Editing Self-adjoint operator (section)

=== Schrödinger operators with singular potentials ===
A more subtle example of the distinction between symmetric and (essentially) self-adjoint operators comes from [[Schrödinger equation|Schrödinger operators]] in quantum mechanics. If the potential energy is singular—particularly if the potential is unbounded below—the associated Schrödinger operator may fail to be essentially self-adjoint. In one dimension, for example, the operator
: <math>\hat{H} := \frac{P^2}{2m} - X^4</math>
is not essentially self-adjoint on the space of smooth, rapidly decaying functions.<ref>{{harvnb|Hall|2013}} Theorem 9.41</ref> In this case, the failure of essential self-adjointness reflects a pathology in the underlying classical system: A classical particle with a <math>-x^4</math> potential escapes to infinity in finite time. This operator does not have a ''unique'' self-adjoint, but it does admit self-adjoint extensions obtained by specifying "boundary conditions at infinity". (Since <math>\hat{H}</math> is a real operator, it commutes with complex conjugation. Thus, the deficiency indices are automatically equal, which is the condition for having a self-adjoint extension.)

In this case, if we initially define <math>\hat{H}</math> on the space of smooth, rapidly decaying functions, the adjoint will be "the same" operator (i.e., given by the same formula) but on the largest possible domain, namely
: <math>\operatorname{Dom}\left(\hat{H}^*\right) = \left\{ \text{twice differentiable functions }f \in L^2(\mathbb{R})\left|\left( -\frac{\hbar^2}{2m}\frac{d^2f}{dx^2} - x^4f(x)\right) \in L^2(\mathbb{R}) \right. \right\}. </math>

It is then possible to show that <math>\hat{H}^*</math> is not a symmetric operator, which certainly implies that <math>\hat{H}</math> is not essentially self-adjoint. Indeed, <math>\hat{H}^*</math> has eigenvectors with pure imaginary eigenvalues,<ref>{{harvnb|Berezin|Shubin|1991}} p. 85</ref><ref>{{harvnb|Hall|2013}} Section 9.10</ref> which is impossible for a symmetric operator. This strange occurrence is possible because of a cancellation between the two terms in <math>\hat{H}^*</math>: There are functions <math>f</math> in the domain of <math>\hat{H}^*</math> for which neither <math>d^2 f/dx^2</math> nor <math>x^4f(x)</math> is separately in <math>L^2(\mathbb{R})</math>, but the combination of them occurring in <math>\hat{H}^*</math> is in <math>L^2(\mathbb{R})</math>. This allows for <math>\hat{H}^*</math> to be nonsymmetric, even though both <math>d^2/dx^2</math> and <math>X^4</math> are symmetric operators. This sort of cancellation does not occur if we replace the repelling potential <math>-x^4</math> with the confining potential <math>x^4</math>.