Editing Self-adjoint operator (section)

=== Constant-coefficient operators ===
We next give the example of differential operators with [[constant coefficient]]s. Let
: <math>P\left(\vec{x}\right) = \sum_\alpha c_\alpha x^\alpha </math>
be a polynomial on '''R'''<sup>''n''</sup> with ''real'' coefficients, where α ranges over a (finite) set of [[multi-index|multi-indices]]. Thus
: <math> \alpha = (\alpha_1, \alpha_2, \ldots, \alpha_n)</math>
and
: <math>x^\alpha = x_1^{\alpha_1} x_2^{\alpha_2} \cdots x_n^{\alpha_n}.</math>

We also use the notation
: <math>D^\alpha = \frac{1}{i^{|\alpha|}} \partial_{x_1}^{\alpha_1}\partial_{x_2}^{\alpha_2} \cdots \partial_{x_n}^{\alpha_n}. </math>

Then the operator ''P''(D) defined on the space of infinitely differentiable functions of compact support on '''R'''<sup>''n''</sup> by
: <math> P(\operatorname{D}) \phi = \sum_\alpha c_\alpha \operatorname{D}^\alpha \phi</math>
is essentially self-adjoint on ''L''<sup>2</sup>('''R'''<sup>''n''</sup>).

{{math theorem|Let ''P'' a polynomial function on '''R'''<sup>''n''</sup> with real coefficients, '''F''' the Fourier transform considered as a unitary map ''L''<sup>2</sup>('''R'''<sup>''n''</sup>) → ''L''<sup>2</sup>('''R'''<sup>''n''</sup>).  Then '''F'''*''P''(D)'''F''' is essentially self-adjoint and its unique self-adjoint extension is the operator of multiplication by the function ''P''.}}

More generally, consider linear differential operators acting on infinitely differentiable complex-valued functions of compact support. If ''M'' is an open subset of '''R'''<sup>''n''</sup>
: <math>P \phi(x) = \sum_\alpha a_\alpha (x) \left[D^\alpha \phi\right](x)</math>
where ''a''<sub>α</sub> are (not necessarily constant) infinitely differentiable functions. ''P'' is a linear operator
: <math> C_0^\infty(M) \to C_0^\infty(M).</math>

Corresponding to ''P'' there is another differential operator, the '''[[formal adjoint]]''' of ''P''
: <math> P^\mathrm{*form} \phi = \sum_\alpha D^\alpha \left(\overline{a_\alpha} \phi\right)</math>

{{math theorem|The adjoint ''P''* of ''P'' is a restriction of the distributional extension of the formal adjoint to an appropriate subspace of <math>L^2</math>.  Specifically:
<math display="block">\operatorname{dom} P^* = \left\{u \in L^2(M): P^{\mathrm{*form}}u \in L^2(M) \right\}.</math>}}