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Quantitative observability for one-dimensional Schrödinger equations with potentials

This paper establishes quantitative observability with an explicit control cost for the one-dimensional Schrödinger equation with real-valued, bounded continuous potentials on thick sets by employing distinct techniques for low- and high-frequency estimates, thereby extending previous large-time observability results to arbitrary short times and generalizing spectral inequalities to bounded continuous potentials.

Original authors: Pei Su, Chenmin Sun, Xu Yuan

Published 2026-07-20

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CC BY 4.0

Original authors: Pei Su, Chenmin Sun, Xu Yuan

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, invisible ocean where tiny particles, like electrons, don't just sit still but ripple and dance. These ripples are described by a famous set of rules called the Schrödinger equation, which acts like a weather forecast for the quantum world. It tells us how these particles move, spread out, and interact with their surroundings. But here's the tricky part: in the real world, we can't see everything at once. We usually only have access to a small patch of this ocean, a specific "control region" where we can measure or influence the waves.

The big question scientists have been asking is: If we can only watch a small, perhaps messy or irregular patch of this quantum ocean, can we still figure out exactly what the entire wave looked like at the very beginning? It's like trying to guess the shape of a whole thunderstorm just by looking at a single, cloudy window. If we can do this, it means we have "observability." This isn't just a math puzzle; it's the key to "controllability." If we can see the whole picture from a small piece, we can also steer the system—like guiding a spaceship or stabilizing a quantum computer—using only that small patch of control. The challenge gets even harder when the "ocean" isn't empty but has hidden currents (potentials) that push the waves around, and when our window isn't a perfect square but a rough, scattered collection of spots.

This paper by Pei Su, Chenmin Sun, and Xu Yuan tackles exactly that messy, difficult scenario. They focus on a one-dimensional version of this quantum ocean (a single line) where the hidden currents are real, continuous, and bounded (they don't go to infinity). Their goal was to prove that even if the control region is "thick"—meaning it's not a solid block but a scattered set of intervals that appear frequently enough along the line—we can still reconstruct the entire initial state of the wave in any amount of time, no matter how short.

The authors prove a "quantitative observability" result. In plain English, this means they didn't just say "yes, it's possible"; they wrote down a specific formula showing exactly how much "effort" or "cost" is required to see the whole wave from the small patch. They found that this cost grows in a predictable way as the time you have to observe gets shorter. Specifically, they showed that the cost explodes like $e^{C/T^2}$ as the time $T$ approaches zero. This is a crucial detail because it answers a lingering question from previous research: does this work for very short times? The answer is a definitive yes.

To solve this, the authors had to split the problem into two distinct parts, like sorting a pile of mixed-up toys into "slow" and "fast" categories. The "low-frequency" parts of the wave are the slow, heavy swells that move gently. For these, the authors extended a known mathematical tool (a spectral inequality) to work with their rough, continuous potentials, proving that these slow waves can't hide in the dark spots of the control region. The "high-frequency" parts are the jittery, fast-moving ripples. For these, they used a different technique involving "resolvent estimates," which essentially measures how the wave reacts to being pushed. They showed that even these fast ripples cannot escape detection if the control region is "thick."

The most creative part of their proof is how they glued these two pieces together. Usually, mathematicians use a "compactness" argument to connect low and high frequencies, but that trick fails when the space is infinite (unbounded) like the line they are studying. Instead, the authors used a clever mathematical transformation called the FBI transform. You can think of this as a magical lens that turns the Schrödinger equation (the quantum wave) into a heat equation (the way heat spreads). Heat spreads in a very predictable way, and the authors used this to show that if a wave disappears in the control region, it must have been zero everywhere to begin with. This "unique continuation" property allowed them to stitch the low-frequency and high-frequency results into one complete proof.

The result is a rigorous, proven statement that for any 1D Schrödinger equation with a bounded, continuous potential, if your control region is "thick" (meaning it has a guaranteed minimum density of coverage everywhere), you can observe and control the system in any time $T > 0$ . They explicitly rule out the idea that you need a long time or a perfect, solid control region; a rough, scattered, but "thick" region is enough. The paper provides a concrete upper bound for the cost of this control, ensuring that while the cost gets very high for very short times, it remains finite and calculable. This work extends previous findings that were limited to empty space or specific types of potentials, showing that the robustness of quantum observation holds up even in more complex, realistic environments.

Technical Summary: Quantitative Observability for One-Dimensional Schrödinger Equations with Potentials

Problem Statement
This note addresses the quantitative observability of the one-dimensional Schrödinger equation on the real line $\mathbb{R}$ with a real-valued, bounded, and continuous potential $V(x)$ . The equation is given by:
$i\partial_t u - \partial_x^2 u + V(x)u = 0, \quad u|_{t=0} = u_0 \in L^2(\mathbb{R}).$
The central question is whether the initial state $u_0$ can be uniquely determined (and bounded in $L^2$ norm) by observing the solution $u(t)$ on a measurable subset $\Omega \subset \mathbb{R}$ over a time interval $(0, T)$ . Specifically, the authors seek to establish an inequality of the form:
$\|u_0\|_{L^2(\mathbb{R})}^2 \leq C(T, V, \Omega) \int_0^T \|u(t)\|_{L^2(\Omega)}^2 dt,$
where the control cost $C(T, V, \Omega)$ is explicitly quantified. The study focuses on "rough" control regions, specifically $(L, \zeta)$ -thick sets, defined as measurable sets satisfying $|\Omega \cap [x, x+L]| \geq \zeta L$ for all $x \in \mathbb{R}$ . While observability on compact manifolds and bounded domains is well-studied, the case of unbounded domains with general bounded potentials and arbitrary short observation times $T > 0$ remains less explored.

