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Semi-Classical Localization of the Schrödinger Resolvent on Closed Riemann Surfaces

Sébastien Campagne

TL;DR

The work addresses the localization of semiclassical Schrödinger resolvents on compact Riemann surfaces with bounded potentials by extending Carleman-based localization techniques to non-smooth potentials. The authors regularize the potential, formulate a Carleman framework on balls, and exploit a uniformization-based reduction to Euclidean, hyperbolic, or spherical model geometries to obtain local-to-global estimates. The main result is a resolvent-type bound of the form $\int_M |u|^2 + |h\nabla u|^2 \le C e^{C\beta(h)^{1/2}/h^{4/3}} \left( \int_U |u|^2 + |h\nabla u|^2 + \int_M |(P_V - E)u|^2 \right)$, where $\beta(h)$ encodes the local regularity of $V$ via a modulus of continuity; in the Hölder case this recovers known bounds and connects to results by Klopp–Vodev. This work thus bridges smooth and non-smooth regimes for semiclassical analysis on closed surfaces and provides a robust framework for localization with merely bounded potentials.

Abstract

This paper investigates the localization properties of solutions to the semi-classical Schrödinger equation on closed Riemann surfaces. Unlike classical studies that assume a smooth potential, our work addresses the challenges arising from irregular potentials, specifically those that are merely bounded. We employ a regularization technique to manage the potential's lack of smoothness and establish a local-to-global estimate. This result provides a quantitative measure of how the local regularity of the potential influences the global concentration of the solution, thereby bridging the gap between smooth and non-continuous regimes in semi-classical analysis.

Semi-Classical Localization of the Schrödinger Resolvent on Closed Riemann Surfaces

TL;DR

The work addresses the localization of semiclassical Schrödinger resolvents on compact Riemann surfaces with bounded potentials by extending Carleman-based localization techniques to non-smooth potentials. The authors regularize the potential, formulate a Carleman framework on balls, and exploit a uniformization-based reduction to Euclidean, hyperbolic, or spherical model geometries to obtain local-to-global estimates. The main result is a resolvent-type bound of the form , where encodes the local regularity of via a modulus of continuity; in the Hölder case this recovers known bounds and connects to results by Klopp–Vodev. This work thus bridges smooth and non-smooth regimes for semiclassical analysis on closed surfaces and provides a robust framework for localization with merely bounded potentials.

Abstract

This paper investigates the localization properties of solutions to the semi-classical Schrödinger equation on closed Riemann surfaces. Unlike classical studies that assume a smooth potential, our work addresses the challenges arising from irregular potentials, specifically those that are merely bounded. We employ a regularization technique to manage the potential's lack of smoothness and establish a local-to-global estimate. This result provides a quantitative measure of how the local regularity of the potential influences the global concentration of the solution, thereby bridging the gap between smooth and non-continuous regimes in semi-classical analysis.
Paper Structure (5 sections, 6 theorems, 79 equations, 2 figures)

This paper contains 5 sections, 6 theorems, 79 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Proof of theorem \ref{['thm: mainthm']}
  3. Reduction of the problem
  4. Carleman estimate on a ball
  5. Conclusion

Key Result

Theorem 1

Let $M$ be a closed Riemann surface, and let $U \subset M$ be an open subset. Let $E \in I \subset \mathbb{R}$ and $V \in L^\infty(M, \mathbb{R})$. Then there exist constants $C > 0$ and $h_0 > 0$ such that for all $0 < h < h_0$ and $u \in H^2(M)$, where and $\kappa > 0$ is a fixed small constant.

Figures (2)

  • Figure 1: Critical point of a radial function: $\phi$ is a radial function at $0$, so $\phi$ is constant on circles centred at 0 and evolves orthogonally to these circles. Consequently, in $0$, the derivative vector should be $0$.
  • Figure 2: Partition of the ball $B(0,R_0)$ to construct $\phi$.

Theorems & Definitions (11)

  • Theorem 1
  • Proposition 2.1
  • proof
  • Theorem 2
  • proof
  • Lemma 2.2
  • proof
  • Corollary 2.3
  • proof
  • Corollary 2.4
  • ...and 1 more