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Semiclassical localization of Schrödinger's eigenfunctions

Sébastien Campagne

TL;DR

The paper addresses the problem of how tightly Schrödinger eigenfunctions can localize on a closed Riemann surface when the potential is only bounded. It adapts a recent Landis-conjecture technique to the semiclassical setting by reducing the geometry to a ball via uniformization, converting Schrödinger eigenfunctions into harmonic functions on a perforated domain, and then using a sequence of quasiconformal mappings to control space distortion. By applying Carleman-type inequalities on contracting holes and carefully reversing the transformations, the authors derive a global bound of the form $\int_{M}(|u_h|^2+|h\nabla u_h|^2) \leq \exp\left(C\frac{\log(1/h)^2}{h}\right) \int_{U}(|u_h|^2+|h\nabla u_h|^2)$, with $C>0$ independent of $u_h$. This provides a quantitative localization result for non-smooth potentials on curved geometries, improving previous bounds for merely bounded potentials and showcasing a robust, geometric-analytic approach that could extend to broader settings.

Abstract

This article addresses the microlocalization of eigenfunctions for the semiclassical Schrödinger operator $-h^2Δ+V$ on closed Riemann surfaces with real bounded potentials. Our primary aim is to establish quantitative bounds on the spatial concentration of these eigenfunctions, extending classical results, typically restricted to smooth potentials, to the more general case where the potential is merely bounded. Our main result provides an explicit exponential bound for the $L^2$-norm of eigenfunctions on the entire surface in terms of their $L^2$-norm on an arbitrary open subset with an exponential weight of $Ch^{-1}\log(h)^2$. This bound improves upon previous estimates for non-smooth potentials that was an exponential weight of $Ch^{-4/3}$. Our proof is based on a recent approach of the Landis conjecture develop by Logunov, Malinnikova, Nadirashvili and Nazarov (2025).

Semiclassical localization of Schrödinger's eigenfunctions

TL;DR

The paper addresses the problem of how tightly Schrödinger eigenfunctions can localize on a closed Riemann surface when the potential is only bounded. It adapts a recent Landis-conjecture technique to the semiclassical setting by reducing the geometry to a ball via uniformization, converting Schrödinger eigenfunctions into harmonic functions on a perforated domain, and then using a sequence of quasiconformal mappings to control space distortion. By applying Carleman-type inequalities on contracting holes and carefully reversing the transformations, the authors derive a global bound of the form , with independent of . This provides a quantitative localization result for non-smooth potentials on curved geometries, improving previous bounds for merely bounded potentials and showcasing a robust, geometric-analytic approach that could extend to broader settings.

Abstract

This article addresses the microlocalization of eigenfunctions for the semiclassical Schrödinger operator on closed Riemann surfaces with real bounded potentials. Our primary aim is to establish quantitative bounds on the spatial concentration of these eigenfunctions, extending classical results, typically restricted to smooth potentials, to the more general case where the potential is merely bounded. Our main result provides an explicit exponential bound for the -norm of eigenfunctions on the entire surface in terms of their -norm on an arbitrary open subset with an exponential weight of . This bound improves upon previous estimates for non-smooth potentials that was an exponential weight of . Our proof is based on a recent approach of the Landis conjecture develop by Logunov, Malinnikova, Nadirashvili and Nazarov (2025).
Paper Structure (19 sections, 21 theorems, 99 equations, 4 figures)

This paper contains 19 sections, 21 theorems, 99 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Reduction of the problem
  4. Some lemmas
  5. Perforation of space
  6. From Schrödinger to Harmonic equation
  7. Divergence type equation on the perforated space
  8. Quasi-conformal mapping
  9. Space distortion
  10. From control on Harmonic functions to one on Scrödinger eigenfunctions
  11. A few steps backwards
  12. Distortions control
  13. Carleman's inequality on contracting balls
  14. Conclusion
  15. Small Poincaré constant
  16. ...and 4 more sections

Key Result

Theorem 1

Let $(M,g)$ be a closed Riemann surface with the volume form $\mu_g$, and let $U \subset M$ be an open subset. Let $E \in I\subset \mathbb R$, and assume that $u_h \in L^2(M,\mathbb R)$ satisfies with $V \in L^\infty(M, \mathbb R)$. Then there exist constants $C > 0$ and $h_0 > 0$, independent of $u_h$, such that for all $0 < h < h_0$, we have

Figures (4)

  • Figure 1: Hyperbolic's disk paved by pentagon (code source https://doi.org/10.5281/zenodo.7559393)
  • Figure 3: Puncturing outside the nodal domains in $B(0, R_0)$ (Figure from logunov2020landisconjectureexponentialdecay).
  • Figure 6: Critical point for a radial function: $\phi$ is a radial function in $0$. $\phi$ is constant on circles centred on $0$ and moves orthogonally to these circles. In $0$ the derivative vector should point in all directions, which is impossible. So $0$ is a critical point of $\phi$.
  • Figure 7: Cutting the ball $B(0,h\epsilon(h))$: $\chi_{1,h}=0$ on $B_{1,h}$ and $=1$ outside $\tilde{B}_{1,h}$ (green), and $\chi_{2,h}=0$ on $B_{2,h}$ and $=1$ outside $\tilde{B}_{2,h}$ (blue).

Theorems & Definitions (38)

  • Theorem 1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • proof
  • Definition 2.5
  • Definition 2.6
  • Lemma 2.7
  • proof
  • ...and 28 more