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The multidimensional L² conjecture for Schrödinger operators
Show that in two or more dimensions, −Δ + V keeps a.c. spectrum of infinite multiplicity on [0, ∞) whenever V² |x|^(1−d) is integrable.
State the problem, from Simon's problems in mathematical physics
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| 1 | +# The multidimensional L² conjecture for Schrödinger operators | |
| 2 | + | |
| 3 | +In one dimension, a square-integrable potential cannot destroy the absolutely continuous (a.c.) spectrum of a Schrödinger operator, the part carried by scattering states. Simon conjectured the analogue in higher dimensions, with $V^2$ weighted by $|x|^{1-d}$. It is open even for potentials known only to decay like $|x|^{-\gamma}$ with $1/2 < \gamma \le 1$. | |
| 4 | + | |
| 5 | +## Problem | |
| 6 | + | |
| 7 | +Let $d \ge 2$ and let $V$ be a bounded real function on $\mathbb{R}^d$ with | |
| 8 | + | |
| 9 | +$$ | |
| 10 | +\int_{\mathbb{R}^d} \frac{V(x)^2}{1 + |x|^{d-1}} \, dx < \infty . | |
| 11 | +$$ | |
| 12 | + | |
| 13 | +**Conjecture (Problem 8).** $H = -\Delta + V$ on $L^2(\mathbb{R}^d)$ has a.c. spectrum of infinite multiplicity on $[0, \infty)$. As for the free Laplacian, the multiplicity of its a.c. spectrum is infinite at almost every energy $E \ge 0$. | |
| 14 | + | |
| 15 | +Simon wrote the weight as $|x|^{1-d}$, which is equivalent for bounded $V$. Boundedness, or some local regularity, is needed only to define $H$ [4]. Any $V$ with $|V(x)| \le C(1+|x|)^{-\gamma}$ and $\gamma > 1/2$ qualifies. | |
| 16 | + | |
| 17 | +## Background | |
| 18 | + | |
| 19 | +This is Problem 8 on Barry Simon's 2000 list. In one dimension, Deift and Killip proved that for $V \in L^2(\mathbb{R})$ the a.c. spectrum of $-d^2/dx^2 + V$ fills $[0,\infty)$ [2]. Simon noted that this gives the conjecture for spherically symmetric $V$ in any dimension, which explains the weight [1]. Classical scattering theory makes the positive spectrum purely a.c. in two cases: short-range $V$ with $\gamma > 1$ (Kato, Agmon), and long-range $V$ with $\gamma > 1/2$ whose derivatives also decay (Hörmander) [5]. The conjecture is about rough potentials, where these tools fail and singular spectrum can coexist with a.c. spectrum [5]. | |
| 20 | + | |
| 21 | +## What is known | |
| 22 | + | |
| 23 | +- With only $|V(x)| \le C(1+|x|)^{-\gamma}$ and $1/2 < \gamma \le 1$, it is not known whether any a.c. spectrum exists (Denisov, 2021) [5]. | |
| 24 | +- In three dimensions the a.c. spectrum fills $[0,\infty)$ in two cases. The first is oscillating $V = \operatorname{div} Q$ with $|Q|$ and $|\operatorname{div} Q|$ at most $C(1+|x|)^{-1/2-\varepsilon}$ (Denisov, 2004) [3]. The second is when $|V|$ and $|x|$ times the angular gradient of $V$ are $O(|x|^{-1/2-\varepsilon})$ (Perelman, 2005) [4]. | |
| 25 | +- The method of Laptev, Naboko and Safronov (2005) gives a.c. spectrum $[0,\infty)$ for $d \ge 3$ under decay conditions plus square integrability of the Fourier transform of $V$ near the origin, which forces $V$ to decay fast or oscillate [4]. Safronov showed that the a.c. spectrum of $-\Delta + \alpha V$ fills $[0,\infty)$ for almost every $\alpha$ if $\int |\nabla W|^2 \, |x|^{1-d} \, dx < \infty$, where $W(r,\theta) = \int_0^r V(\rho,\theta)\, d\rho$ [6]. | |
| 26 | +- On a rooted binary tree the analogue holds. For bounded $V$ with $\sum_n 2^{-n} \sum_{|X| = n} V(X)^2 < \infty$, the a.c. spectrum contains the free spectrum; $2^n$ plays the role of $|x|^{d-1}$ [4]. | |
| 27 | +- In 2012 Denisov judged that, with the asymptotic (WKB-type) methods then available, the prognosis for the full conjecture was "rather negative" [4]. | |
| 28 | + | |
| 29 | +## What would count as progress | |
| 30 | + | |
| 31 | +- In some dimension $d \ge 2$, a proof of some a.c. spectrum for every bounded $V$ with $|V(x)| \le C(1+|x|)^{-\gamma}$, for a fixed $\gamma \in (1/2, 1]$, without oscillation or regularity assumptions. | |
| 32 | +- The full conjecture in a single dimension, for example $d = 3$. | |
| 33 | +- A counterexample: a bounded $V$ satisfying the condition, with no a.c. spectrum on some interval of positive energies. | |
| 34 | + | |
| 35 | +## References | |
| 36 | + | |
| 37 | +1. B. Simon, "Schrödinger operators in the twenty-first century", in *Mathematical Physics 2000*, Imperial College Press, 2000, pp. 283–288. [doi:10.1142/9781848160224_0014](https://doi.org/10.1142/9781848160224_0014) | |
| 38 | +2. P. Deift and R. Killip, "On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials", *Comm. Math. Phys.* 203 (1999), 341–347. [doi:10.1007/s002200050615](https://doi.org/10.1007/s002200050615) | |
| 39 | +3. S. A. Denisov, "Absolutely continuous spectrum of multidimensional Schrödinger operator", *Int. Math. Res. Not.* 2004, no. 74, 3963–3982. [arXiv:math/0408376](https://arxiv.org/abs/math/0408376) | |
| 40 | +4. S. A. Denisov, "Multidimensional L² conjecture: a survey", in *Recent Trends in Analysis*, Theta Ser. Adv. Math. 16, Theta Foundation, 2013, pp. 101–112. [arXiv:1203.4255](https://arxiv.org/abs/1203.4255) | |
| 41 | +5. S. A. Denisov, "Subharmonic functions in scattering theory", *C. R. Math. Acad. Sci. Paris* 359 (2021), 757–762. [doi:10.5802/crmath.217](https://doi.org/10.5802/crmath.217) | |
| 42 | +6. O. Safronov, "Absolutely continuous spectrum of a typical Schrödinger operator with a slowly decaying potential", preprint, 2011. [arXiv:1111.5552](https://arxiv.org/abs/1111.5552) | |
| 43 | + | |
| 44 | +## Source | |
| 45 | + | |
| 46 | +Imported from [Simon's problems in mathematical physics 2000-8](https://en.wikipedia.org/wiki/Simon_problems#The_2000_list), curated by Barry Simon. Open according to the source as of 2026-10-20. The statement above is written in our own words; see the source for the original wording, history and discussion. | |