Editing Differential operator (section)

==Adjoint of an operator==
{{See also|Hermitian adjoint}}
Given a linear differential operator <math>T</math>
<math display="block">Tu = \sum_{k=0}^n a_k(x) D^k u</math>
the [[Hermitian adjoint|adjoint]] of this operator is defined as the operator <math>T^*</math> such that
<math display="block">\langle Tu,v \rangle = \langle u, T^*v \rangle</math>
where the notation <math>\langle\cdot,\cdot\rangle</math> is used for the [[scalar product]] or [[inner product]].  This definition therefore depends on the definition of the scalar product (or inner product).

=== Formal adjoint in one variable ===

In the functional space of [[square-integrable function]]s on a [[real number|real]] [[interval (mathematics)|interval]] {{open-open|''a'', ''b''}}, the scalar product is defined by
<math display="block">\langle f, g \rangle = \int_a^b  \overline{f(x)} \,g(x) \,dx , </math>

where the line over ''f''(''x'') denotes the [[complex conjugate]] of ''f''(''x'').  If one moreover adds the condition that ''f'' or ''g'' vanishes as <math>x \to a</math> and <math>x \to b</math>, one can also define the adjoint of ''T'' by
<math display="block">T^*u = \sum_{k=0}^n (-1)^k D^k \left[ \overline{a_k(x)} u \right].</math>

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 <math>T^*</math> is defined according to this formula, it is called the '''formal adjoint''' of ''T''.

A (formally) '''[[self-adjoint operator|self-adjoint]]''' operator is an operator equal to its own (formal) adjoint.

=== Several variables ===

If Ω is a domain in '''R'''<sup>''n''</sup>, and ''P'' a differential operator on Ω, then the adjoint of ''P'' is defined in [[Lp space|''L''<sup>2</sup>(Ω)]] by duality in the analogous manner:

:<math>\langle f, P^* g\rangle_{L^2(\Omega)} = \langle P f, g\rangle_{L^2(\Omega)}</math>

for all smooth ''L''<sup>2</sup> functions ''f'', ''g''.  Since smooth functions are dense in ''L''<sup>2</sup>, this defines the adjoint on a dense subset of ''L''<sup>2</sup>:  P<sup>*</sup> is a [[densely defined operator]].

=== Example ===
The [[Sturm&ndash;Liouville theory|Sturm&ndash;Liouville]] operator is a well-known example of a formal self-adjoint operator.  This second-order linear differential operator ''L'' can be written in the form

: <math>Lu = -(pu')'+qu=-(pu''+p'u')+qu=-pu''-p'u'+qu=(-p) D^2 u +(-p') D u + (q)u.</math>

This property can be proven using the formal adjoint definition above.<ref>
: <math>\begin{align}
L^*u & {} = (-1)^2 D^2 [(-p)u] + (-1)^1 D [(-p')u] + (-1)^0 (qu) \\
 & {} = -D^2(pu) + D(p'u)+qu \\
 & {} = -(pu)''+(p'u)'+qu \\
 & {} = -p''u-2p'u'-pu''+p''u+p'u'+qu \\
 & {} = -p'u'-pu''+qu \\
 & {} = -(pu')'+qu \\
 & {} = Lu
\end{align}</math></ref>

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