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Subprincipal Control of Pseudospectral Quasimodes, II

Pelle Brook Borgeke

TL;DR

The paper investigates semiclassical quasimodes that depend on the subprincipal symbol $b$ for pseudodifferential operators with double characteristics ($p=dp=0$). Using a microlocal normal form and a model operator $P(h)=hD_1(hD_1+\mathcal Q(\xi))+B(x,hD_x)$, it analyzes how the imaginary part $\beta$ of $b$ (and its derivative $\partial_{\xi}\beta$) controls pseudospectrum formation along limit bicharacteristics, yielding quasimodes under $\beta$- and $\partial_{\xi}\beta$-conditions. A Factorization Theorem clarifies when subprincipal control can be annihilated (notably at $j=k$), affecting the existence of quasimodes in tangential-type operators, and the analysis extends to nondegenerate quadratic forms $q(x,\xi)$ in the tangential setting. The results provide a detailed taxonomy of when spectral instability arises, linking pseudospectrum to microlocal geometry and factorization properties, with implications for solvability theory in semiclassical regimes.

Abstract

In this paper, we continue the analysis of the effects of semiclassical sub principal controlled quasimodes, approximate solutions to P(h)u(h,b), depending on the subprincipal symbol b, which can give spectral insta bility (pseudospectrum). We consider a pseudodifferential operator, which has double zeros for the principal symbol, p. This means that p = dp = 0 in a small neighborhood. In the first paper in this series, we considered operators with transversal inter sections of bicharacteristics. Now we study operators with tangential in tersections of bicharacteristics, as well as with double characteristics for p. We put the pseudodifferential operator on normal form microlocally, and use a model operator, P(h) to test for quasimodes. We demonstrate two cases where this happens. We shall also continue with more advanced cases, when the operators are factorable to P(h) = P2(h)P1(h,B), thus annihilating the subprincipal control over the quasimodes.

Subprincipal Control of Pseudospectral Quasimodes, II

TL;DR

The paper investigates semiclassical quasimodes that depend on the subprincipal symbol for pseudodifferential operators with double characteristics (). Using a microlocal normal form and a model operator , it analyzes how the imaginary part of (and its derivative ) controls pseudospectrum formation along limit bicharacteristics, yielding quasimodes under - and -conditions. A Factorization Theorem clarifies when subprincipal control can be annihilated (notably at ), affecting the existence of quasimodes in tangential-type operators, and the analysis extends to nondegenerate quadratic forms in the tangential setting. The results provide a detailed taxonomy of when spectral instability arises, linking pseudospectrum to microlocal geometry and factorization properties, with implications for solvability theory in semiclassical regimes.

Abstract

In this paper, we continue the analysis of the effects of semiclassical sub principal controlled quasimodes, approximate solutions to P(h)u(h,b), depending on the subprincipal symbol b, which can give spectral insta bility (pseudospectrum). We consider a pseudodifferential operator, which has double zeros for the principal symbol, p. This means that p = dp = 0 in a small neighborhood. In the first paper in this series, we considered operators with transversal inter sections of bicharacteristics. Now we study operators with tangential in tersections of bicharacteristics, as well as with double characteristics for p. We put the pseudodifferential operator on normal form microlocally, and use a model operator, P(h) to test for quasimodes. We demonstrate two cases where this happens. We shall also continue with more advanced cases, when the operators are factorable to P(h) = P2(h)P1(h,B), thus annihilating the subprincipal control over the quasimodes.
Paper Structure (5 sections, 11 theorems, 87 equations)

This paper contains 5 sections, 11 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. Short review from the first article, and formulas for the tangential case
  3. Operators with tangential intersection of bicharacteristics, $\beta$-condition
  4. Operators with ${\partial_{\xi}\beta}$-condition, Annihilation of ${\beta}$-condition in Theorem of Factorization
  5. Summation and Discussion of Results in this Series, Background and Bibliography

Key Result

Theorem 1

Let $P(x,hD_x;h^n B_{n\geq 0}(x,hD_x))$ have a real principal symbol $p(x,\xi)$ that microlocally factorizes $p=p_1p_2$ in the neighborhood $\Omega$ of $(x_0, \xi_0)$. Assume that $p^{-1}(z)$ is a union of two hypersurfaces with transversal involutive intersection at $\Sigma_2(P(h))$ and that $d^2_{

Theorems & Definitions (20)

  • Theorem
  • Lemma
  • Lemma
  • Lemma 2.1
  • proof
  • Remark 2.2
  • Remark 2.3
  • Theorem 3.1
  • Proposition 3.2
  • proof
  • ...and 10 more