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Optimal bounds for embedded eigenvalues of one-dimensional discrete Schrödinger operators with decaying potentials

This paper establishes the sharp asymptotic threshold for the decay of real-valued potentials in one-dimensional discrete Schrödinger operators that determines the existence or nonexistence of embedded eigenvalues.

Original authors: Jifeng Chu, Wencai Liu, Kang Lyu

Published 2026-09-04

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Jifeng Chu, Wencai Liu, Kang Lyu

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

In the quiet, abstract world of mathematical physics, researchers study how particles move through space when they encounter obstacles. Imagine a particle traveling along a one-dimensional line, like a bead sliding on a wire. If the wire is perfectly smooth, the particle moves freely, its energy spreading out in a continuous, predictable way. However, if the wire is rough or uneven, the particle can get trapped, settling into a specific, stationary state known as an eigenvalue. The central question for decades has been: how rough can the wire be before it stops trapping the particle? If the bumps are too small, the particle slides right over them; if they are large enough, they can hold the particle in place. The challenge lies in finding the exact tipping point—the precise size of the bumps that allows a particle to become trapped without the wire becoming so chaotic that the particle's behavior becomes entirely unpredictable.

A team of mathematicians has now solved this puzzle for a specific, difficult type of roughness found in discrete systems, where the wire is not a smooth line but a series of distinct steps. They focused on a scenario where the bumps on the wire get smaller as the particle travels further away, fading away at a specific, critical rate. For a long time, scientists knew that if the bumps faded too quickly, no trapping could occur. They also knew that if the bumps were large enough, trapping was possible. But the exact boundary between these two worlds remained a mystery, particularly when the particle's energy matched a specific, repeating pattern within the system. This new work identifies that exact boundary with absolute precision, revealing that the ability to trap a particle depends on a delicate, hidden relationship between the size of the bumps and the arithmetic nature of the particle's energy.

The researchers examined a model where the wire consists of a sequence of points, and the "roughness" is a potential energy that changes from point to point. They discovered that the answer depends entirely on whether the particle's energy corresponds to a rational number or an irrational number. In the world of numbers, a rational number can be written as a simple fraction, like one-half or three-quarters, while an irrational number, like the square root of two, cannot be written as a simple fraction and has a decimal expansion that never repeats. The team found that when the energy corresponds to an irrational number, the rules are relatively straightforward: the bumps must be smaller than a specific, universal limit to avoid trapping the particle. However, when the energy corresponds to a rational number, the situation becomes far more complex and depends on the specific denominator of that fraction.

The most significant breakthrough in this paper concerns the case where the denominator of the fraction is an odd number. Previous studies had established a range within which the answer must lie, but they could not pinpoint the exact value. The authors of this paper proved that the limit is determined by a specific, intricate formula involving trigonometric functions that describe the shape of the bumps over one cycle. They demonstrated that if the bumps are even slightly larger than this precise value, it is possible to construct a wire that traps the particle. Conversely, if the bumps are even slightly smaller, the particle will never be trapped, no matter how the wire is arranged. This result closes a long-standing gap in mathematical physics, showing that the transition from "no trapping" to "trapping" is sharp and exact.

To reach this conclusion, the researchers had to overcome a fundamental difficulty that had stumped them for years. In systems with even denominators, the bumps have a natural symmetry that allows the researchers to balance the forces acting on the particle, keeping it in a stable state. But with odd denominators, this symmetry is broken. The forces that help trap the particle also tend to push it out of the stable position, creating a kind of instability where the particle is constantly being nudged away from the very spot where it needs to be to stay trapped. The authors solved this by developing a new method to track the particle's movement over long periods. They realized that while the particle might be pushed away from the ideal spot in one moment, the very act of being pushed creates a compensating effect later on. By carefully pairing these opposing movements, they showed that the net effect still respects the strict limit they had calculated.

The paper also addresses the critical case where the bumps are exactly at the limit. Here, the particle is on the verge of being trapped. The authors showed that even at this exact threshold, it is possible to construct a wire that successfully traps the particle, provided the bumps are arranged with a specific, slowly varying pattern. They proved that by making tiny, precise adjustments to the size of the bumps as the particle travels further, one can accumulate enough small gains to eventually hold the particle in place. This construction is delicate; if the adjustments are too large, the particle escapes, but if they are too small, the particle never settles. The authors found the perfect balance, proving that the limit is not just a theoretical boundary but a reachable reality.

This work completes a comprehensive picture of how particles behave in these discrete, fading environments. It confirms that the rules governing the existence of trapped states are not arbitrary but are dictated by the deep arithmetic properties of the system. The findings provide a definitive answer to a question that has driven research in spectral theory for decades, showing that the boundary between order and chaos in these quantum systems is far more precise and mathematically rich than previously imagined. By resolving the odd-denominator case, the authors have removed the last major uncertainty in this field, offering a clear and complete understanding of the conditions required for a particle to become trapped in a fading potential.

Technical Summary: Optimal Bounds for Embedded Eigenvalues of One-Dimensional Discrete Schrödinger Operators with Decaying Potentials

Problem Statement
This paper investigates the spectral properties of one-dimensional discrete Schrödinger operators $H = \Delta + V$ acting on $\ell^2(\mathbb{N})$ , where $\Delta$ is the discrete Laplacian and $V$ is a real-valued potential decaying at the critical Coulomb scale, specifically $V(n) = O(1/(1+n))$ . The central problem is to determine the sharp asymptotic constant $C(E)$ governing the existence of eigenvalues embedded in the interior of the essential spectrum $[-2, 2]$ .

