Symmetric operator

Definition

A symmetric operator is a linear operator defined on a dense domain in a Hilbert space that satisfies the property \( \langle Ax, y \rangle = \langle x, Ay \rangle \) for all vectors \( x \) and \( y \) in its domain. This means that the operator is equal to its adjoint on that domain. Symmetric operators are important because they can be linked to self-adjointness and the behavior of differential operators, which are crucial in understanding various aspects of quantum mechanics and mathematical physics.

5 Must Know Facts For Your Next Test

1. Symmetric operators can have different properties depending on whether they are densely defined or not; only densely defined symmetric operators can be considered for self-adjointness.
2. The Kato-Rellich theorem provides conditions under which a symmetric operator can be perturbed while retaining essential self-adjointness, which is vital for understanding stability in quantum systems.
3. Not all symmetric operators are self-adjoint; an operator must meet additional criteria related to its domain to achieve self-adjointness.
4. A symmetric operator can have complex eigenvalues if it is not self-adjoint, highlighting the significance of the symmetry condition in ensuring real spectral properties.
5. Essential self-adjointness characterizes the boundary behavior of symmetric operators, indicating whether they can be extended uniquely to a self-adjoint operator on their closure.

Review Questions

• How does the definition of a symmetric operator relate to the concept of an adjoint operator?
  • The definition of a symmetric operator includes the property that \( \langle Ax, y \rangle = \langle x, Ay \rangle \) for all vectors in its domain. This relationship implies that the action of the operator is closely tied to its adjoint, as symmetry means the operator's action produces equivalent inner products when switching arguments. Thus, understanding symmetric operators requires examining how they interact with their adjoints and what this says about their spectral properties.
• Discuss how the Kato-Rellich theorem connects symmetric operators to essential self-adjointness.
  • The Kato-Rellich theorem establishes conditions under which perturbations of symmetric operators remain essentially self-adjoint. It highlights how small modifications to a symmetric operator do not disrupt its fundamental properties, allowing researchers to analyze more complex systems without losing control over the original operator's characteristics. This connection is crucial for studying stability and continuity in various applications, particularly in quantum mechanics where perturbations often arise.
• Evaluate the implications of having a symmetric operator that is not self-adjoint regarding its spectrum and physical interpretations.
  • When a symmetric operator is not self-adjoint, it may lead to complex eigenvalues, which have significant implications for physical interpretations in quantum mechanics. Non-self-adjoint operators suggest the presence of unobservable states or instabilities in a system. This situation complicates the analysis of the system's behavior because it may no longer exhibit expected physical properties like conservation laws or real energy values. Thus, recognizing whether an operator is self-adjoint or merely symmetric is critical for ensuring that predictions made by models remain physically valid.

Related terms

Adjoint Operator: The adjoint operator of a given linear operator is defined such that it satisfies the relationship \( \langle Ax, y \rangle = \langle x, A^*y \rangle \), providing a way to relate operators to their duals.

Self-Adjoint Operator: An operator is self-adjoint if it is equal to its adjoint on its entire domain, implying not only symmetry but also that it has a complete set of eigenfunctions and real eigenvalues.

Closed Operator: A closed operator is one where the graph of the operator is closed in the product space of the Hilbert space, ensuring that limits of convergent sequences within its domain also belong to the domain.