semibound

Semibound (semibounded) is a term used in functional analysis to describe linear operators on a Hilbert space whose spectrum is bounded from below, or equivalently whose associated quadratic form is bounded below.

Definition and basic properties

Let H be a complex or real Hilbert space and A be a densely defined linear operator with domain D(A) ⊂ H. A is semibounded below if there exists m ∈ R such that for all x ∈ D(A),

⟨Ax, x⟩ ≥ m||x||^2.

If A is symmetric (or self-adjoint), this inequality holds with the lower bound m, and the spectrum of A is contained in [m, ∞) in the self-adjoint case. The bound m is called a lower bound of A or the bottom of the spectrum.

Quadratic form viewpoint

A is semibounded below exactly when its associated quadratic form q(x) = ⟨Ax, x⟩ is bounded below on D(A). In many contexts, the theory of closed semibounded quadratic forms provides a bridge to self-adjoint operators via the representation theorem, yielding a unique self-adjoint operator associated with a closed semibounded form.

Extensions and spectral theory

A densely defined, semibounded symmetric operator has self-adjoint extensions. The Friedrichs extension is a canonical self-adjoint extension that preserves the lower bound and preserves the form domain. Semibounded operators arise frequently in quantum mechanics (energy operators), PDEs, and spectral theory.

Examples

The Laplacian with Dirichlet boundary conditions on a bounded domain is semibounded below (positive). In finite dimensions, any Hermitian matrix is semibounded, with lower bound equal to its smallest eigenvalue.

See also

Semibounded operator, self-adjoint extension, Friedrichs extension, spectral theory, quadratic forms.