Deficiency indices

Definition

Deficiency indices are a pair of non-negative integers that characterize the extension properties of a symmetric operator. Specifically, they provide information on the dimensions of the deficiency spaces associated with an operator, which helps determine whether the operator can be extended to a self-adjoint operator or if it has self-adjoint extensions.

5 Must Know Facts For Your Next Test

1. Deficiency indices are denoted by two numbers, usually represented as (n+, n-), where n+ is the dimension of the positive deficiency space and n- is the dimension of the negative deficiency space.
2. If both deficiency indices are zero, the symmetric operator is essentially self-adjoint, meaning it has a unique self-adjoint extension.
3. If one index is positive and the other is zero, there exists at least one self-adjoint extension, but it is not unique.
4. If both indices are positive, this indicates that there are infinitely many self-adjoint extensions of the symmetric operator.
5. The study of deficiency indices provides a bridge between symmetric operators and their possible self-adjoint extensions, forming an important concept in operator theory.

Review Questions

• How do deficiency indices relate to the extension of symmetric operators?
  • Deficiency indices play a crucial role in understanding how symmetric operators can be extended to self-adjoint operators. They provide insight into the dimensions of the positive and negative deficiency spaces, which inform us about the existence and uniqueness of such extensions. If both deficiency indices are zero, the operator can be uniquely extended; if one index is non-zero, at least one extension exists; and if both are non-zero, multiple extensions are possible.
• What implications do deficiency indices have on determining whether an operator is essentially self-adjoint?
  • Deficiency indices directly influence whether an operator is considered essentially self-adjoint. An operator with deficiency indices (0,0) indicates that it can be uniquely extended to a self-adjoint operator. Conversely, if either index is greater than zero, it implies that there exist self-adjoint extensions but potentially with more than one possibility. Thus, analyzing these indices is key for understanding the self-adjoint nature of symmetric operators.
• Evaluate the significance of deficiency indices in the context of quantum mechanics and functional analysis.
  • In quantum mechanics and functional analysis, deficiency indices are essential for determining how physical observables represented by symmetric operators can be treated mathematically. They help classify operators based on their extension properties, affecting how we interpret measurements and states in quantum systems. By understanding these indices, we gain deeper insights into the stability and behavior of quantum systems when modeled using unbounded operators, enhancing our ability to predict and manipulate physical phenomena.

Related terms

symmetric operator:

A symmetric operator is a linear operator that is equal to its adjoint on its domain, meaning it preserves inner products in a certain way.

self-adjoint operator:

A self-adjoint operator is one that is equal to its adjoint and has a domain equal to the domain of its adjoint, making it a specific type of symmetric operator with stronger properties.

deficiency space: A deficiency space is the kernel of the operator's adjoint minus a complex number, which plays a key role in understanding the extensions of symmetric operators.