Densely defined

Definition

A densely defined operator is one whose domain is dense in the Hilbert space it acts upon, meaning that the closure of its domain equals the entire space. This property is crucial when dealing with unbounded operators, as it often leads to the existence of adjoint operators. Understanding whether an operator is densely defined helps in determining its properties, such as whether it can be extended and how it interacts with other operators.

5 Must Know Facts For Your Next Test

1. For an operator to be densely defined, its domain must intersect with every non-empty open set of the Hilbert space.
2. Densely defined operators can often be extended to larger domains, potentially allowing them to become closed operators.
3. The adjoint of a densely defined operator exists if the operator is densely defined and has a well-defined action on its domain.
4. Many important physical and mathematical operators, like the position and momentum operators in quantum mechanics, are densely defined.
5. If an operator is not densely defined, it may fail to have certain desirable properties, such as a well-defined adjoint.

Review Questions

• How does the concept of being densely defined relate to the existence of adjoint operators?
  • The concept of being densely defined is integral to the existence of adjoint operators because for an adjoint to exist, the original operator must have a domain that is dense in the Hilbert space. This density ensures that any element of the Hilbert space can be approximated by elements from the domain of the operator, allowing for a well-defined action when taking the adjoint. If an operator lacks this property, it may not be possible to define its adjoint properly.
• Discuss how the properties of densely defined operators influence their use in quantum mechanics.
  • In quantum mechanics, densely defined operators are crucial because they often correspond to observable physical quantities. For instance, position and momentum operators are densely defined in the Hilbert space of square-integrable functions. Their dense definition allows for meaningful physical interpretations and calculations involving wave functions. Furthermore, having a dense domain allows for extensions and helps establish connections between different states in quantum systems.
• Evaluate how understanding densely defined operators can impact your approach to solving functional analysis problems involving unbounded operators.
  • Understanding densely defined operators significantly enhances your approach to solving problems involving unbounded operators because it clarifies their limitations and potential. Recognizing that a densely defined operator can be extended or possess a well-defined adjoint informs how you manipulate these operators within proofs and applications. This knowledge also helps you identify when an operator's behavior can lead to issues like non-closedness or failure to apply certain results from bounded operator theory, ultimately guiding your problem-solving strategies.

Related terms

Hilbert Space: A complete inner product space that allows for the generalization of Euclidean spaces, providing a framework for quantum mechanics and functional analysis.

Unbounded Operator: An operator that is not necessarily bounded on its entire domain, which means it does not have a finite upper limit on the magnitude of its output values.

Adjoint Operator: An operator associated with a given operator that reflects the inner product structure, critical for analyzing properties like self-adjointness and spectra.