Self-Adjoint

A densely defined operator on a Hilbert space is self-adjoint if it equals its adjoint operator, including equality of their domains. For a differential operator, this condition must be distinguished from formal self-adjointness of the differential expression. Consider the second-order expression

(1)

where and are real functions of on the region of interest with continuous derivatives and with on . This means that there are no singular points in . Then the formal adjoint is defined by

			(2)

In order for the differential expression to be formally self-adjoint, i.e.,

(3)

the second terms in (1) and (2) must be equal, so

(4)

This also guarantees that the third terms are equal, since

(5)

so (2) becomes

			(6)

The differential operators corresponding to the Legendre differential equation and the equation of simple harmonic motion are formally self-adjoint, while those corresponding to the Laguerre differential equation and Hermite differential equation are not formally self-adjoint in their displayed forms.

A second-order ordinary differential equation of the form above can be put in formally self-adjoint form using an integrating factor, yielding a Sturm-Liouville equation. For the homogeneous equation in the special case , (6) gives

(7)

(8)

(9)

(10)

where is a constant of integration.

A choice of domain and boundary conditions is additionally required to obtain a self-adjoint operator. For a Sturm-Liouville expression, suitable boundary conditions make the boundary form

(11)

Formal self-adjointness alone does not imply self-adjointness of the associated operator.