Mathematical Open Problems: Choose a Precise Statement

A mathematical quantum-research problem becomes actionable when every quantifier, operator domain and limit is explicit. The right graduate entry is usually a restricted lemma, a counterexample or a quantitative improvement within a known framework. A famous conjecture supplies motivation; it does not by itself define a feasible project.

Helpful background. Lieb–Robinson bounds illustrates how locality, norm and constants enter a usable theorem.

Choose the mathematical obstruction

Obstruction	Topic assessment	Bounded contribution
Finite-size evidence does not control an infinite limit	Many-body spectral gaps	A uniform estimate or a verified finite-size criterion
A dynamical generator has unproved domain or positivity properties	Open-system rigor	A theorem for an explicit generator class
Proposed reduced states may be incompatible	Quantum marginals	A new compatibility certificate or counterexample
An asymptotic scattering statement lacks a hypothesis	Mathematical scattering	A restricted completeness or decay result
A computational claim confuses preparation complexity and hardness	Quantum complexity	A parameter-preserving reduction or separation

Established: undecidability results obstruct a universal spectral-gap algorithm for sufficiently broad Hamiltonian families. They do not make every concrete gap problem unanswerable. Active: restricted families with additional geometry, symmetry or positivity. Conjectural: extrapolations from a finite sequence when no uniform theorem controls the limit. Primary undecidability result.

Write the quantifiers before choosing a technique

Compare “for each finite size there exists a positive gap” with “there exists a positive lower bound for all sufficiently large sizes.” The latter is the uniform statement needed for a gapped thermodynamic family. Specify boundary conditions and whether the ground-state sector is degenerate.

Similarly, norm convergence, strong convergence and convergence of one observable are different conclusions. A theorem for bounded operators does not transfer unchanged to an unbounded Hamiltonian. Decide which topology and domain serve the physical application.

Entry project: a restricted proof and its failure boundary

Choose one theorem used by a linked assessment. Reprove a finite-dimensional or one-dimensional special case, retaining explicit constants. Then test whether one hypothesis can be weakened. Deliver a written proof, a dependency graph of lemmas and a small counterexample search. A laptop can assist with matrix examples; the central resource is sustained work with the relevant analysis or operator theory.

The baseline is the published theorem under unchanged hypotheses. Failure tests remove compactness, a uniform gap, finite range or positivity one at a time. Numerical positivity must be certified if it is used in a proof. A solver’s failure to find a counterexample establishes no theorem. Research begins with a new valid restriction, improved bound, or counterexample that identifies the sharp boundary of a known claim.

Novelty checks and entry references

Search the exact hypotheses and equivalent formulations, including corrections and later versions. For complexity-adjacent projects, the 2024 quantum-PCP status note is a concrete warning that a claimed implication may have been withdrawn. Status note.

Reviewed 11 October 2026; selected methodological anchors span 2015–2024. The topic owners contain more specific evidence. Update the project when a new theorem settles the exact restricted statement, not merely when a paper shares its title.

References

• T. S. Cubitt, D. Pérez-García and M. M. Wolf, “Undecidability of the spectral gap,” Nature 528, 207–211 (2015). Full version.
• A. Natarajan and C. Nirkhe, “The status of the quantum PCP conjecture (games version),” arXiv:2403.13084 (2024). Status note.