Rigol, Dunjko and Olshanii's 2008 demonstration that a generic isolated quantum many-body system relaxes to the standard statistical-mechanical prediction — with thermalization occurring at the level of individual eigenstates, as conjectured by Deutsch and Srednicki — explicitly posed the foundations of the subject as open. The open problems raised by that letter are the absence of a general proof of the eigenstate thermalization hypothesis (ETH) for generic interacting systems, with arguments existing only for restricted classes; the existence of a quantum analogue of the Kolmogorov-Arnold-Moser (KAM) threshold separating thermalizing from non-thermalizing regimes under integrability breaking; the identification of the conserved quantities that the asymptotic statistical description must retain (completeness of generalized Gibbs ensembles); together with the microscopic origin of irreversibility, finite-size ensemble validity, and the engineering of 'unthermalizable' states. Resolving the three load-bearing cores (a)-(c) below — and illuminating the auxiliary facets they subsume — is the objective of this task.

Background

An isolated quantum many-body system evolves unitarily under a linear, time-independent Schrödinger equation with a discrete spectrum, so classical dynamical chaos cannot be the mechanism by which such a system forgets its initial conditions. Thermalization here means the relaxation of expectation values of macroscopic (few-body) observables to stationary values that are universal across widely differing initial conditions and reproducible by a statistical-mechanical ensemble. Experiments such as the one-dimensional Bose gas 'quantum Newton's cradle' of Kinoshita, Wenger and Weiss showed that when the dynamics is integrable — possessing, beyond the energy, an extensive set of conserved quantities such as the occupations of single-particle momentum modes — this relaxation fails. For integrable systems, Rigol, Dunjko, Yurovsky and Olshanii proposed the generalized Gibbs ensemble (GGE): a density matrix ρ^GGE∝exp⁡(−∑kλkI^k)\hat{\rho}_{\mathrm{GGE}}\propto\exp(-\sum_k\lambda_k\hat{I}_k) built from the conserved quantities I^k\hat{I}_k, generalizing the canonical and microcanonical (fixed-energy) ensembles in which only the energy appears.

For generic (nonintegrable) isolated systems, Rigol, Dunjko and Olshanii demonstrated in 2008 that relaxation does reach the standard statistical-mechanical prediction, and identified the mechanism as eigenstate thermalization: each individual many-body eigenstate already carries the thermal information. This is the eigenstate thermalization hypothesis (ETH), conjectured by Deutsch and by Srednicki in the early 1990s: for a few-body observable A^\hat{A}, the diagonal matrix elements AααA_{\alpha\alpha} in the energy eigenbasis are smooth functions of the eigenenergy EαE_\alpha equal to the microcanonical average, while the off-diagonal elements AαβA_{\alpha\beta} are exponentially small in the system size. If ETH holds, a single generic eigenstate suffices to compute thermal averages, and time evolution plays only an auxiliary role: relaxation is dephasing over the initial state's spread on eigenstates that all agree on AααA_{\alpha\alpha}.

The classical analogue of the boundary question is governed by the Kolmogorov-Arnold-Moser (KAM) theorem: a small perturbation of an integrable classical system leaves most invariant tori of quasiperiodic motion intact, and chaos invades phase space only gradually as the perturbation grows, with a well-defined threshold in finite-dimensional systems. Whether quantum many-body systems possess an analogous 'KAM-like threshold' — a sharp boundary, in the thermodynamic limit, between a regime in which an extensive set of (possibly deformed) conserved quantities survives integrability breaking and the asymptotic state is GGE-like, and a thermalizing regime — was posed as open by the 2008 letter. The same letter asked which conserved quantities a statistical description must retain (a GGE with few integrals versus the exponentially many eigenstate projectors that determine the exact asymptotic 'diagonal ensemble' (the dephased stationary state built from the initial state's weights on the eigenbasis)), and listed as further open problems the microscopic origin of irreversibility within linear, time-independent eigenstate dynamics, finite-size effects in canonical versus microcanonical predictions at experimentally realistic sizes, and the engineering of 'unthermalizable' states of matter.

Subsequent work mapped robust classes in which thermalization fails: many-body localization (MBL), in which strong quenched (time-independent random) disorder localizes all eigenstates and blocks transport; quantum many-body scars, special nonthermal eigenstates embedded in otherwise thermal spectra; and Hilbert-space fragmentation, in which kinetic constraints split the state space into dynamically disconnected components. These classes delimit, but do not answer, the question of what happens for generic clean interacting systems.

Problem Statement

The problem posed by Rigol, Dunjko and Olshanii's 2008 letter is to complete the foundations of thermalization for isolated quantum many-body systems with generic interactions. Its load-bearing cores are:

(a) ETH validity. Prove or refute the eigenstate thermalization hypothesis for generic locally interacting, nonintegrable many-body Hamiltonians in the thermodynamic limit — every eigenstate in a microcanonical energy window reproduces the microcanonical expectation values and fluctuations of all few-body observables, while the off-diagonal matrix elements AαβA_{\alpha\beta} between distinct eigenstates are exponentially small in the system size. Here 'generic' denotes a clean, nonintegrable Hamiltonian with no fine-tuned couplings, no additional conserved densities beyond the energy, and no fine-tuned kinetic constraint or symmetry — the absence of the fine-tuning, disorder or constraints that define the known non-thermalizing classes. Only a result that decides this class settles this part: a proof that ETH holds for every Hamiltonian in the class, or an explicit Hamiltonian in the class for which ETH provably fails, resolves (a), whereas a proof for a structured subclass or a sufficient condition does not. Establishing the precise structural conditions on Hamiltonians under which ETH holds and fails is a distinct, additional deliverable — a classification of the boundary of the hypothesis — and is not an alternative completion standard for the generic statement. Existing arguments cover only restricted classes: random-matrix perturbations of integrable Hamiltonians, semiclassical chaotic limits, typical translation-invariant Hamiltonians with all-to-all interactions, correlation-clustering assumptions, or specific solvable models.

(b) Quantum KAM threshold. Determine whether a quantum analogue of the Kolmogorov-Arnold-Moser threshold exists: as the strength of an integrability-breaking perturbation is varied, is there a sharp boundary — sharp in the thermodynamic limit — between a regime in which an extensive set of (possibly deformed) conserved quantities survives and the asymptotic state is a generalized Gibbs ensemble, and a thermalizing regime; or is there only a crossover, with the thermalization time diverging smoothly as the perturbation vanishes?

(c) Conserved quantities of the asymptotic state. For systems in which thermalization fails or is only delayed, identify the complete set of conserved (or quasi-conserved) quantities that the asymptotic statistical description must retain — when a generalized Gibbs ensemble built from a limited set of (quasi)local integrals is complete, when exponentially many eigenstate projectors are in principle required, and whether deformed, approximately conserved charges can define genuine steady states under weak integrability breaking rather than merely long-lived transient ones.

These three questions jointly delineate the conditions for, and the mechanism of, thermalization in generic isolated quantum systems; the answer should make explicit how the three parts connect (the conserved-quantity structure of (c) is what survives across the boundary asked for in (b), and its trivial collapse to energy alone is necessary for, but not sufficient to guarantee, (a)). The source's auxiliary open problems — the microscopic origin of macroscopic irreversibility within linear, time-independent eigenstate dynamics; the finite-size domain in which canonical and microcanonical ensembles agree at experimentally realistic sizes; and the systematic engineering of 'unthermalizable' many-body states — are facets of the same objective that a completed theory should illuminate, but the completion boundary is set by (a)-(c).

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