Deficiency Indices

Deficiency indices count the missing directions that prevent a closed symmetric operator from being self-adjoint. In differential problems they are computed by solving the adjoint equation at nonreal spectral parameters and imposing square integrability. They diagnose whether boundary data are unnecessary, selectable, or insufficient to produce a self-adjoint realization.

Helpful background. Adjoints of Unbounded Operators determines the maximal adjoint domain. Self-Adjoint Extensions constructs the extensions once their existence is known.

Deficiency spaces and the range defect

For a densely defined closed symmetric operator $S$, fix $\eta>0$ and define

$$\mathcal N_+=\ker(S^*-i\eta), \qquad \mathcal N_-=\ker(S^*+i\eta), \qquad n_\pm=\dim\mathcal N_\pm.$$

The scale $\eta$ has the units of $S$. Dimensions can be finite or infinite, and are constant as the nonreal parameter varies within the corresponding upper or lower half-plane. The plus/minus labels must be read together with the sign convention.

The identity

$$\operatorname{Ran}(S+i\eta)^\perp=\mathcal N_+$$

shows what the count means: $n_+$ is the number of independent directions not reached by the shifted operator $S+i\eta$. For closed symmetric $S$ these shifted ranges are closed, so this is a literal codimension, not just a failure of some algebraic inverse formula.

The classification is:

• $(n_+,n_-)=(0,0)$: $S$ is self-adjoint.
• $n_+=n_->0$: self-adjoint extensions exist and are not unique.
• $n_+\neq n_-$: no self-adjoint extension exists on the same Hilbert space.

For a nonclosed symmetric seed, apply this classification to its closure. Its adjoint is unchanged by closure, so the deficiency equations can often be solved directly from the seed.

First-order differentiation on three spaces

Set $\hbar=\eta=1$ for this example, and start from $-i\,d/dx$ on compactly supported smooth functions. The maximal adjoint equation gives

$$f_+(x)=C_+e^{-x}, \qquad f_-(x)=C_-e^x.$$

On the full line, neither solution is square-integrable. Thus $(n_+,n_-)=(0,0)$ and the seed is essentially self-adjoint.

On a finite interval $(0,L)$, both solutions belong to $H^1(0,L)$ and $L^2(0,L)$. Thus $(n_+,n_-)=(1,1)$, and a one-phase family of self-adjoint momenta exists. In boundary coordinates it becomes the quasiperiodic condition $f(L)=e^{i\theta}f(0)$.

On the half-line $(0,\infty)$, only $e^{-x}$ is square-integrable. Thus $(n_+,n_-)=(1,0)$, and no self-adjoint extension exists there. Reversing the sign of the differential expression interchanges the labels but does not alter the failure of equality.

This calculation counts global square-integrable solutions. Counting formal local solutions without their behavior at every endpoint gives the wrong answer.

Second-order differentiation changes the count

Consider the minimal operator $S=-d^2/dx^2$ on $(0,\infty)$. For the plus deficiency equation,

$$-f''=if, \qquad r^2=-i,$$

and exactly one of the two exponential solutions has negative real part in its exponent. For the minus equation $r^2=i$, again exactly one solution decays. Thus the deficiency indices are $(1,1)$.

Unlike half-line momentum, the half-line kinetic Hamiltonian therefore admits self-adjoint realizations. Its Robin and Dirichlet conditions are constructed in Self-Adjoint Extensions.

On a finite interval all solutions of either constant-coefficient second-order equation are square-integrable, so the indices are $(2,2)$. The full extension family is parameterized by $U(2)$ and includes boundary conditions coupling the two endpoints. Restricting attention to separate Dirichlet, Neumann, or Robin conditions at each end describes only part of that family.

On the full line, no nonzero global solution can decay at both ends for either nonreal equation, giving $(0,0)$. Joining separate decaying expressions at zero introduces a derivative jump and does not solve the free adjoint equation there.

What the indices do not determine

The indices establish existence and parameter count, but do not give the spectrum of a particular extension. They also do not decide positivity, locality of boundary coupling, time-reversal invariance, or compatibility with a physical wall model. Those are further conditions on the extension.

For singular potentials, the endpoint analysis can be subtler than exponent counting. A solution must lie in the actual maximal adjoint domain; distributional singularities and boundary behavior cannot be ignored. Limit-point and limit-circle theory systematizes such second-order problems.

A larger Hilbert-space dilation can sometimes represent dynamics associated with an operator that has no same-space self-adjoint extension. That is a different construction and does not contradict the index criterion.

Exercises

Compute the indices of the direct sum of two half-line momentum operators, first with the same sign and then with opposite signs.

Solution

Adjoints and deficiency kernels split as direct sums. Two copies of $-i\,d/dx$ give $(2,0)$ and have no self-adjoint extension. A copy of $-i\,d/dx$ and one of $+i\,d/dx$ give $(1,1)$ and do admit extensions, with boundary conditions coupling their endpoint values. Adding a channel can balance the missing directions.

Why is $e^{-|x|/\sqrt2}$ times a suitable phase not automatically a deficiency vector for the whole-line free Laplacian?

Solution

Matching decaying solutions from two half-lines generally produces a jump in the first derivative at zero. The distributional second derivative then contains a delta function, so it is not in $L^2$ and does not satisfy the free adjoint equation. A point interaction would be a different operator-domain problem.

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 — explicit deficiency equations for momentum and Hamiltonians.
• M. Gallone and A. Michelangeli, Self-Adjoint Extension Schemes and Modern Applications to Quantum Hamiltonians, Springer, 2023, doi:10.1007/978-3-031-10885-3, author manuscript — deficiency-space classification and singular models.
• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Springer, 2012, doi:10.1007/978-94-007-4753-1 — extension theory and Sturm–Liouville operators.