# What really is the self-adjoint extension?

Going through the Quantum mechanics book by Capri, am time and again held with some stupid doubts on this topic of self-adjointness. We have for the momentum operator in finite domain,

$$p = -i\hbar \frac{\partial }{\partial x} \\ D_p = \big\{f(x),f'(x)\in \mathrm{L_2}(0,L) , f(0) = f(L) = 0 \big\}$$ Now we go on to define, the adjoint of the $p$ operator, with the larger domain

$$p^{\dagger} = -i\hbar \frac{\partial }{\partial x} \\ D_{p^{\dagger}} = \big\{f(x),f'(x)\in \mathrm{L_2}(0,L) , f(0) = \mathrm e^{i\theta}f(L)\big\}$$

Now, he says, we can see that $p^{\dagger}$ is self-adjoint, which I believe means that $p^\dagger = p ^{{\dagger}^{\dagger}}$ ($D_{p^\dagger} = D_{p ^{{\dagger}^{\dagger}}}$). So we have a self-adjoint extension of $p$. So my question now is that, is it $p ^{{\dagger}^{\dagger}}$ that is the extension of p or what is it that is its extension ?

But he also goes on to say that a symmetric operator $A$ is essentially self-adjoint if $A^{{\dagger}^{\dagger}}$ is self-adjoint. But is not this true for the above case, given that $p^\dagger = p ^{{\dagger}^{\dagger}}$ and a $p^\dagger$ is self-adjoint ? But however the above case is not essentially self-adjoint since there are infinite ways of choosing $\theta$.

A closed extension $A_c$ of an operator $A$ is an operator whose action is the same as $A$, the domains satisfy $D(A_c)\supset D(A)$ and $A_c$ is closed.
Given that, the smallest closed extension of a symmetric (densely defined) operator is its double adjoint $A^{**}$. We call it the closure of $A$, and denote it by $\overline{A}$.
An operator $A$ is called essentially self-adjoint if its closure $\overline{A}$ is self-adjoint. This also implies that $\overline{A}$ is the unique self-adjoint extension of $A$ (a self-adjoint extension is a closed extension of $A$ that is self-adjoint). This is a nice feature, since in general a symmetric, densely defined, operator may have zero, one or infinite self-adjoint extensions.
In the case above, it depends on the definition of $D_{p^*}$. If it is dependent on a fixed $\theta$, then something seems not right, but maybe the domain is of functions that satisfy the condition for any $\theta$, and then it could be effectively self-adjoint (and therefore $p$ ess. s-a).
• What you mean by $A_c$ is closed ? I think from the literature I have gone through $p$ is not essentially self-adjoint in the domain $(0,L)$. When you say same action, is not the action of $p^\dagger$ same as that of $p$ in this case ? Apr 20, 2015 at 6:38
• An operator $A$ on a Hilbert space $\mathscr{H}$ is closed if the graph $\{(\phi,A\phi), \phi\in D(A)\}$ is a closed subspace of $\mathscr{H}\times \mathscr{H}$. Every self-adjoint operator is closed. It is possible that $p$ is not essentially self-adjoint, and that would probably mean that the $p^*$ he defines is a self-adjoint extension of $p$ but not its adjoint (if else it would mean that $p$ is essentially self-adjoint). Apr 20, 2015 at 7:21
• So in this case $p^*$ is the s-a extension by that logic. Thanks ! Am sorry, am from a physics background, what is a graph ? Apr 20, 2015 at 7:34
• No problem, the graph is this set I have defined above. It is called like that because if $\mathscr{H}=\mathbb{R}$ (the reals) it is exactly the cartesian graph of the function $A$ ;-) Apr 20, 2015 at 7:44