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Semiclassical estimates for measure potentials on the real line

Andrés Larraín-Hubach, Jacob Shapiro

Abstract

We prove an explicit weighted estimate for the semiclassical Schrödinger operator $P = - h^2 \partial^2_x + V(x;h)$ on $L^2(\mathbb{R})$, with $V(x;h)$ a finite signed measure, and where $h >0$ is the semiclassical parameter. The proof is a one dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues for $P$, and a limiting absorption resolvent estimate with sharp $h$-dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support.

Semiclassical estimates for measure potentials on the real line

Abstract

We prove an explicit weighted estimate for the semiclassical Schrödinger operator on , with a finite signed measure, and where is the semiclassical parameter. The proof is a one dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues for , and a limiting absorption resolvent estimate with sharp -dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support.
Paper Structure (7 sections, 12 theorems, 117 equations)

This paper contains 7 sections, 12 theorems, 117 equations.

Table of Contents

  1. Introduction and statement of results
  2. Review of BV
  3. Self-adjointness for $V$ a measure
  4. Weighted estimate
  5. Exterior estimate
  6. Uniform resolvent estimate and resonance free strips
  7. wave decay

Key Result

Theorem 1.1

Fix $\delta > 0$. For all $E = E(h) > 0$ (which may depend on $h$), $\varepsilon \in [0,1]$, $h > 0$, and $u \in \mathcal{D}$ with $(|x| + 1)^{(1 + \delta)/2} (P - E\pm i\varepsilon) u \in L^2(\mathbb{R})$, Here, and $\| V\| \mathrel{\vcenter{\hbox{\scriptsize.}\hbox{\scriptsize.}}} = |V|(\mathbb{R})$, with$|V|$ the total variation of $V$, defined by where $\{V^+, V^-\}$ is

Theorems & Definitions (22)

  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Remark 1.5
  • Remark 1.6
  • Theorem 1.7: exterior estimate
  • Theorem 1.8
  • Proposition 2.1: integration by parts
  • Proposition 2.2: funadmental theorem of calculus
  • ...and 12 more