Friedrichs extension

Definition

Friedrichs extension is a method used to obtain a self-adjoint extension of a symmetric operator defined on a dense domain in a Hilbert space. It plays a crucial role in the spectral theory of operators by ensuring that certain symmetric operators can be extended to self-adjoint operators, which are essential for defining physical observables in quantum mechanics. This extension is particularly significant when considering deficiency indices and essential self-adjointness, as it provides a systematic approach to deal with symmetric operators that might not be initially self-adjoint.

5 Must Know Facts For Your Next Test

1. Friedrichs extension can be applied when dealing with symmetric operators that have equal deficiency indices, typically (1,1), which allows for the construction of a unique self-adjoint extension.
2. The existence of a Friedrichs extension guarantees that there is a well-defined spectral theory associated with the operator, making it possible to apply results from quantum mechanics effectively.
3. The process to find a Friedrichs extension involves utilizing the closure of the symmetric operator and ensuring that the resulting extension is densely defined and self-adjoint.
4. In quantum mechanics, the Friedrichs extension ensures that observables represented by certain symmetric operators yield real eigenvalues corresponding to measurable quantities.
5. If an operator is essentially self-adjoint, its Friedrichs extension coincides with its unique self-adjoint extension, emphasizing the connection between these concepts.

Review Questions

• How does Friedrichs extension relate to symmetric operators and their properties?
  • Friedrichs extension specifically deals with symmetric operators, allowing them to be extended to self-adjoint operators under certain conditions. For symmetric operators that have equal deficiency indices, Friedrichs extension provides a systematic way to obtain their self-adjoint extensions. This connection is essential because only self-adjoint operators have well-defined spectra, which are necessary for physical applications like quantum mechanics.
• What role do deficiency indices play in determining whether a symmetric operator admits a Friedrichs extension?
  • Deficiency indices are crucial in understanding the structure of symmetric operators and their extensions. They help determine if a symmetric operator can be extended to a self-adjoint operator through Friedrichs extension. When both deficiency indices are equal, typically (1,1), this indicates that a unique self-adjoint extension exists. Thus, deficiency indices provide key information on whether an appropriate Friedrichs extension can be constructed.
• Evaluate how the concept of essential self-adjointness interacts with Friedrichs extension in spectral theory.
  • Essential self-adjointness directly impacts how we view Friedrichs extension within spectral theory. If a symmetric operator is essentially self-adjoint, it means that its closure yields a unique self-adjoint extension without needing additional conditions. In such cases, the Friedrichs extension becomes redundant because it is already encompassed by the unique self-adjoint nature of the closure. Therefore, understanding essential self-adjointness helps clarify when and how Friedrichs extensions are applied within the broader framework of spectral theory.

Related terms

Symmetric operator: An operator that satisfies the property ⟨Ax, y⟩ = ⟨x, Ay⟩ for all x and y in its domain, where ⟨ , ⟩ denotes the inner product.

Self-adjoint operator: An operator that is equal to its adjoint, meaning its domain coincides with the domain of its adjoint and it satisfies the same inner product properties.

Deficiency indices: A pair of integers (n+, n−) that indicate the dimensions of the kernel of the adjoint operator related to the symmetric operator, providing insight into the self-adjointness and possible extensions of the operator.