What Counts as Rigorous

A rigorous statement specifies objects and hypotheses clearly enough that its conclusion has a definite meaning, then supplies a proof or identifies the theorem that supplies it. Rigor does not require every theorem to be reproved from elementary axioms on every page. It requires that imported results, remaining assumptions, and the strength of the conclusion be visible.

A theorem has objects, hypotheses, and a conclusion

The sentence “Hermitian Hamiltonians give unitary evolution” leaves several questions open in infinite dimension. A precise version begins with a complex Hilbert space and a densely defined self-adjoint operator $H$. The spectral calculus then defines

$$U(t)=e^{-itH/\hbar},\qquad t\in\mathbb R,$$

and proves that $U$ is a strongly continuous unitary group. On $D(H)$ its strong derivative is $-iHU(t)\psi/\hbar$. The converse identification of a self-adjoint generator is Stone’s Theorem.

The domain, self-adjointness, topology, and restriction on differentiability are substantive parts of the claim. Removing them yields a stronger or ambiguous statement. A proof for bounded matrices does not establish the same assertion for an arbitrary symmetric differential expression.

Equality must identify where both sides exist

For bounded maps on one Hilbert space, equality of their action on a dense subspace extends by continuity. For unbounded maps, equality on a dense subspace need not identify the full operators. One must specify domains or show that the shared subspace is a core for the relevant closures.

A displayed formula can also be an equality of quadratic forms, distributions, kernels, or matrix elements. These are legitimate statements when labeled. They are not interchangeable. For example, a finite spectral mean can exist on a form domain even when $H\psi$ is not a Hilbert-space vector; interpreting the mean as a literal inner product $\langle\psi,H\psi\rangle$ would then conceal a domain issue.

The Operator Theory Minimum and Measure Theory Minimum supply elementary tests of these distinctions.

A proof sketch must expose the missing work

A proof sketch may explain the main reduction, establish a representative calculation, and invoke a substantial construction theorem. It should say which step is imported. For example, “use the spectral theorem” can be an honest proof step after the operator has been shown self-adjoint and the relevant function’s domain has been checked. It is circular if spectral properties are then used to establish that same unproved self-adjointness.

A useful proof sketch distinguishes an example from the general mechanism. Checking the free-particle Fourier transform illustrates spectral calculus; it does not prove the spectral theorem for every self-adjoint operator. Conversely, a general existence theorem does not compute the spectrum of a particular interacting Hamiltonian. Hall and Teschl provide both theorem statements and model calculations in which this difference can be traced explicitly.

Approximations can be rigorous

An approximation becomes a theorem when the assumptions, norm, parameter regime, and error are controlled. For bounded self-adjoint $H$ and $K$, write $U_H(t)=e^{-itH/\hbar}$ and define $U_K$ similarly. The Duhamel identity gives, for $t\ge0$,

$$\begin{gathered} U_H(t)-U_K(t) =-\frac{i}{\hbar}\int_0^t U_H(t-s)(H-K)U_K(s)\,ds, \\ \|U_H(t)-U_K(t)\| \le\frac{t}{\hbar}\|H-K\|. \end{gathered}$$

Differentiate the product $e^{-i(t-s)H/\hbar}e^{-isK/\hbar}$ with respect to $s$ and integrate to obtain the identity. Unitarity bounds both exponential factors by one. The conclusion is uniform over unit vectors but grows with time. It is therefore useful only when the displayed bound is sufficiently small for the task.

For unbounded Hamiltonians, a formally identical calculation needs domain and integration justification; boundedness of the difference can provide one route, but arbitrary potentials require more care. A perturbative series without a remainder estimate can still be a valuable formal approximation, provided it is described as such.

Mathematical validity and physical adequacy are separate questions

A theorem can be correct for a model whose assumptions fail in an experiment. A perfectly self-adjoint one-particle Hamiltonian does not prove that a one-particle description is adequate at all energies. Similarly, a no-go theorem excludes models satisfying specified assumptions, not every imaginable interpretation of quantum theory.

Conventions, such as the sign of a Fourier transform, must be consistent but are not empirical laws. Model assumptions, such as a Markov approximation, need physical justification. Theorem hypotheses determine a logical implication. Keeping these categories separate is part of a rigorous explanation.

Exercise: identify the quantifiers in an error estimate

Suppose $\|(U_\epsilon(t)-U(t))\psi\|\le C_\psi\epsilon$ for each fixed $\psi$ and each fixed $t$, with unspecified dependence on both. Does this prove an error bound uniform over normalized states and all times?

Solution

No. Uniformity over states would require control of the constants on the unit sphere, and uniformity in time requires a specified time range and common bound. A fixed-state, fixed-time result is meaningful, but it cannot silently be promoted to operator-norm or long-time convergence. The quantifiers determine the strength of the approximation theorem.

References

• Hall, Brian C., Quantum Theory for Mathematicians, Graduate Texts in Mathematics 267, Springer, 2013, doi:10.1007/978-1-4614-7116-5 — precise spectral statements and their quantum applications.
• Teschl, Gerald, Mathematical Methods in Quantum Mechanics: With Applications to Schrödinger Operators, 2nd ed., Graduate Studies in Mathematics 157, American Mathematical Society, 2014, author’s edition — proof structure, operator domains, and controlled perturbations.