[Paper Review] Non-selfadjoint perturbations of selfadjoint operators in 2 dimensions IIIa. One branching point

Michael Hitrik, Johannes Sjoestrand|ArXiv.org|Jun 13, 2005

Spectral Theory in Mathematical Physics12 references4 citations

TL;DR

This paper analyzes the spectral asymptotics of non-selfadjoint perturbations of selfadjoint semiclassical pseudodifferential operators in two dimensions, focusing on eigenvalues near a non-degenerate saddle point of the flow-averaged imaginary part of the perturbation. Using microlocal analysis and transition matrix asymptotics, it derives precise eigenvalue distributions in narrow rectangles near the imaginary axis, showing that eigenvalues cluster according to a logarithmic scaling governed by the perturbation's flow average, with explicit asymptotic formulas in the regime $ h^2 \ll \epsilon \ll h^{1/2} $. The key contribution is a complete description of eigenvalue distribution in the branching singularity case, resolving a generic case left open in prior works.

ABSTRACT

This is the third in a series of works devoted to spectral asymptotics for non-selfadjoint perturbations of selfadjoint $h$-pseudodifferential operators in dimension 2. We assume that the unperturbed operator has a periodic Hamilton flow, and study the remaining generic case; when the flow average of the perturbation has a saddle point

Motivation & Objective

• To complete the spectral analysis of non-selfadjoint perturbations of selfadjoint 2D semiclassical operators by resolving the remaining generic case: eigenvalues associated with a non-degenerate saddle point of the flow-averaged perturbation.
• To extend prior results on periodic classical flows by analyzing the case where the flow average of the imaginary part of the perturbation has a non-degenerate saddle point, rather than a non-critical value or non-degenerate extremum.
• To derive precise asymptotic formulas for eigenvalue distribution in narrow rectangles near the imaginary axis, under the parameter regime $ h^2 \ll \epsilon \ll h^{1/2} $, where $ \epsilon $ is the perturbation strength and $ h $ is the semiclassical parameter.
• To establish the connection between the eigenvalue distribution and the geometry of the classical flow, particularly through the transition matrix at a branching level, using microlocal and complex analysis techniques.

Proposed method

• Reduction of the spectral problem to a one-dimensional pseudodifferential operator via microlocal normal forms and symplectic reduction near the energy surface.
• Exponential decoupling of the problem using FBI transforms and complex WKB methods to separate incoming and outgoing solutions near the critical set.
• Computation of the transition matrix at the branching level using the solution of a model problem involving logarithmic singularities and complex phase integrals.
• Asymptotic analysis of the transition matrix using the method of steepest descent and uniform estimates in the complex plane, particularly near the branching point.
• Solution of the one-dimensional spectral problem by studying zeros of sums of exponential functions with complex phases, leading to eigenvalue counting via the argument principle.
• Justification of the global spectral picture using a Grushin problem formulation, ensuring consistency across the entire energy band and validating the local asymptotics.

Experimental results

Research questions

• RQ1 How do eigenvalues of non-selfadjoint perturbations of selfadjoint 2D semiclassical operators distribute when the flow-averaged imaginary part of the perturbation has a non-degenerate saddle point?
• RQ2 What is the precise asymptotic distribution of eigenvalues in the rectangle $[-1/C, 1/C] + i\epsilon[F_0 - 1/C, F_0 + 1/C]$ for $ C \gg 1 $, when $ \epsilon F_0 $ is a saddle point value of the flow average?
• RQ3 How does the transition matrix at the branching level behave asymptotically in the semiclassical limit, and what role does it play in determining the global eigenvalue distribution?
• RQ4 What is the correct scaling for the eigenvalue spacing and imaginary part in the regime $ h^2 \ll \epsilon \ll h^{1/2} $, and how does it differ from the non-saddle cases?
• RQ5 Can the eigenvalue distribution be rigorously justified using a global Grushin problem formulation, ensuring consistency beyond local models?

Key findings

• Eigenvalues are asymptotically distributed in narrow rectangles of width $ \mathcal{O}(\epsilon) $ centered at $ \mathcal{O}(1) $ in the real part and $ \epsilon F_0 + \mathcal{O}(\epsilon/C) $ in the imaginary part, with $ F_0 $ being the saddle point value of the flow-averaged perturbation.
• The imaginary part of the eigenvalues is asymptotically given by $ \mathrm{Im}\, \lambda \sim \frac{F(x)}{\ln(1/|x|)} $ when $ |F(x)| \ll |x|\ln(1/|x|) $, and $ \mathrm{Im}\, \lambda \sim \frac{F(x)}{\ln(1/|F(x)|)} $ when $ |F(x)| \gg |x|\ln(1/|x|) $, with $ x $ the real part.
• The transition matrix at the branching level is shown to have a specific asymptotic form involving logarithmic terms, which governs the eigenvalue spacing and determines the spectral density.
• The eigenvalue counting function in the region $ |\mu| \gg h $ is derived via the argument principle applied to the determinant of the transition matrix, yielding a precise count up to $ \mathcal{O}(1) $ error.
• For $ |\mu| \leq \mathcal{O}(h) $, the skeleton of eigenvalues is shown to be governed by the zeros of a sum of exponentials, with the leading-order behavior determined by the flow-averaged perturbation $ F(x) $.
• The parameter range $ h^2 \ll \epsilon \ll h^{1/2} $ is shown to be optimal for the asymptotic description, with improvements possible in special cases such as the barrier top resonance.