Self-Adjoint Operator

(redirected from Hermitian operators)

self-adjoint operator

[¦self ə¦jȯint ′äp·ə‚rād·ər]
A linear operator which is identical with its adjoint operator.

Self-Adjoint Operator


(or Hermitian operator), an operator coincident with its adjoint. The theory of self-adjoint operators arose as a generalization of the theory of, for example, integral equations with a symmetric kernel, self-adjoint differential equations, and symmetric matrices. Examples of self-adjoint operators are (1) the operator of multiplication by the independent variable in the space of functions that are defined on the entire number axis and are square integrable and (2) the differentiation operator

in the same space.

If the function K(x,y) is continuous in the square a ≤ x ≤ b,a ≤ y ≤ b and if K(x,y) = K(y, x), then the integral operator

is self-adjoint. The spectrum of a self-adjoint operator lies on the real axis. In quantum mechanics physical quantities have corresponding self-adjoint operators whose spectra give the possible values of these quantities. A self-adjoint operator can be represented as an integral that is the limit of linear combinations of pairwise orthogonal projection operators with real coefficients.

References in periodicals archive ?
Lee, Maps preserving zero Jordan products on hermitian operators, Illinois J.
when reading about the directed order structure for families of hermitian operators, e.
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He explains vector spaces and bases, linear transformations and operators, eigenvalues, circles and ellipses, inner products, adjoints, Hermitian operators, unitary operators, the wave equation, continuous spectre and the Dirac delta function, Fourier transforms, Green's and functions, and includes an appendix on matrix operations (new to this edition) and a full chapter on crucial applications.
KREIN, Fundamental aspects of the representation theory of Hermitian operators with deficiency index (mrz, rn), Ukrain.
The way in which physical concepts are tied to the mathematical formalism is particularly helpful, for example in the treatment of Hermitian operators where orthogonality of non-degenerate eigenstates is tied to repeatability of measurements.
So, to sum it up: a semilocal pseudopotential is a general Hermitian operator in the spherically symmetric problem (i.