Self-Adjoint Extensions

A self-adjoint extension enlarges the domain of a symmetric operator while retaining its action on the original domain. The extension parameters often encode boundary conditions or contact physics. Self-adjointness constrains these parameters but does not necessarily select one physical model.

Helpful background. Self-Adjoint Operators states the extension criterion. Essential Self-Adjointness treats the unique-closure case. Here the goal is to construct the larger domains when an extension choice remains.

Deficiency-space construction of the domain

Let $S$ be densely defined, closed, and symmetric. Choose $\eta>0$ with the same units as $S$ and write

$$\mathcal N_\pm=\ker(S^*\mp i\eta).$$

If the two spaces have equal dimension, every unitary map $U:\mathcal N_+\to\mathcal N_-$ defines a self-adjoint extension by

$$\begin{aligned} D(S_U)&=\{f+u+Uu:f\in D(S),\ u\in\mathcal N_+\},\\ S_U(f+u+Uu)&=Sf+i\eta u-i\eta Uu. \end{aligned}$$

The decomposition is unique. Conversely, every self-adjoint extension of $S$ is obtained in this way. If the original symmetric seed is not closed, replace it by its closure before using this formula.

The underlying decomposition is $D(S^*)=D(S)\dotplus\mathcal N_+\dotplus\mathcal N_-$, with graph-norm topology. In the boundary form

$$\mathcal B(f,g)=\langle f,S^*g\rangle-\langle S^*f,g\rangle,$$

the plus and minus deficiency components contribute with opposite signs. The graph of a unitary $U$ pairs them with equal norms, eliminating this form. Maximality of the resulting neutral subspace gives equality with the adjoint domain, not merely symmetry. This is the structural content of von Neumann’s theorem; the full decomposition proof is given in the references.

Boundary conditions for a half-line Hamiltonian

Consider the minimal operator generated by $-d^2/dx^2$ on $C_c^\infty(0,\infty)$ in $L^2(0,\infty)$. We suppress the positive factor $\hbar^2/(2m)$ until discussing energies. Its adjoint acts on $H^2(0,\infty)$, and integration by parts gives

$$\mathcal B(f,g) =\overline{f(0)}g'(0)-\overline{f'(0)}g(0).$$

The endpoint at infinity gives no independent boundary data for this operator. The finite endpoint has one complex value and one complex derivative. The self-adjoint domains are

$$D(H_\alpha)=\{f\in H^2(0,\infty):f'(0)=\alpha f(0)\}, \qquad \alpha\in\mathbb R,$$

together with the Dirichlet condition $f(0)=0$.

Reality of $\alpha$ makes the boundary form vanish. To establish self-adjointness, test a potential adjoint vector against arbitrary $g$ satisfying the condition. The term becomes $(\alpha\overline{f(0)}-\overline{f'(0)})g(0)$; it vanishes for all $g(0)$ only when the same condition holds for $f$. This proves equality of domains.

Merely showing zero probability current on individual eigenfunctions would be insufficient. The sesquilinear boundary form must vanish for every pair of vectors, and the resulting domain must be maximal.

Extension parameters change measurable physics

For $\alpha<0$, the normalized function

$$f_\alpha(x)=\sqrt{-2\alpha}\,e^{\alpha x}$$

obeys the Robin condition and is square-integrable. Restoring units, its energy is

$$E_\alpha=-\frac{\hbar^2\alpha^2}{2m}.$$

For $\alpha\geq0$, there is no negative bound state. At positive energy $E=\hbar^2k^2/(2m)$, write an incoming and reflected wave as $f(x)=e^{-ikx}+r(k)e^{ikx}$. The boundary condition gives

$$r(k)=\frac{ik+\alpha}{ik-\alpha}, \qquad |r(k)|=1.$$

Neumann reflection has $r=1$; the Dirichlet limit has $r=-1$. Thus two self-adjoint extensions can have different bound states and reflection phases even though the bulk differential expression is identical.

The parameter $\alpha$ has dimensions of inverse length. A microscopic wall model or low-energy scattering measurement can determine it. Selecting an extension is therefore model input when essential self-adjointness has not removed the choice.

Extensions, closure, and positivity

Closing the minimal graph does not choose one Robin parameter. Its closed domain has both $f(0)=0$ and $f'(0)=0$, which is still too small to be self-adjoint. An extension adds boundary data; closure adds only limits of the existing graph.

If the initial symmetric operator is semibounded, the Friedrichs construction supplies a distinguished extension through closure of its quadratic form. For the minimal half-line Laplacian that extension is Dirichlet. Other extensions may still be self-adjoint and semibounded. The choice is not universally a consequence of self-adjointness alone; see Friedrichs Extension.

Exercises

Derive the reflection coefficient above and verify conservation of incoming and outgoing flux.

Solution

At zero, $f(0)=1+r$ and $f'(0)=ik(r-1)$. Solving $ik(r-1)=\alpha(1+r)$ gives the displayed $r$. For real $\alpha$ and real positive $k$, numerator and denominator have equal modulus, so $|r|^2=1$. The plane-wave currents have equal magnitudes and opposite signs.

For a one-dimensional deficiency space, explain how many extensions exist and why the auxiliary scale $\eta$ is not a physical coupling.

Solution

A unitary between one-dimensional complex spaces is multiplication by a phase after choosing normalized bases, so there is a circle of extensions. Changing $\eta$ changes the deficiency vectors and their parameterization; it does not change the set of self-adjoint extensions of the original operator. Physical data are encoded by the resulting domain, such as its Robin parameter.

References

• G. Bonneau, J. Faraut, and G. Valent, “Self-adjoint extensions of operators and the teaching of quantum mechanics,” American Journal of Physics 69, 322–331, 2001, doi:10.1119/1.1328351, author manuscript — boundary conditions and spectra.
• M. Gallone and A. Michelangeli, Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians, Springer Monographs in Mathematics, Springer, 2023, doi:10.1007/978-3-031-10885-3, author manuscript — extension domains and quantum applications.
• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, 2012, doi:10.1007/978-94-007-4753-1 — von Neumann and boundary-space extension theory.