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Semiclassical Defect Measures and Observability Estimate for Schrödinger Operators with Homogeneous Potentials of Order Zero

Keita Mikami

TL;DR

The paper develops a novel semiclassical framework to study Schrödinger operators with homogeneous potentials of order zero by introducing a cut-off–based quantization $ ext{Op}_{j}(a)$ and corresponding defect measures that capture asymptotic behavior at infinity. A reduced Hamiltonian dynamics on $ eal imes T^{*}S^{n-1}$ is employed to describe energy surfaces and directional localization, leading to a necessary observability condition and insights into quasimodes escaping to infinity. The authors prove that defect measures localize in direction and construct explicit quasimodes with nonvanishing measures in Section 4, demonstrating that mass can concentrate along specific directions even in nontrapping settings. The final section translates these analytic findings into observability/controllability results, showing failure or validity of observability on sets $\\Omega$ depending on a directional geometric control condition at infinity. Overall, the work extends semiclassical defect measure techniques to infinity, linking spectral asymptotics, dynamical systems, and observability for homogeneous-at-infinity potentials.

Abstract

We study the asymptotic behavior as |x| \to \infty of Schrödinger operators with homogeneous potentials. For this purpose, we use methods from semiclassical analysis and investigate semiclassical defect mesures. We prove their localization in direction which we apply in order to obtain a necessary condition of observability.

Semiclassical Defect Measures and Observability Estimate for Schrödinger Operators with Homogeneous Potentials of Order Zero

TL;DR

The paper develops a novel semiclassical framework to study Schrödinger operators with homogeneous potentials of order zero by introducing a cut-off–based quantization and corresponding defect measures that capture asymptotic behavior at infinity. A reduced Hamiltonian dynamics on is employed to describe energy surfaces and directional localization, leading to a necessary observability condition and insights into quasimodes escaping to infinity. The authors prove that defect measures localize in direction and construct explicit quasimodes with nonvanishing measures in Section 4, demonstrating that mass can concentrate along specific directions even in nontrapping settings. The final section translates these analytic findings into observability/controllability results, showing failure or validity of observability on sets depending on a directional geometric control condition at infinity. Overall, the work extends semiclassical defect measure techniques to infinity, linking spectral asymptotics, dynamical systems, and observability for homogeneous-at-infinity potentials.

Abstract

We study the asymptotic behavior as |x| \to \infty of Schrödinger operators with homogeneous potentials. For this purpose, we use methods from semiclassical analysis and investigate semiclassical defect mesures. We prove their localization in direction which we apply in order to obtain a necessary condition of observability.

Paper Structure

This paper contains 7 sections, 21 theorems, 67 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Admissible cut-off function
  4. Induced Dynamical System
  5. Proof of Theorem 1.2 and Theorem 1.4
  6. Example of asymptotic eigenvectors whose defect measure does not vanish
  7. Proof of Theorem 1.5

Key Result

Theorem 1.1

(Existence of semiclassical defect measure for $\mathrm{Op}_j$) Let $u_h\in L^2(\mathbb{R}^n)$ be a bounded sequence in $h$. There exists a sequence of positive numbers $h_m$ and a positive finite Radon measure $\mu_{j}$ on $\mathbb{R} \times T^*S^{n-1}$ such that $h_m \to 0$ as $m \to \infty$ and for all $a \in C^{\infty}_0(\mathbb{R} \times T^*S^{n-1})$.

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Remark
  • Theorem 1.5
  • Definition 2.1
  • Theorem 2.1
  • Remark
  • Theorem 2.2
  • ...and 32 more