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Semi-Classical Behavior of the Spectral Function

Ivana Alexandrova

TL;DR

This paper investigates the semi-classical spectral function of a Schrödinger operator with short-range potential on R^n. It proves that, after suitable localization, the spectral function is a semi-classical Fourier integral operator associated with the forward and backward Hamiltonian-flow relations of the principal symbol. Under a geometric non-degeneracy assumption, the phase in the corresponding oscillatory integral can be explicitly computed via a phase function tied to the flow. The work builds on Vainberg's high-energy asymptotics and frames the spectral function in terms of Lagrangian manifolds and microlocal oscillatory representations, offering explicit phase and symbol structures for high-frequency behavior in quantum scattering.

Abstract

We study the semi-classical behavior of the spectral function of the Schr\"{o}dinger operator with short range potential. We prove that the spectral function is a semi-classical Fourier integral operator quantizing the forward and backward flow relations of the system. Under a certain geometric condition we explicitly compute the phase in an oscillatory integral representation of the spectral function.

Semi-Classical Behavior of the Spectral Function

TL;DR

This paper investigates the semi-classical spectral function of a Schrödinger operator with short-range potential on R^n. It proves that, after suitable localization, the spectral function is a semi-classical Fourier integral operator associated with the forward and backward Hamiltonian-flow relations of the principal symbol. Under a geometric non-degeneracy assumption, the phase in the corresponding oscillatory integral can be explicitly computed via a phase function tied to the flow. The work builds on Vainberg's high-energy asymptotics and frames the spectral function in terms of Lagrangian manifolds and microlocal oscillatory representations, offering explicit phase and symbol structures for high-frequency behavior in quantum scattering.

Abstract

We study the semi-classical behavior of the spectral function of the Schr\"{o}dinger operator with short range potential. We prove that the spectral function is a semi-classical Fourier integral operator quantizing the forward and backward flow relations of the system. Under a certain geometric condition we explicitly compute the phase in an oscillatory integral representation of the spectral function.

Paper Structure

This paper contains 3 sections, 3 theorems, 33 equations.

Table of Contents

  1. Introduction
  2. The Semi-Classical Spectral Function
  3. Elements of Semi-Classical Analysis

Key Result

Theorem 1

Let \rho_0\in\Lambda_{R}^{+}(\lambda) be such that \gamma\left(\pi_2\left(\rho_0\right)\right) is non-trapped. Let \|R(\lambda\pm i0, h)\|_{\alpha, -\alpha}=\mathcal{O}(h^{s}),s\in\mathbb{R}. Then there exist open sets W_{\pm}\in T^{*}\mathbb{R}^{n}\times T^{*}\mathbb{R}^{n}, such that

Theorems & Definitions (8)

  • Definition 1
  • Theorem 1
  • proof : Proof of Theorem \ref{['tspffio']}
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Definition 2