[Paper Review] Inverse spectral problem for analytic $(Z/2Z)^n$-symmetric domains in $R^n$

Hamid Hezari, Steve Zelditch|arXiv (Cornell University)|Feb 9, 2009

Spectral Theory in Mathematical Physics36 references18 citations

TL;DR

This paper establishes that bounded, real analytic domains in ℝⁿ with (ℤ/2ℤ)ⁿ symmetry and a fixed-length bouncing ball orbit are uniquely determined by their Dirichlet or Neumann eigenvalues, provided non-degeneracy conditions on the classical dynamics (eigenvalues of the Poincaré map) are satisfied. The key contribution is a positive solution to the higher-dimensional inverse spectral problem for non-spherical domains, extending earlier results for planar domains and proving spectral rigidity under symmetry and dynamical non-degeneracy.

ABSTRACT

In this paper we show that bounded analytic domains in $\R^n$ with mirror symmetries across all coordinate axes are spectrally determined among other such domains. Our approach builds on finding concrete formulas for the wave invariants at a bouncing ball orbit. The wave invariants are calculated from a stationary phase expansion applied to a well-constructed microlocal parametrix for the trace of the resolvent.

Motivation & Objective

• To resolve the inverse spectral problem for bounded, real analytic domains in ℝⁿ with (ℤ/2ℤ)ⁿ symmetry.
• To determine whether such domains are uniquely determined by their Dirichlet or Neumann eigenvalues.
• To extend previous results on planar domains to higher dimensions, avoiding reliance on spherical symmetry.
• To establish spectral rigidity under generic non-degeneracy conditions on bouncing ball orbits.
• To prove that the spectrum uniquely determines the domain's geometry among symmetric analytic domains with fixed axis length.

Proposed method

• Uses the semi-classical resolvent and wave group to extract spectral invariants from the Laplacian's spectrum.
• Analyzes wave invariants associated with bouncing ball orbits using the Poincaré map and its eigenvalues.
• Applies the Balian-Bloch formula to express wave invariants in terms of Hessian and curvature data at periodic orbits.
• Imposes non-degeneracy conditions: no eigenvalue 1 in the Poincaré map and linear independence of angles over ℚ.
• Uses Taylor expansion of the boundary defining function near the bouncing ball orbit and recovers coefficients via linear independence of trigonometric functions.
• Employs mathematical induction and spectral invariants to reconstruct the domain's shape from the spectrum.

Experimental results

Research questions

• RQ1 Can bounded, real analytic domains in ℝⁿ with (ℤ/2ℤ)ⁿ symmetry be uniquely determined by their Dirichlet or Neumann eigenvalues?
• RQ2 Does the spectrum of the Laplacian on such domains determine their geometry, even when not a ball?
• RQ3 What dynamical conditions on periodic orbits ensure spectral rigidity in higher dimensions?
• RQ4 Can wave invariants associated with bouncing ball orbits be used to reconstruct the domain's boundary up to isometry?
• RQ5 Is spectral determination possible for non-spherical, symmetric domains in dimensions greater than two?

Key findings

• The spectrum of the Dirichlet or Neumann Laplacian uniquely determines the domain among all bounded, real analytic domains with (ℤ/2ℤ)ⁿ symmetry and fixed bouncing ball orbit length.
• The non-degeneracy condition — that 1 is not an eigenvalue of the Poincaré map and that the angles are linearly independent over ℚ — ensures spectral rigidity.
• The wave invariants at the bouncing ball orbit are spectral invariants that encode the Taylor coefficients of the boundary defining function.
• The linear independence of functions of the form (cot(rα/2))^γ over ℕ implies that all even-order Taylor coefficients of the boundary function can be recovered from the spectrum.
• The proof uses the semi-classical resolvent and wave group to extract invariants, avoiding reliance on heat trace methods.
• The result is the first positive inverse spectral result for higher-dimensional Euclidean domains that are not balls.