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Subprincipal Controlled Quasimodes and Spectral Instability

Pelle Brooke Borgeke

TL;DR

This paper addresses spectral instability for a semiclassical operator with double characteristics by focusing on the subprincipal symbol $b$ when the principal symbol factors as $p=p_1p_2$ and satisfies $p=dp=0$ in a neighborhood. It develops a transport-equation framework via a semiclassical normal-form analysis, using a conjugated WKB ansatz and a three-parameter scaling to derive a leading expansion in which the subprincipal term governs solvability; a sign change of $\operatorname{Im} b$ along a limit bicharacteristic yields quasimodes and thus infinite-order pseudospectrum, while a factorization $P(h)=h^2P_1P_2$ can annihilate this subprincipal control. The main result establishes that, under the sign-change condition, quasimodes exist with residuals $\|P(h)u(h)\|$ of order $h^N$ for all $N$, implying resolvent blow-up and pseudospectral stability issues. A companion factorization example demonstrates that the $\beta$-condition can be eliminated in certain models, guiding future work on more complex tangential/intersections.

Abstract

Here we explore, in a series of articles, semiclassical quasimodes u(h,b), approximative solutions P(h)u(h,b)\sim 0, depending on $0<h<1$, and on b, the subprincipal symbol. We study a pseudodifferential operator with transversal intersections of bicharacteristics, where the principal symbol has double multiplicity, $p=dp=0$, in a small neigborhood $Ω$. Because of this fact, we instead study the subprincipal symbol b, and we can conclude that we get transport equations depending on b where sign changes for the imaginary part of b give approximative solutions with small support. These modes are used to estimate spectral instability, or the pseudospectrum. We also investigate the possibility that we can factorize the model operator as $P(h)=h^2P_1P_2,$ in this way actually annihilating the subprincipal symbol, thus there is no condition for the imaginary part of b. In a follow-up article, we examine different cases for more complex operators with tangential intersections of bicharacteristics, thereby generalizing the findings here.

Subprincipal Controlled Quasimodes and Spectral Instability

TL;DR

This paper addresses spectral instability for a semiclassical operator with double characteristics by focusing on the subprincipal symbol when the principal symbol factors as and satisfies in a neighborhood. It develops a transport-equation framework via a semiclassical normal-form analysis, using a conjugated WKB ansatz and a three-parameter scaling to derive a leading expansion in which the subprincipal term governs solvability; a sign change of along a limit bicharacteristic yields quasimodes and thus infinite-order pseudospectrum, while a factorization can annihilate this subprincipal control. The main result establishes that, under the sign-change condition, quasimodes exist with residuals of order for all , implying resolvent blow-up and pseudospectral stability issues. A companion factorization example demonstrates that the -condition can be eliminated in certain models, guiding future work on more complex tangential/intersections.

Abstract

Here we explore, in a series of articles, semiclassical quasimodes u(h,b), approximative solutions P(h)u(h,b)\sim 0, depending on , and on b, the subprincipal symbol. We study a pseudodifferential operator with transversal intersections of bicharacteristics, where the principal symbol has double multiplicity, , in a small neigborhood . Because of this fact, we instead study the subprincipal symbol b, and we can conclude that we get transport equations depending on b where sign changes for the imaginary part of b give approximative solutions with small support. These modes are used to estimate spectral instability, or the pseudospectrum. We also investigate the possibility that we can factorize the model operator as in this way actually annihilating the subprincipal symbol, thus there is no condition for the imaginary part of b. In a follow-up article, we examine different cases for more complex operators with tangential intersections of bicharacteristics, thereby generalizing the findings here.
Paper Structure (7 sections, 10 theorems, 111 equations)

This paper contains 7 sections, 10 theorems, 111 equations.

Table of Contents

  1. Introduction
  2. The proof method
  3. Proof of the Theorem
  4. Factorization annihilating the $\beta$-condition
  5. Preliminaries
  6. The Spectrum and the Psedospectrum
  7. Terminology and Notations

Key Result

Lemma 2.1

The parameters $\alpha, \beta$ and $\gamma$ in our system of transport equations form a partition of the unity of the semiclassical parameter $1<h \leq 1$ so that $h=h^{\alpha + \beta + \gamma}$. They are real numbers, used in the exponent of the semiclassical variable to balance the system of equat which is a plane in $\mathbb{R}^3$ with vertices in $(\alpha, \beta, \gamma)= (1,0,0), (0,1,0), (0,

Theorems & Definitions (33)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Example 2.3
  • Example 2.4
  • Lemma 2.5
  • Lemma 2.6
  • proof
  • Definition 2.7
  • ...and 23 more