Dispersive estimates for Schrödinger operators with negative Coulomb-like potentials in one dimension

Abstract: In this paper, we consider the dispersive estimates for Schrödinger operators with Coulomb-like decaying potentials, such as $V(x)=-c|x|<sup>{-μ}$ for $|x|\gg 1$ with $0<μ<2$, in one dimension. As an application, we establish both the standard and orthonormal Strichartz estimates for this model. One of the difficulties here is that perturbation arguments, which are typically applicable to rapidly decaying potentials, are not available. To overcome this, we derive a WKB expression for the spectral density and use a variant of the degenerate stationary phase formula to exploit its oscillatory behavior in the low-energy regime.

• The paper establishes global-in-time dispersive L¹→L∞ estimates with a decay rate of |t|⁻¹/² for 1D Schrödinger operators with negative Coulomb-like potentials.
• It introduces a refined WKB expansion and degenerate stationary phase analysis to accurately control low-energy spectral contributions in the presence of infinitely many bound states.
• The resulting Strichartz estimates inform scattering theory and improve well-posedness results for nonlinear dispersive PDEs in one dimension.

Dispersive Dynamics for 1D Schrödinger Operators with Negative Coulomb-like Potentials

Introduction and Problem Setting

The paper "Dispersive estimates for Schrödinger operators with negative Coulomb-like potentials in one dimension" (2603.29731) investigates the long-time dispersive behavior for the one-dimensional Schrödinger operator

$$P = -\partial_x^2 + V(x), \qquad V(x) \sim -c|x|^{-\mu}, \quad 0 < \mu < 2, \quad c>0,$$

focusing on the case of negative, slowly-decaying potentials (Coulomb-like). Such potentials are significant as one-dimensional analogues to the hydrogen atom, and are mathematically challenging due to their slow spatial decay and their non-perturbative long-range character.

The principal goal is to establish dispersive $L^1 \to L^\infty$ estimates and resulting Strichartz estimates for the unitary group $e^{-itP}$, with the essential difficulty that standard perturbative/short-range techniques are inapplicable due to the non-integrable tail of $V(x)$. The analysis is relevant both for direct scattering theory and for nonlinear applications, where the dispersive decay rate dictates critical regularity and well-posedness thresholds.

Main Results

The main achievement is a uniform-in-time dispersive estimate for the absolutely continuous component of the propagator in one spatial dimension:

$$\| e^{-itP} E_{\mathrm{ac}}(P) \|_{L^1 \to L^\infty} \lesssim |t|^{-1/2}, \qquad t \neq 0,$$

where $E_{\mathrm{ac}}(P)$ projects onto the absolutely continuous subspace. This result holds for a large class of real-valued $V \in C^\infty(\mathbb{R})$ satisfying $-V(x) \gtrsim \langle x \rangle^{-\mu}$, with $\mu\in(0,2)$, globally or outside a compact set. As an immediate corollary, both standard Strichartz and orthonormal Strichartz estimates are obtained for all admissible pairs.

A salient point is that, due to the slow decay and negativity of $V$, $L^1 \to L^\infty$0 exhibits infinitely many negative eigenvalues (hence the spectral projection); zero energy states retain scattering character, and the low-energy spectral analysis is subtle.

Analytical Techniques and Proof Strategy

The analysis circumvents the breakdown of perturbative/Born series arguments by leveraging the spectral theorem and a precise WKB expansion for the absolutely continuous spectral measure. The propagator is represented as

$L^1 \to L^\infty$1

with $L^1 \to L^\infty$2 admitting an explicit WKB-type oscillatory kernel involving complex phases adapted to the negative, slowly-decaying $L^1 \to L^\infty$3.

Two core technical innovations are central:

• Refined WKB Construction: The one-dimensional setting allows for a detailed ODE analysis using Liouville transforms. The authors construct global Jost solutions and spectral representations with explicit phase functions $L^1 \to L^\infty$4, and symbolically controlled amplitudes.
• Degenerate Stationary Phase Analysis: The oscillatory integrals governing the kernel exhibit degeneracies (critical points of high order) in the low-energy regime. Rather than a simple stationary phase, the authors utilize a quantitative degenerate stationary phase theorem, controlling contributions near degenerate critical points using detailed expansions and the vanishing properties of the amplitude.

Much of the analysis is regime-dependent, splitting the integral into high- and low-energy regions where either standard or degenerate stationary phase yields the optimal decay. In particular, the worst-case scenario corresponds to the low-energy sector, where the stationary phase point can coalesce at zero, requiring the use of higher-order expansion in the phase and exploiting amplitude vanishing.

Explicit Claims and Contrasts with Earlier Work

A notable assertion is that no previous result established global-in-time dispersive or Strichartz estimates for negative Coulomb-like potentials in one dimension—all earlier dispersive bounds for such decay rates either assumed positive potentials or were restricted to higher dimensions or special symmetry classes.

Furthermore, the optimality of the decay rate $L^1 \to L^\infty$5 (matching the free Schrödinger case) is conjectured to hold for this regime, which is non-trivial due to the presence of infinitely many bound states and absence of zero-energy resonances. In contrast, for higher dimensions ($L^1 \to L^\infty$6), such decay fails: only local-in-space decay ($L^1 \to L^\infty$7), at best, holds.

An extension is provided to more general $L^1 \to L^\infty$8 which may change sign or be positive on compact sets, provided the tail satisfies the negative decay.

Theoretical and Practical Implications

This work clarifies the dispersive landscape for one-dimensional Schrödinger flows with long-range attractive interactions. It highlights several phenomena:

• The long-range negative tail, despite supporting infinitely many eigenstates, does not destroy optimal dispersive decay on the absolutely continuous subspace.
• The main technical obstruction is in the control of "almost-threshold" energies, where the stationary phase degenerates; these are handled using delicate amplitude-vanishing arguments.
• The result asserts that the main mechanism for dispersive decay breakdown in higher dimensions is truly dimensional, rather than a consequence of the slow tail itself.

In practical terms, these dispersive (and thus Strichartz) bounds underpin the local and global well-posedness theory for nonlinear dispersive equations with Coulomb-like interactions in one dimension. They also provide a template for constructing nonperturbative spectral representations for other classes of long-range potentials and may inform numerics for ionization/scattering in model quantum systems.

Future Directions

Several directions are noted by the authors:

• The explicit analysis of repulsive (positive) Coulomb tails using these spectral/WKB techniques.
• Understanding threshold behavior and possible exceptions in higher dimensions.
• Extensions to fractional cases and related operators (e.g., on cones, with singularities).
• Nonlinear applications, where the sharpness of the dispersive rate affects critical exponents, global existence, and norm growth properties.

Conclusion

The paper provides a comprehensive resolution to the problem of dispersive decay for 1D Schrödinger operators with negative Coulomb-like potentials, delineating the impact of slow, attractive tails on quantum evolution. The methods blend spectral theory, refined ODE asymptotics, and non-classical oscillatory integral analysis, yielding results with both rigorous spectral-theoretic content and clear applications to nonlinear PDE theory.