Methodology
The proof of the main theorem relies on a three-step strategy that combines spectral analysis, resolvent estimates, and a quantitative gluing argument:

Low-Frequency Spectral Inequality:
The authors establish a spectral inequality for the low-frequency part of the solution. This involves proving a propagation of smallness for solutions to a 2D elliptic equation derived from the Schrödinger operator.
- They utilize a gauge transformation to reduce the non-divergence form elliptic equation $-\Delta \phi + V(x)\phi = 0$ to a divergence form $\nabla \cdot (A(z)\nabla \tilde{\phi}) = 0$ .
- Leveraging a recent result by [38] on the propagation of smallness for elliptic equations on the plane, they derive an $L^2$ -propagation of smallness estimate.
- This leads to a spectral inequality (Lemma 3.8) of the form $\|\Pi_\mu f\|_{L^2} \leq C e^{3\mu} \|\Pi_\mu f\|_{L^2(\Omega)}$ , where $\Pi_\mu$ is the spectral projector associated with frequencies up to $\mu$ . This result extends the spectral inequality of Lebeau-Moyano [27] from real-analytic potentials to bounded continuous potentials.
High-Frequency Resolvent Estimate:
For the high-frequency part, the authors derive a resolvent estimate (Lemma 4.1) of the Hautus type:
$\|f\|_{L^2}^2 \leq \frac{C}{\mu} \|(H - \mu)f\|_{L^2}^2 + C \|f\|_{L^2(\Omega)}^2.$
While similar estimates exist for the free Schrödinger operator ( $V=0$ ) in [19], the authors provide a distinct, more elementary proof for the case with a potential $V \in L^\infty(\mathbb{R})$ . This proof utilizes a quantitative analysis of the Logvinenko-Sereda uncertainty principle, specifically estimating the integral of $\cos^2(\lambda(x-x_0))$ over thick sets (Lemma 4.2).
Quantitative Gluing via FBI Transformation:
The most delicate part of the analysis is combining the low-frequency and high-frequency estimates to obtain the full observability inequality. In non-compact settings where the operator does not have a compact resolvent, standard compactness-uniqueness arguments fail.
- The authors employ a quantitative unique continuation property derived from the observability of the associated heat semigroup.
- They introduce the Fourier-Bros-Iagolnitzer (FBI) transformation to map the 1D Schrödinger equation to a 1D heat equation.
- By establishing backward-in-time observability for the heat semigroup (Proposition 3.10) and applying the FBI transform, they derive a quantitative unique continuation estimate (Proposition 5.1) that links the Schrödinger solution to the heat solution.
- This allows them to "glue" the estimates, avoiding the need for compact resolvent arguments.

Key Results
The main result (Theorem 1.2) states that for any $(L, \zeta)$ -thick set $\Omega$ and any potential $V \in C(\mathbb{R}) \cap L^\infty(\mathbb{R})$ , there exists a constant $C = C(V, L, \zeta)$ such that for any $T > 0$ :
$\|u_0\|_{L^2(\mathbb{R})}^2 \leq C e^{\frac{C}{T^2}} \int_0^T \|u(t)\|_{L^2(\Omega)}^2 dt.$
Key features of this result include:

Arbitrary Short Time: The observability holds for any $T > 0$ , not just large times.
Explicit Control Cost: The blow-up rate of the control cost is explicitly given as $e^{C/T^2}$ .
General Potentials: The result applies to bounded continuous potentials, extending previous work that was limited to free equations or real-analytic potentials.

As a consequence of the Hilbert Uniqueness Method, the authors also prove the exact controllability of the associated inhomogeneous Schrödinger equation (Corollary 1.4).

Significance and Claims
The authors position this work as a significant extension of existing literature in several ways:

Extension of Huang-Wang-Wang [20]: They extend the large-time observability result for the 1D free Schrödinger equation to any short time $T > 0$ and generalize it to operators with bounded continuous potentials. This answers a specific question raised in [20] regarding short-time observability.
Extension of Lebeau-Moyano [27]: They extend the spectral inequality, previously established for real-analytic potentials, to the case of bounded continuous potentials in one dimension.
Non-Compact Setting: The paper addresses the observability problem on an unbounded domain ( $\mathbb{R}$ ) with rough control regions, a setup noted as being much less studied than the compact manifold case.
Methodological Contribution: The authors highlight their use of the FBI transformation to bridge the gap between heat and Schrödinger observability in non-compact settings, offering a quantitative alternative to the classical compactness-uniqueness strategy which is inapplicable here due to the lack of a compact resolvent.

The paper concludes by noting that the results can be extended to potentials in $L^\infty(\mathbb{R})$ via a standard density argument, though the primary analysis is presented for continuous potentials to simplify notation.

Technical Summary: Quantitative Observability for One-Dimensional Schrödinger Equations with Potentials

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