Differential operator: Encyclopedia II - Differential operator - Adjoint of an operator

Differential operator - Adjoint of an operator

Given a linear differential operator

the adjoint of this operator is defined as the operator T ^* such that

where the notation is used for the scalar product or inner product. This definition therefore depends on the definition of the scalar product. In the functional space of square integrable functions, the scalar product is defined by

If one moreover adds the condition that f and g vanish for and , one can also define the adjoint of T by

This formula does not explicitly depend on the definition of the scalar product. It is therefore sometimes chosen as a definition of the adjoint operator. When T ^* is defined according to this formula, it is called the formal adjoint of T.

A self-adjoint operator is an operator adjoint of itself.

The Sturm-Liouville operator is a well-known example of formal self-adjoint operator. This second order linear differential operators L can be written in the form

This property can be proven using the formal adjoint definition above.

This operator is central to Sturm-Liouville theory where the eigenfunctions (analogues to eigenvectors) of this operator are considered.

Adapted from the Wikipedia article "Adjoint of an operator", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki