Extending Bourgain–Klein's log-Hölder bounds on the density of states outer-measure to continuum Schrödinger operators in dimensions $d\ge 4$

Bourgain and Klein (Invent. Math. 2013) proved log-Hölder continuity of the density of states outer-measure $\eta^*$ for continuum Schrödinger operators $H=-\Delta+V$ with real bounded $V$ in dimensions $d=1,2,3$ — $\eta^*([E,E+\varepsilon])\le C_{d,V_\infty,E_0}(\log 1/\varepsilon)^{-\kappa_d}$ with $\kappa_1=1$, $\kappa_2=1/4$, $\kappa_3=1/8$ — and in all dimensions for discrete operators. The restriction to $d\le 3$ comes from the exponent $Q^{4/3}$ in their quantitative unique continuation estimate (Theorem 3.4): a version with exponent $Q^\beta$ would give dimensions $d<\beta/(\beta-1)$, and they expected $\beta=1+$ for real potentials, which would yield all dimensions with $\kappa_d\approx 1/2-$. The $d\ge 4$ extension is open: no proof or counterexample is known in the literature. The conjectured route is now refuted in $d\ge 3$: Frank and Ivanisvili (2026) constructed real bounded potentials in every $n\ge 3$ with nontrivial solutions of $-\Delta u+Vu=0$ on $\mathbb{R}^n$ decaying like $e^{-c_n|x|^{4/3}}$, so no quantitative unique continuation estimate with exponent $\beta<4/3$ holds for real potentials and the exponent $4/3$ is optimal. Any resolution in $d\ge 4$ — a proof of log-Hölder bounds, which would in particular settle the $d\ge 4$ cases of Simon's Problem 14 on continuity of the integrated density of states, or a counterexample — requires a new mechanism.

Background

Continuum Schrödinger operators $H=-\Delta+V$ on $L^2(\mathbb{R}^d)$ with real bounded potentials describe quantum particles in disordered media. For ergodic $V$ (statistically homogeneous under translations, e.g. periodic or random), the integrated density of states (IDS) $N(E)$ counts eigenvalues of the infinite-volume operator per unit volume below energy $E$; it is the distribution function of the density of states (DOS) measure $\eta$. Whether $N$ is continuous — equivalently, whether a fixed energy can carry extensive per-volume eigenvalue mass — is a basic regularity question: open since the mid-1980s for multi-dimensional continuum ergodic Schrödinger operators with merely bounded potentials, where it is Problem 14 in Simon's 2000 problem list. For random alloy potentials — potentials built as sums of randomly translated copies of a single-site function — whose single-site distribution is sufficiently regular, continuity follows from Wegner estimates, i.e. uniform bounds on the expected eigenvalue count of the finite-volume operator in small energy windows obtained by spectral averaging (Combes–Hislop–Klopp proved the IDS has as much regularity as the concentration function of the single-site distribution); Germinet–Klein extended log-Hölder continuity to arbitrary single-site distributions, including Bernoulli, but only in the region of applicability of the multiscale analysis, the iterative scheme that establishes localization on a growing sequence of scales. These random-model methods fail for Bernoulli-type disorder outside the localization regime and say nothing for arbitrary deterministic $V$.

A stronger, quantitative statement is log-Hölder continuity: $\eta([E,E+\varepsilon])\le C\,(\log 1/\varepsilon)^{-1}$. It was proved in the early 1980s for one-dimensional continuum ergodic Schrödinger operators (Craig–Simon, via subharmonicity of the Lyapunov index, the asymptotic exponential growth rate of solutions of the one-dimensional equation) and for ergodic discrete Schrödinger and Jacobi operators in every dimension (Craig–Simon, Delyon–Souillard), but the multi-dimensional continuum case with arbitrary bounded $V$ resisted all Wegner-free methods. Since for general $V$ the infinite-volume DOS measure need not exist, Bourgain and Klein introduced the density of states outer-measure: for a box $\Lambda_L(x)=x+(-L/2,L/2)^d$ and boundary condition $\wp\in\{D,N,P\}$ (Dirichlet, Neumann, periodic) set $\eta_{\Lambda,\wp}(B)=|\Lambda|^{-1}\,\mathrm{tr}\,\chi_B(H_{\Lambda,\wp})$ and $\eta_\wp^*(B)=\limsup_{L\to\infty}\sup_{x\in\mathbb{R}^d}\eta_{\Lambda_L(x),\wp}(B)$. This is always defined and dominates the DOS measure when the latter exists, so its continuity implies continuity of the IDS; Bourgain–Klein show the choices of $\wp$ agree on intervals under continuity, so one writes $\eta^*$. Their Theorem 1.1 states that for $d=1,2,3$, $\eta^*([E,E+\varepsilon])\le C_{d,V_\infty,E_0}(\log 1/\varepsilon)^{-\kappa_d}$ with $\kappa_1=1$, $\kappa_2=1/4$, $\kappa_3=1/8$; their Theorem 1.2 gives the discrete analogue in every dimension with $\kappa=1$.

The mechanism is quantitative unique continuation (QUC): a propagation-of-smallness bound for real solutions of $(-\Delta+V)\psi=\zeta$ with $\|V\|_\infty\le K$, controlling the mass of $\psi$ on a bounded set $\Theta$ by its mass on a small ball at distance $Q$ from $\Theta$, at multiplicative cost roughly $(1+K^3)^{C Q^{4/3}}$ (Theorem 3.4, inequality (3.62) of the source — a parameter-explicit form of Bourgain–Kenig's estimate for the continuous Anderson–Bernoulli model, the continuum alloy model with two-valued single-site disorder; the proof follows Germinet–Klein and rests on a Carleman-type inequality, a weighted integral estimate for elliptic operators that is the standard tool for quantitative unique continuation). The exponent $4/3$ cannot be improved for complex potentials: Meshkov constructed a bounded complex potential with a nontrivial solution decaying like $e^{-c|x|^{4/3}}$. In the DOS argument, eigenfunctions in a narrow energy window must each carry their mass, and a multi-center counting (packing) argument bounds how many of them can avoid a fixed set of centers; the resulting bound converges precisely in dimensions $d<\beta/(\beta-1)$ when the QUC cost is $Q^\beta$. With $\beta=4/3$ this is $d<4$.

The authors state this arithmetic explicitly and conjecture that 'something like Theorem 3.4 holds with $\beta=1+$' for real potentials — a quantitative form of Landis' problem on unique continuation at infinity for real potentials (can a solution of the stationary Schrödinger equation with bounded potential decay much faster than exponentially at infinity?) — predicting Theorem 1.1 in all dimensions with $\kappa_d=(\beta-d(\beta-1))/(2\beta)\approx 1/2-$ for $d\ge 2$. The subsequent literature: the bounds were extended to singular real potentials but still only for $d\le 3$ (Klein–Tsang); the real Landis problem was resolved positively in the plane (Kenig–Silvestre–Wang; Logunov–Malinnikova–Nadirashvili–Nazarov), but in 2026 Frank and Ivanisvili constructed, for every $n\ge 3$, real bounded potentials with nontrivial solutions of $-\Delta u+Vu=0$ on all of $\mathbb{R}^n$ decaying like $e^{-c|x|^{4/3}}$. This refutes any QUC estimate with $\beta<4/3$ for real potentials in $d\ge 3$ and closes the conjectured route to $d\ge 4$; whether the $d\ge 4$ DOS outer-measure bounds hold by some other mechanism, or fail, is open.

Problem Statement

Let $d\ge 4$, let $V\in L^\infty(\mathbb{R}^d)$ be real-valued with $\|V\|_\infty=V_\infty<\infty$, and let $H=-\Delta+V$ on $L^2(\mathbb{R}^d)$. For a box $\Lambda_L(x)=x+(-L/2,L/2)^d$ with boundary condition $\wp\in\{D,N,P\}$ (Dirichlet, Neumann, or periodic) write $H_{\Lambda,\wp}$ for the corresponding self-adjoint restriction and $\eta_{\Lambda,\wp}(B)=|\Lambda|^{-1}\,\mathrm{tr}\,\chi_B(H_{\Lambda,\wp})$ for Borel $B\subset\mathbb{R}$, and define the density of states outer-measure $\eta_\wp^*(B)=\limsup_{L\to\infty}\sup_{x\in\mathbb{R}^d}\eta_{\Lambda_L(x),\wp}(B)$. Decide whether there exist, for each $d\ge 4$ (in particular for $d=4$), constants $\kappa_d>0$ such that for every $E_0$, all $E\le E_0$ and all $0<\varepsilon\le 1/2$, $\eta_\wp^*([E,E+\varepsilon])\le C_{d,V_\infty,E_0}(\log 1/\varepsilon)^{-\kappa_d}$ for $\wp=D,N,P$; a weaker related question is whether $\eta^*$ is continuous, i.e. whether $\eta^*([E-\varepsilon,E+\varepsilon])\to 0$ as $\varepsilon\downarrow 0$. For ergodic $V$ the density of states measure exists and is dominated by the outer-measure, so a positive answer implies continuity of the integrated density of states and settles the $d\ge 4$ cases of Simon's Problem 14 for continuum Schrödinger operators with bounded potentials. A negative answer is a bounded real potential in some $d\ge 4$ for which $\eta^*$ is discontinuous or admits no log-Hölder bound of the above form. The question stands with the single-solution route closed: by the Frank–Ivanisvili counterexamples, no quantitative unique continuation estimate of the Bourgain–Kenig type — a bound controlling the mass of a real solution of $(-\Delta+V)\psi=\zeta$ with $\|V\|_\infty\le K$ on a bounded set by its mass on a small ball at distance $Q$, at multiplicative cost $(1+K^3)^{C Q^\beta}$ — can hold with exponent $\beta<4/3$ for real potentials in any $d\ge 3$, so a proof cannot proceed through individual-solution decay bounds of that form; the conjectured exponents $\kappa_d\approx 1/2-$ are unsupported.

References

1. Jean Bourgain and Abel Klein, 'Bounds on the density of states for Schrödinger operators', Inventiones mathematicae 194 (2013), 41–72. DOI: 10.1007/s00222-012-0440-1, https://doi.org/10.1007/s00222-012-0440-1; arXiv:1112.1716, https://arxiv.org/abs/1112.1716. The source paper; contains Theorems 1.1, 1.2, 3.4, 3.7 and inequality (3.62).
2. Jean Bourgain and Carlos E. Kenig, 'On localization in the continuous Anderson–Bernoulli model in higher dimension', Inventiones mathematicae 161 (2005), 389–426. DOI: 10.1007/s00222-004-0435-7, https://doi.org/10.1007/s00222-004-0435-7. Origin of the $Q^{4/3}$-type quantitative unique continuation estimate (Lemma 3.10 there) that the source paper states explicitly as its Theorem 3.4.
3. Abel Klein and C. S. Sidney Tsang, 'Local behavior of solutions of the stationary Schrödinger equation with singular potentials and bounds on the density of states of Schrödinger operators', Communications in Partial Differential Equations 41 (2016), 1040–1055. DOI: 10.1080/03605302.2016.1179316, https://doi.org/10.1080/03605302.2016.1179316; arXiv:1408.7111, https://arxiv.org/abs/1408.7111. Extends the log-Hölder density of states outer-measure bounds to singular real potentials, still only in $d=1,2,3$.
4. Abel Klein and C. S. Sidney Tsang, 'Quantitative unique continuation principle for Schrödinger operators with singular potentials', Proceedings of the American Mathematical Society 144 (2016), 665–679. DOI: 10.1090/proc12734, https://doi.org/10.1090/proc12734; arXiv:1408.1992, https://arxiv.org/abs/1408.1992. Its Theorem 1.1 recovers the source's Theorem 3.4 with the same exponent $4/3$ in the bounded-potential limit $p\to\infty$.
5. Rupert L. Frank and Paata Ivanisvili, 'Counterexamples to the Landis conjecture in dimensions three and higher', preprint (August 2026). arXiv:2608.00802, https://arxiv.org/abs/2608.00802. Theorem 1 constructs, for every $n\ge 3$, a nonzero real $u\in C^\infty(\mathbb{R}^n)$ and a real $V\in C^\infty(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)$ with $-\Delta u+Vu=0$ on $\mathbb{R}^n$ and $|u(x)|\le C_n e^{-c_n|x|^{4/3}}$ for all $x$.
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7. Carlos E. Kenig, Luis Silvestre, and Jenn-Nan Wang, 'On Landis' conjecture in the plane', Communications in Partial Differential Equations 40 (2015), 766–789. DOI: 10.1080/03605302.2014.978015, https://doi.org/10.1080/03605302.2014.978015; arXiv:1404.2496, https://arxiv.org/abs/1404.2496. Quantitative Landis theorem for real potentials in two dimensions.
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9. V. Z. Meshkov, 'On the possible rate of decay at infinity of solutions of second order partial differential equations', Mathematics of the USSR-Sbornik 72 (1992), 343–361. DOI: 10.1070/sm1992v072n02abeh001414, https://doi.org/10.1070/sm1992v072n02abeh001414. Bounded complex potential with a nontrivial solution decaying like $e^{-c|x|^{4/3}}$, proving the exponent $4/3$ sharp for complex potentials.
10. Aingeru Fernández-Bertolin, Luz Roncal, and Diana Stan, 'Landis' conjecture: a survey', RIMS Kôkyûroku Bessatsu B99 (2026), 35–78. DOI: 10.17983/301255, https://doi.org/10.17983/301255; arXiv:2412.00788, https://arxiv.org/abs/2412.00788 (preprint December 2024). Records that the real-potential Landis problem in dimensions three and higher was open as of December 2024.
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12. Abel Klein, 'Unique continuation principle for spectral projections of Schrödinger operators and optimal Wegner estimates for non-ergodic random Schrödinger operators', Communications in Mathematical Physics 323 (2013), 1229–1246. DOI: 10.1007/s00220-013-1795-x, https://doi.org/10.1007/s00220-013-1795-x; arXiv:1209.4863, https://arxiv.org/abs/1209.4863. Scale-free unique continuation for spectral projectors.
13. Barry Simon, 'Schrödinger operators in the twenty-first century', in Mathematical Physics 2000, Imperial College Press, London (2000), 283–288. DOI: 10.1142/9781848160224_0014, https://doi.org/10.1142/9781848160224_0014. Source of Problem 14 (continuity of the integrated density of states for multi-dimensional continuum ergodic Schrödinger operators), called there a 15-year-old open problem.
14. Jean-Michel Combes, Peter D. Hislop, and Frédéric Klopp, 'An optimal Wegner estimate and its application to the global continuity of the integrated density of states for random Schrödinger operators', Duke Mathematical Journal 140 (2007), 469–498. DOI: 10.1215/s0012-7094-07-14032-8, https://doi.org/10.1215/s0012-7094-07-14032-8. The Wegner-estimate route to IDS continuity for random operators.
15. Peter D. Hislop and Christoph A. Marx, 'Dependence of the density of states outer measure on the potential for deterministic Schrödinger operators on graphs with applications to ergodic and random models', Journal of Functional Analysis 281 (2021), article 109186. DOI: 10.1016/j.jfa.2021.109186, https://doi.org/10.1016/j.jfa.2021.109186; arXiv:2006.15230, https://arxiv.org/abs/2006.15230. Regularity of the density of states outer-measure as a function of the potential — a different question from the energy-window bounds considered here.
16. Dhriti Ranjan Dolai, M. Krishna, and Anish Mallick, 'Regularity of the density of states of random Schrödinger operators', Communications in Mathematical Physics 378 (2020), 299–328. DOI: 10.1007/s00220-020-03740-1, https://doi.org/10.1007/s00220-020-03740-1; arXiv:1904.11854, https://arxiv.org/abs/1904.11854. Regularity of the density of states for continuum alloy models in the localization region, requiring partition-of-unity single-site structure.
17. François Germinet and Abel Klein, 'A comprehensive proof of localization for continuous Anderson models with singular random potentials', Journal of the European Mathematical Society 15 (2013), 53–143. DOI: 10.4171/jems/356, https://doi.org/10.4171/jems/356. Contains the quantitative unique continuation result (Theorem A.1 there) on which the source's proof of Theorem 3.4 is based, and proves log-Hölder IDS continuity for the continuum Anderson model with arbitrary single-site distribution in the localization region.
18. Walter Craig and Barry Simon, 'Subharmonicity of the Lyaponov index', Duke Mathematical Journal 50 (1983), 551–560. DOI: 10.1215/s0012-7094-83-05025-1, https://doi.org/10.1215/s0012-7094-83-05025-1. Implies log-Hölder continuity (exponent 1) of the IDS for one-dimensional continuum ergodic Schrödinger operators.
19. Walter Craig and Barry Simon, 'Log Hölder continuity of the integrated density of states for stochastic Jacobi matrices', Communications in Mathematical Physics 90 (1983), 207–218. DOI: 10.1007/bf01205503, https://doi.org/10.1007/bf01205503. Log-Hölder continuity of the IDS for ergodic discrete operators in all dimensions.
20. François Delyon and Bernard Souillard, 'Remark on the continuity of the density of states of ergodic finite difference operators', Communications in Mathematical Physics 94 (1984), 289–291. DOI: 10.1007/bf01209306, https://doi.org/10.1007/bf01209306. Elementary proof of density of states continuity in the discrete ergodic case.