For a given energy $E = 2\cos(\pi k)$ with $k \in (0, 1)$ , the authors seek the critical threshold $C(k)$ such that:

If $\limsup_{n\to\infty} |nV(n)| < C(k)$ , then $E$ is not an eigenvalue.
For any $\epsilon > 0$ , there exists a potential with $\limsup_{n\to\infty} |nV(n)| < C(k) + \epsilon$ such that $E$ is an eigenvalue.

While the continuous analogue of this problem is well-understood (with the critical coefficient being $2/\pi$ for irrational quasi-momenta), the discrete case exhibits a complex dependence on the arithmetic nature of $k$ . Previous work established sharp bounds for irrational $k$ and rational $k$ with even denominators, but the case of rational $k = p/q$ with odd $q \ge 3$ remained unresolved, with a gap between the known lower and upper bounds.

Methodology
The analysis relies on the modified Prüfer transformation, which converts the difference equation into radial ( $R(n)$ ) and angular ( $\theta(n)$ ) variables. The existence of an embedded eigenvalue corresponds to the square-summability of the radial solution $R(n) \in \ell^2$ .

The core difficulty in the odd-denominator case ( $q$ odd) is the lack of a "half-period symmetry" available in the even case. In the even case, potential values that maximize radial decay can be chosen to cancel angular drift, allowing the phase to be locked at an optimal value. In the odd case, the potential choice that improves radial decay inevitably induces a non-zero angular drift, preventing the phase from remaining fixed.

To resolve this, the authors employ a multi-step methodology:

Trigonometric Identities and Compensation: They introduce two trigonometric functions, $P(\phi)$ (measuring radial decay gain) and $T(\phi)$ (measuring angular drift). They establish a key identity relating the excess of $P(\phi)$ over the sharp constant $qB_q$ to the drift $T(\phi)$ . This reveals that periods with large radial gains are accompanied by significant angular motion, while periods with small gains have opposite drifts.
Partition of Slowly Varying Sequences: The authors develop a novel partition theory for sequences satisfying $x_{n+1} - x_n = O(1/(1+n))$ . They classify the accumulation sets of the fractional parts of such sequences into four cases: a single point, a proper arc, a union of two arcs (crossing the identification point), or the full circle.
Dynamic Compensation: Using these partitions, they group indices into windows where the Prüfer angle remains in specific regions. For the "full circle" case (where the angle explores the entire domain), they pair indices near $\phi$ and $1-\phi$ . Due to the symmetry properties of $P$ and the antisymmetry of $T$ , the radial gains in these paired windows can be balanced against the angular drifts, ensuring that the total accumulated decay does not exceed the sharp bound.
Critical Case Construction: For the existence part (showing the bound is sharp), they construct potentials using rapidly growing blocks. They introduce vanishing corrections to the potential coefficients to ensure the radial decay exponent is slightly super-critical within blocks, while the block lengths grow fast enough to make transition errors summable. This requires coordinating three scales: the period of oscillation, the construction block, and the global scale.

Key Contributions and Results

Resolution of the Odd-Denominator Case: The primary contribution is the determination of the sharp critical coefficient for rational $k = p/q$ where $q \ge 3$ is odd. The authors prove that:
$C(k) = \frac{\sin(\pi k)}{B_q}, \quad \text{where} \quad B_q = \frac{1 + \cos(\frac{\pi}{q})}{q \sin(\frac{\pi}{q})}.$
This closes the gap between previous lower and upper bounds, replacing the lower bound with the matching upper bound.
Complete Characterization of $C(k)$ : Combined with prior results, the paper provides a complete formula for $C(k)$ for all $k \in (0, 1)$ :
- Irrational $k$ : $C(k) = \frac{\sin(\pi k)}{2/\pi}$ .
- Rational $k = p/q$ ( $q$ even): $C(k) = \frac{\sin(\pi k)}{A_q}$ , where $A_q = \frac{2}{q \sin(\pi/q)}$ .
- Rational $k = p/q$ ( $q$ odd): $C(k) = \frac{\sin(\pi k)}{B_q}$ .
This formula precisely quantifies the discontinuity of $C(k)$ at rational quasi-momenta. As $k'$ approaches a rational $p/q$ , $C(k')$ converges to $\sin(\pi k)/A_0$ , whereas the value at the rational point is determined by $A_q$ (even $q$ ) or $B_q$ (odd $q$ ).
Sharpness at the Critical Coefficient: The paper proves that the critical coefficient is not merely a threshold for non-existence but is actually attained. For any prescribed boundary condition and any $E \in (-2, 2)$ , there exists a potential $V$ with $\limsup |nV(n)| = C(E)$ such that $E$ is an embedded eigenvalue. This requires a delicate construction where the potential does not exceed the limit but accumulates sufficient decay through vanishing corrections on increasingly long scales.
Partition Theory for Slowly Varying Sequences: The paper introduces a rigorous partition result for sequences with $O(1/n)$ variation. This result classifies the accumulation behavior of such sequences on the torus $\mathbb{R}/\mathbb{Z}$ and provides the necessary combinatorial framework to handle the interplay between radial decay and angular evolution in the discrete setting.

Significance
The paper claims to complete the optimal single eigenvalue coefficient problem for one-dimensional discrete Schrödinger operators with Coulomb-decaying potentials. It resolves the specific difficulty of the odd-denominator case, which was previously intractable due to the coupling between radial decay and angular motion. By establishing a "dynamical compensation principle," the authors demonstrate that the sharp constant is determined not by pointwise optimization of the potential, but by the global balance of radial gains and angular drifts over the orbit. This work unifies the understanding of embedded eigenvalues across irrational and rational quasi-momenta, highlighting the profound role of arithmetic properties in discrete spectral theory.

Technical Summary: Optimal Bounds for Embedded Eigenvalues of One-Dimensional Discrete Schrödinger Operators with Decaying Potentials

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