Editing Self-adjoint operator (section)

=== A symmetric operator that is not essentially self-adjoint ===
We first consider the Hilbert space <math>L^2[0, 1]</math> and the differential operator
: <math>D: \phi \mapsto \frac{1}{i} \phi'</math>
defined on the space of continuously differentiable complex-valued functions on [0,1], satisfying the boundary conditions
:<math>\phi(0) = \phi(1) = 0.</math>

Then ''D'' is a symmetric operator as can be shown by [[integration by parts]]. The spaces ''N''<sub>+</sub>, ''N''<sub>−</sub> (defined below) are given respectively by the [[distribution (mathematics)|distribution]]al solutions to the equation
: <math>\begin{align}
  -i u' &=  i u \\
  -i u' &= -i u
\end{align}</math>
which are in ''L''<sup>2</sup>[0, 1]. One can show that each one of these solution spaces is 1-dimensional, generated by the functions ''x'' → ''e''<sup>''−x''</sup> and ''x'' → ''e''<sup>''x''</sup> respectively. This shows that ''D'' is not essentially self-adjoint,<ref>{{harvnb|Hall|2013}} Section 9.6</ref> but does have self-adjoint extensions. These self-adjoint extensions are parametrized by the space of unitary mappings ''N''<sub>+</sub> → ''N''<sub>−</sub>, which in this case happens to be the unit circle '''T'''.

In this case, the failure of essential self-adjointenss is due to an "incorrect" choice of boundary conditions in the definition of the domain of <math>D</math>. Since <math>D</math> is a first-order operator, only one boundary condition is needed to ensure that <math>D</math> is symmetric. If we replaced the boundary conditions given above by the single boundary condition
: <math>\phi(0) = \phi(1)</math>,
then ''D'' would still be symmetric and would now, in fact, be essentially self-adjoint. This change of boundary conditions gives one particular essentially self-adjoint extension of ''D''. Other essentially self-adjoint extensions come from imposing boundary conditions of the form <math>\phi(1) = e^{i\theta}\phi(0)</math>.

This simple example illustrates a general fact about self-adjoint extensions of symmetric differential operators ''P'' on an open set ''M''. They are determined by the unitary maps between the eigenvalue spaces          
: <math> N_\pm = \left\{u \in L^2(M): P_\operatorname{dist} u =  \pm i u\right\} </math>
where ''P''<sub>dist</sub> is the distributional extension of ''P''.