What really is the self-adjoint extension?

Question

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 $\theta$.

Answer 1

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).