# Spectral determination of analytic bi-axisymmetric plane domains

@article{Zelditch2000SpectralDO,
title={Spectral determination of analytic bi-axisymmetric plane domains
},
author={Steve Zelditch},
journal={Geometric \& Functional Analysis GAFA},
year={2000},
volume={10},
pages={628-677}
}
• S. Zelditch
• Published 2000
• Mathematics
• Geometric & Functional Analysis GAFA
Abstract. Let {\cal D}L denote the class of bounded, simply connected real analytic plane domains with re ection symmetries across two orthogonal axes, of which one has length L. Under generic conditions, we prove that if $\Omega_1\Omega_2\,\in\,{\cal D}_L$ and if the Dirichlet spectra coincide, Spec $(\Omega_1)$ = Spec $(\Omega_2)$, then $\Omega_1 = \Omega_2$ up to rigid motion.

### Inverse spectral problem for analytic plane domains II: $\Z_2$-symmetric domains

This is part of a series of papers on the inverse spectral problem for bounded analytic plane domains. Here, we use the trace formula established in the first paper (Balian-Bloch trace formula') to

### Inverse spectral problem for analytic $(Z/2Z)^n$-symmetric domains in $R^n$

• Mathematics
• 2009
We prove that bounded real analytic domains in R n , with the symmetries of an ellipsoid and with one axis length fixed, are determined by their Dirichlet or Neumann eigenvalues among other bounded

### Inverse spectral problem for analytic plane domains II: domains with symmetry

We give two positive results on the inverse spectral problem for simply connected analytic domains with discrete symmetries. First, we consider domains with one mirror isometry and with an invariant

### On conjugacy of convex billiards

• Mathematics
• 2012
Given a strictly convex domain $\Omega$ in $\R^2$, there is a natural way to define a billiard map in it: a rectilinear path hitting the boundary reflects so that the angle of reflection is equal to

### Inverse spectral problem for analytic domains, II: ℤ2-symmetric domains

This paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit. The algorithm is based on a new expression for the

### INVERSE SPECTRAL PROBLEM FOR ANALYTIC DOMAINS II: Z2- SYMMETRIC DOMAINS

This paper develops and implements a new algorithm for calculating wave trace invariants of a bounded plane domain around a periodic billiard orbit. The algorithm is based on a new expression for the

### C∞ spectral rigidity of the ellipse

• Mathematics
• 2012
We prove that ellipses are infinitesimally spectrally rigid among C 1 domains with the symmetries of the ellipse. An isospectral deformation of a plane domain 0 is a one-parameter family t of plane

### Inverse Spectral Problem for Analytic $${{(\mathbb{Z}/2 \mathbb{Z})^{n}}}$$ -Symmetric Domains in $${{\mathbb{R}^{n}}}$$

• Mathematics
• 2010
We prove that bounded real analytic domains in $${\mathbb{R}^{n}}$$, with the symmetries of an ellipsoid and with one axis length fixed, are determined by their Dirichlet or Neumann eigenvalues among

### Sojourn times, manifolds with infinite cylindrical ends, and an inverse problem for planar waveguides

AbstractWe prove that two particular entries in the scattering matrix for the Dirichlet Laplacian on ℝ × (−γ, γ) $$\mathcal{O}$$ determine an analytic strictly convex obstacle $$\mathcal{O}$$.

### Marked Length Spectral determination of analytic chaotic billiards with axial symmetries

• Mathematics, Physics
• 2019
We consider billiards obtained by removing from the plane finitely many strictly convex analytic obstacles satisfying the non-eclipse condition. The restriction of the dynamics to the set of

## References

SHOWING 1-10 OF 27 REFERENCES

### The inverse spectral problem for surfaces of revolution

This paper concerns the inverse spectral problem for analytic simple surfaces of revolution. By simple' is meant that there is precisely one critical distance from the axis of revolution. Such

### The propagation of singularities along gliding rays

• Mathematics
• 1977
SummaryLetP be a second-order differential operator with real principal symbol and fibre-simple characteristics on a manifold with boundary non-characteristic forP. LetB be a differential operator

### Normal form of the wave group and inverse spectral theory

This talk will describe some results on the inverse spectral problem on a compact Riemannian manifold (possibly with boundary) which are based on V.Guillemin's strategy of normal forms. It consists

### Wave invariants at elliptic closed geodesics

This paper concerns spectral invariants of the Laplacian on a compact Riemannian manifold (M,g) known as wave invariants. If U(t) denotes the wave group of (M,g), then the trace Tr U(t) is singular

### Some planar isospectral domains

• Mathematics
• 1994
We give a number of examples of isospectral pairs of plane domains, and a particularly simple method of proving isospectrality. One of our examples is a pair of domains that are not only isospectral

### Number of quasimodes of “bouncing-ball” type

• Mathematics
• 1984
A new two-scale expansion is proposed for eigenfunctions of “bouncing-ball” type and the corresponding eigenvalues of the Laplace operator with the Dirichlet condition in a domain of the plane. The

### Geometry of reflecting rays and inverse spectral problems

• Mathematics
• 1992
Part 1 Preliminaries from differential topology and microlocal analysis: jets and transversality theorems generalized bicharacteristics wave front sets of distributions. Part 2 Reflecting rays:

### Lectures on Celestial Mechanics

• Mathematics
• 1971
The Three-Body Problem: Covarinace of Lagarangian Derivatives.- Canonical Transformation.- The Hamilton-Jacobi Equation.- The Cauchy-Existence Theorem.- The n-Body Poblem.- Collision.- The