The multidimensional L² conjecture for Schrödinger operators

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 $1/2 < \gamma \le 1$.

Problem

Let $d \ge 2$ and let $V$ be a bounded real function on $\mathbb{R}^d$ with

$$\int_{\mathbb{R}^d} \frac{V(x)^2}{1 + |x|^{d-1}} \, dx < \infty .$$

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 $E \ge 0$.

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.

Background

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].

What is known

• 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].
• 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].
• 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].
• 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].
• In 2012 Denisov judged that, with the asymptotic (WKB-type) methods then available, the prognosis for the full conjecture was "rather negative" [4].

What would count as progress

• 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.
• The full conjecture in a single dimension, for example $d = 3$.
• A counterexample: a bounded $V$ satisfying the condition, with no a.c. spectrum on some interval of positive energies.

References

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
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
3. S. A. Denisov, "Absolutely continuous spectrum of multidimensional Schrödinger operator", Int. Math. Res. Not. 2004, no. 74, 3963–3982. arXiv:math/0408376
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
5. S. A. Denisov, "Subharmonic functions in scattering theory", C. R. Math. Acad. Sci. Paris 359 (2021), 757–762. doi:10.5802/crmath.217
6. O. Safronov, "Absolutely continuous spectrum of a typical Schrödinger operator with a slowly decaying potential", preprint, 2011. arXiv:1111.5552

Source

Imported from Simon's problems in mathematical physics 2000-8, 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.