Domains of Hamiltonians

A Hamiltonian domain records both regularity and boundary conditions. When an interaction is added, the resulting operator may keep the original domain, require a new one, or only be definable through a quadratic form. The Kato–Rellich theorem gives a precise criterion for the first possibility. We will prove it and apply it to the Coulomb potential. Quadratic Form Methods treats perturbations controlled only through energy estimates.

Helpful background. Closed and Closable Operators introduces graph norms and cores. Resolvent Operators supplies inverse estimates. Self-Adjoint Extensions treats boundary choices.

Relative bounds control an interaction on the domain

Let $H_0$ be self-adjoint. An operator $V$ is $H_0$-bounded if $D(H_0)\subseteq D(V)$ and there exist finite constants $a,b\geq0$ such that

$$\|V\psi\|\leq a\|H_0\psi\|+b\|\psi\|, \qquad \psi\in D(H_0).$$

The relative bound is the infimum of admissible $a$, allowing $b$ to depend on $a$. Relative bound zero does not mean $V=0$ or that $V$ is bounded on $\mathcal H$; it means arbitrarily small coefficients of $\|H_0\psi\|$ are possible.

Kato–Rellich theorem. If $V$ is symmetric and satisfies this estimate with some $a<1$, then

$$H=H_0+V,\qquad D(H)=D(H_0),$$

is self-adjoint. Every operator core for $H_0$ is also a core for $H$.

Symmetry is required on the perturbation domain. The estimate alone does not make a complex absorbing potential self-adjoint. Nor can a formal expression such as multiplication by a distribution automatically be inserted as $V$: the expression must define an actual operator on $D(H_0)$.

Resolvent proof and preserved cores

For $\lambda>0$, self-adjoint spectral estimates give

$$\|(H_0\mp i\lambda)^{-1}\|\leq\lambda^{-1}, \qquad \|H_0(H_0\mp i\lambda)^{-1}\|\leq1.$$

Therefore

$$\|V(H_0\mp i\lambda)^{-1}\|\leq a+b/\lambda<1$$

when $\lambda$ is sufficiently large. The factorization on $D(H_0)$,

$$H\mp i\lambda =\bigl[I+V(H_0\mp i\lambda)^{-1}\bigr](H_0\mp i\lambda),$$

has both factors onto. The first is boundedly invertible by its Neumann series. Thus both nonreal range equations for the symmetric operator $H$ are surjective, which implies self-adjointness.

The estimates also give equivalence of graph norms. One direction follows from the triangle inequality; for the other,

$$(1-a)\|H_0\psi\| \leq\|H\psi\|+b\|\psi\|.$$

After adjoining $\|\psi\|$ to each norm, both graph norms bound each other. A subspace dense in one graph norm is consequently dense in the other. This is why an established core remains a core after the perturbation. Hilbert-space density alone would not establish this result.

Coulomb singularity without a new operator domain

Use units in which the kinetic coefficient is one, and consider $H_0=-\Delta$ on $H^2(\mathbb R^3)$. For real $g$, let $V(x)=-g/|x|$ with its maximal multiplication domain. The three-dimensional Hardy estimate is

$$\|\psi/|x|\|\leq2\|\nabla\psi\|, \qquad \psi\in H^1(\mathbb R^3).$$

For smooth compactly supported functions, integrate the divergence of $x/|x|^2$ against $|\psi|^2$. Since that divergence is $1/|x|^2$ away from zero, integration by parts and Cauchy–Schwarz give

$$\|\psi/|x|\|^2 \leq2\|\psi/|x|\|\,\|\nabla\psi\|.$$

A punctured-ball calculation has a vanishing inner boundary term in the limit; density extends the estimate to $H^1$.

For $\psi\in H^2$, the Fourier transform or integration by parts gives

$$\|\nabla\psi\|^2 =\langle\psi,H_0\psi\rangle \leq\|\psi\|\,\|H_0\psi\|.$$

Hence, for every $\varepsilon>0$,

$$\|V\psi\| \leq |g|\varepsilon\|H_0\psi\| +|g|\varepsilon^{-1}\|\psi\|.$$

The relative bound is zero. Kato–Rellich gives the self-adjoint Coulomb Hamiltonian on $H^2(\mathbb R^3)$, and $C_c^\infty(\mathbb R^3)$ remains a core. This calculation establishes the operator realization. The bound-state spectrum is a separate problem treated in Hydrogen Atom as a Schrödinger Operator.

The conclusion is dimension- and potential-dependent. One cannot replace $1/|x|$ by an arbitrary stronger singularity and retain the argument without a new estimate.

Which domain question is being answered?

An operator domain determines when $H\psi$ is a Hilbert-space vector and therefore when the autonomous orbit is strongly differentiable. A form domain determines when the energy form is finite. A core is a smaller space on which the full operator can be recovered by graph closure. These roles should be distinguished even if a smooth test space occurs in all three discussions.

Kato–Rellich is a sufficient theorem, not a universal domain recipe. Failure of its small relative-bound hypothesis does not prove that no self-adjoint Hamiltonian exists. Form perturbations, extension theory, or more specialized estimates may still work. Conversely, a time-dependent family with the same self-adjoint domain needs time regularity before a propagator theorem applies; see Kato Theory.

Exercises

1. Let $V$ be bounded and self-adjoint. Show that its addition preserves $D(H_0)$ and every core of $H_0$.

Solution

Use $a=0$ and $b=\|V\|$. All hypotheses of Kato–Rellich hold. The graph-norm estimates above then prove the core statement as well as self-adjointness on the original domain.

2. Explain why the theorem cannot simply include relative bound one, using $V=-H_0$ for an unbounded self-adjoint $H_0$.

Solution

On $D(H_0)$, the sum is the zero operator, but its domain is still the proper dense subspace $D(H_0)$. Its adjoint is zero on all of $\mathcal H$, so the sum is not self-adjoint on the claimed domain. Its closure is the bounded zero operator on $\mathcal H$. Formal cancellation does not preserve domain equality.

References

• K. Schmüdgen, Unbounded Self-adjoint Operators on Hilbert Space, Graduate Texts in Mathematics 265, Springer, 2012, publisher edition — perturbations, domains, and self-adjointness criteria.
• G. Teschl, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, Theorem 6.4 and Sections 10.1–10.2, author’s text — Kato–Rellich and Coulomb Hamiltonians.