semibounded

Semibounded is a term used in functional analysis to describe a property of operators or forms that are bounded from below. For a densely defined linear operator A on a Hilbert space H, A is semibounded if there exists a real number m such that for all x in the domain of A, the inner product satisfies ⟨Ax, x⟩ ≥ m||x||^2. Equivalently, the associated quadratic form q_A(x) = ⟨Ax, x⟩ is semibounded from below: there exists m with q_A(x) ≥ m||x||^2 for all x in the form domain. If A is symmetric or self-adjoint and semibounded, its lower bound m provides information about the spectrum and stability of the operator.

In the self-adjoint case, semiboundedness implies that the numerical range and the spectrum lie in a half-line [m, ∞). For densely defined symmetric operators, semiboundedness often leads to the existence of a self-adjoint extension that preserves the lower bound, notably the Friedrichs extension. The corresponding closed quadratic form inherits the lower bound, enabling variational methods and spectral analysis.

Examples include the Laplacian with Dirichlet boundary conditions on a bounded domain, which is semibounded from below by 0, and Schrödinger operators of the form −Δ + V with V(x) ≥ V_min, which are semibounded from below by V_min. Semiboundedness is central in stability considerations, the study of spectra, and the application of the Rayleigh–Ritz method in PDEs and quantum mechanics.

See also: lower bound, spectrum, Friedrichs extension, quadratic form, variational methods.

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