint operator contains values for which the operator can be inverted, leading to important implications in spectral theory.

Review Questions

• How does the definition of an unbounded self-adjoint operator differ from that of a bounded operator, and what implications does this have for its spectral properties?
  • An unbounded self-adjoint operator differs from a bounded operator mainly in that it does not have a finite operator norm. This lack of boundedness means that its spectrum can be unbounded as well. In contrast, bounded operators have a compact spectrum. The implications of this difference are significant; for instance, unbounded self-adjoint operators may exhibit more complex spectral properties and require careful consideration when applying the spectral theorem.
• Discuss how the functional calculus can be applied to unbounded self-adjoint operators and why this is significant in mathematical analysis.
  • The functional calculus for unbounded self-adjoint operators allows us to apply continuous functions to these operators, extending our understanding beyond mere multiplication. This is significant because it facilitates the analysis of physical systems described by such operators, especially in quantum mechanics where observables are represented as unbounded self-adjoint operators. It enables us to define new operators derived from existing ones and to study their properties systematically.
• Evaluate the role of the spectral theorem for unbounded self-adjoint operators in understanding their functional calculus and implications for quantum mechanics.
  • The spectral theorem for unbounded self-adjoint operators provides a foundation for understanding how these operators can be diagonalized in a sense, allowing us to relate them to their eigenvalues and eigenvectors. This connection is vital in quantum mechanics as it links observable quantities to measurable outcomes. By establishing the structure of these operators through the spectral theorem, we gain insights into how quantum systems behave and can predict phenomena based on their spectral properties.

Related terms

Hilbert Space: A complete inner product space that provides the framework for various areas of mathematics and physics, where self-adjoint operators are typically defined.

Adjoint Operator: An operator that represents the 'transpose' of a linear transformation in a certain sense, allowing for the definition of self-adjointness.

Spectral Theorem: A fundamental result that characterizes self-adjoint operators in terms of their eigenvalues and eigenvectors, which can be extended to unbounded operators.