[Paper Review] 0-th Order Pseudo-differential Operator on the Circle

Zhongkai Tao|arXiv (Cornell University)|Sep 13, 2019

Spectral Theory in Mathematical Physics8 references4 citations

TL;DR

This paper studies 0-th order pseudodifferential operators on the circle, proving that their spectrum is absolutely continuous with possibly finitely many embedded eigenvalues in intervals disjoint from critical values of the principal symbol. Using Mourre estimates and spectral theory, it establishes the structure of the essential, absolutely continuous, and pure point spectra, and provides explicit examples of operators with embedded eigenvalues at zero via Weyl quantization.

ABSTRACT

In this paper we consider 0-th order pseudodifferential operators on the circle. We show that inside any interval disjoint from critical values of the principal symbol, the spectrum is absolutely continuous with possibly finitely many embedded eigenvalues. We also give an example of embedded eigenvalues.

Motivation & Objective

• To analyze the spectral structure of 0-th order pseudodifferential operators on the circle, particularly the distribution of essential, absolutely continuous, and pure point spectra.
• To extend the Mourre estimate framework to 0-th order operators on compact manifolds, focusing on the circle.
• To establish conditions under which the spectrum is absolutely continuous with finitely many embedded eigenvalues.
• To construct explicit examples of operators with embedded eigenvalues at zero using Weyl quantization and cutoff functions.
• To generalize results from classical symbol theory to non-classical 0-th order operators via microlocal analysis.

Proposed method

• The paper uses Mourre estimates to analyze the spectral type of self-adjoint 0-th order pseudodifferential operators on the circle.
• It defines the essential spectrum via the asymptotic behavior of the principal symbol at infinity in fiber directions.
• It applies the Weyl quantization map to construct operators from symbols and derives eigenvalue conditions via integral representations.
• It constructs a self-adjoint conjugate operator A based on the derivative of the symbol to generate a Mourre estimate.
• It uses almost-analytic extensions and compactness arguments to control error terms in spectral projections.
• It applies the Cayley transform to extend results to unitary operators, proving continuous spectrum with finitely many embedded eigenvalues.

Experimental results

Research questions

• RQ1 Under what conditions is the spectrum of a 0-th order pseudodifferential operator on the circle absolutely continuous with finitely many embedded eigenvalues?
• RQ2 How do critical values of the principal symbol affect the spectral decomposition of such operators?
• RQ3 Can explicit examples of embedded eigenvalues be constructed for 0-th order operators on the circle?
• RQ4 What role does the Weyl quantization play in realizing eigenfunctions and eigenvalues?
• RQ5 How does the Mourre estimate framework apply to non-classical 0-th order symbols on compact manifolds?

Key findings

• The essential spectrum of a 0-th order pseudodifferential operator on a compact manifold is the set of limit points of the principal symbol along divergent fiber directions.
• On the circle, the purely point spectrum is contained in the set of critical values of the principal symbol, defined as values where the derivative with respect to x vanishes at ±1 in the fiber direction.
• In any interval disjoint from the critical values, the pure point spectrum is finite, and the absolutely continuous spectrum is the union of intervals [a_-, a^-] and [a_+, a^+] minus the critical set.
• An explicit example is constructed where the Weyl quantization of a(x,ξ) = sin(2πx)(1−χ(ξ)) has an embedded eigenvalue at 0 with eigenfunction e^{2kπix} for integer k.
• For unitary operators with homogeneous principal symbols on the circle, the spectrum is absolutely continuous with finitely many embedded eigenvalues in intervals avoiding critical values.
• The Mourre estimate method applies to unitary operators via the Cayley transform, yielding continuous spectrum with finitely many embedded eigenvalues under the same geometric conditions.