Counterexamples to the Landis conjecture in dimensions three and higher

Abstract: In every dimension three and higher we construct a nonzero function that satisfies the stationary Schrödinger equation with bounded, real-valued potential and decays like $\exp(-c|x|^{4/3})$. This gives a counterexample to the Landis conjecture in dimension three and higher.

• The paper constructs a smooth, real-valued nontrivial solution decaying as exp(-c|x|^(4/3)), thereby refuting the Landis conjecture in dimensions n ≥ 3.
• It employs a two-fold method combining a radial decay envelope with tangential oscillations and harmonic repair to achieve bounded potentials.
• The approach highlights geometric differences between higher dimensions and n=2, emphasizing the importance of tangential frequency separation for unique continuation.

Counterexamples to the Landis Conjecture in Dimensions Three and Higher

Introduction and Context

The paper "Counterexamples to the Landis conjecture in dimensions three and higher" (2608.00802) addresses a longstanding open problem in the unique continuation theory for elliptic PDEs, specifically the strong unique continuation property at infinity for solutions of stationary Schrödinger equations with real-valued, bounded potentials. Landis conjectured in the 1960s that if a solution $u$ to $-\Delta u + V u = 0$ with $V \in L^\infty(\mathbb{R}^n)$ decays faster than any exponential with a rate proportional to $\|V\|_\infty^{1/2}$, then $u \equiv 0$. Although counterexamples with fast-decaying solutions in the case of complex-valued potentials have been known since Meshkov's 1991 construction, the existence of smooth, real-valued counterexamples—particularly in odd dimensions—remained unresolved.

This work constructs, for each $n \geq 3$, a smooth real-valued solution $u \not\equiv 0$ and a smooth, bounded real potential $V$ so that $u$ solves the stationary Schrödinger equation and decays like $\exp(-c|x|^{4/3})$ as $-\Delta u + V u = 0$0, thereby disproving the Landis conjecture in these dimensions.

Mathematical Construction

The construction is two-fold: building an approximate solution with fast decay, followed by a harmonic repair on regions where the oscillatory factor approaches zero, ensuring the quotient $-\Delta u + V u = 0$1 remains bounded.

Approximate Solution: Radial Envelope and Tangential Oscillations

The function $-\Delta u + V u = 0$2 is assembled as $-\Delta u + V u = 0$3, where $-\Delta u + V u = 0$4 is a radial decay envelope, with $-\Delta u + V u = 0$5 for large $-\Delta u + V u = 0$6, and $-\Delta u + V u = 0$7 is a carefully patched local superposition of real plane waves. The $-\Delta u + V u = 0$8 exponent is critical; it matches the sharp decay threshold already established by Meshkov for the complex case and arises from balancing radial and tangential cancellation errors under variable-scale partitions of unity. The construction uses a macroscopic covering of space by balls with radii growing as $-\Delta u + V u = 0$9, providing local frames to freeze tangential frequencies, and a combinatorial coloring argument to ensure phase separation across overlapping balls, which is feasible only for $V \in L^\infty(\mathbb{R}^n)$0.

This approach is purely real and hinges on the geometry of high-dimensional spheres, as only for $V \in L^\infty(\mathbb{R}^n)$1 can a sufficient number of separated directions in the tangent space be assigned to overlapping patches to defeat potential local cancellations.

Harmonic Replacement and Final Smoothing

On a microscopic scale—for regions where $V \in L^\infty(\mathbb{R}^n)$2 is small—the direct use of $V \in L^\infty(\mathbb{R}^n)$3 would admit regions where the potential $V \in L^\infty(\mathbb{R}^n)$4 could be unbounded. To remedy this, the construction replaces $V \in L^\infty(\mathbb{R}^n)$5 with a harmonic function in each such small region via a Dirichlet problem, carefully leveraging spectral estimates and weighted Poincaré inequalities for the Laplacian on sparse subregions. This "harmonic repair" ensures that the solution and potential remain smooth and globally controlled.

The global solution is then mollified at the microscopic scale with a radial kernel with spatially varying radius, preserving harmonicity away from interfaces and removing non-smoothness at boundaries between the original and repaired regions.

Main Result and Numerical Sharpness

The constructed function $V \in L^\infty(\mathbb{R}^n)$6 satisfies the following precise properties:

• $V \in L^\infty(\mathbb{R}^n)$7, $V \in L^\infty(\mathbb{R}^n)$8
• $V \in L^\infty(\mathbb{R}^n)$9
• $\|V\|_\infty^{1/2}$0 in $\|V\|_\infty^{1/2}$1
• $\|V\|_\infty^{1/2}$2 for all $\|V\|_\infty^{1/2}$3

The exponent $\|V\|_\infty^{1/2}$4 is sharp in the sense that Meshkov proved any solution decaying like $\|V\|_\infty^{1/2}$5 for arbitrarily large $\|V\|_\infty^{1/2}$6 must be identically zero, both in the real and complex settings [(2608.00802), Meshkov 1991]. This construction achieves the maximal possible decay for a nontrivial solution when potentials are real and bounded.

Theoretical Implications and Relation to Previous Work

This paper definitively refutes Landis's conjecture in all dimensions $\|V\|_\infty^{1/2}$7. Notably, the method does not extend to $\|V\|_\infty^{1/2}$8 due to the absence of sufficient tangential directions, a fact tightly connected to the geometry of the sphere: the sphere $\|V\|_\infty^{1/2}$9 in $u \equiv 0$0 (i.e., two-point set) does not admit enough separation for coloring arguments, unlike $u \equiv 0$1 for $u \equiv 0$2. Thus, strong unique continuation at infinity (of the exponential decay type) remains possible in two dimensions under real quantum potentials, in line with the striking recent positive results of Logunov, Malinnikova, Nadirashvili, and Nazarov [LMNN].

The result is structurally reminiscent of Meshkov’s and Filonov-Krymskii's explicit constructions but surpasses them by providing a genuinely real-valued, smooth, isotropic solution in the physically significant setting of real potentials, and for all high dimensions, both odd and even. The technical novelty lies in the harmonic repair and mollification techniques that propagate the correct decay and maintain smoothness without loss of control on the potential.

Broader Impact and Future Directions

The negative resolution of Landis's conjecture for $u \equiv 0$3 has ramifications for the spectral theory of Schrödinger operators and unique continuation problems. It demarcates sharp borders for the strong unique continuation property at infinity and quantitatively refines the understanding of how localization and decay can co-exist with bounded real potentials in unbounded domains.

Future research could involve:

• Investigation of such counterexamples under additional qualitative constraints (e.g., decay or positivity conditions on potential $u \equiv 0$4),
• Extension of repair and mollification arguments to more general (non-Euclidean) geometries,
• Quantitative refinements and stability bounds for unique continuation rates,
• Further exploration of unique continuation phenomena in random or pseudodifferential settings.

Conclusion

This paper (2608.00802) provides a mathematically rigorous construction of smooth, real-valued solutions to stationary Schrödinger equations with bounded real potentials decaying at the maximal possible rate in dimensions $u \equiv 0$5, thus giving a conclusive negative answer to the Landis conjecture in all higher dimensions. The techniques unify and generalize previous constructions, mark a fundamental distinction between high and low dimensions, and fully elucidate the geometric and analytic mechanisms underlying strong unique continuation at infinity.