arXiv subjects

Amir Vig

Publications and source records attributed to Amir Vig.

7 recordsLinked to original sources

arXiv2025

Spectral rigidity of two-dimensional Liouville tori

It is a folklore conjecture that Liouville metrics are the only Riemannian metrics with integrable geodesic flow on the two-dimensional torus. In this paper, we study Laplace-isospectral deformations, within a fixed conformal class, of generic Liouville metrics and prove two rigidity results: First, any isospectral deformation that is affine in $\epsilon$ within the fixed conformal class is trivial. Second, any isospectral deformation that is analytic in the perturbation parameter and whose Taylor coefficients are trigonometric polynomials in the spatial variables remains Liouville and is obtained by componentwise rearrangement of the original metric. For the first result, the proof combines a wave-trace noncancellation result, which allows us to recover suitable length data from the Laplace spectrum, with a second-variation analysis of the energy functional along closed geodesics. For the second result, we prove that, within the Liouville class considered here, the relevant length-isospectrality condition is equivalent to componentwise rearrangement.

math.DG↗

arXiv2024

Deformational spectral rigidity of axially-symmetric symplectic billiards

Symplectic billiards were introduced by Albers and Tabachnikov as billiards in strictly convex bounded domains of the plane with smooth boundary having a specific law of reflection. This paper proves a rigidity result for symplectic billiards which is similar to a previous result on classical billiards formulated by De Simoi, Kaloshin and Wei. Namely, it states that close to an ellipse, a sufficiently smooth one-parameter family of axially symmetric domains either contains domains with different area-spectra or is trivial, in a sense that the domains differ by area-preserving affine transformations of the plane. The paper also prove that in the general setting - that is even if the domains are not close to an ellipse - any sufficiently smooth one-parameter family of axially symmetric domains which preserves the area-spectrum is tangent to a finite dimensionnal space.

math.DS↗

arXiv2024

Silent Orbits and Cancellations in the Wave Trace

This paper shows that the wave trace of a bounded and strictly convex planar domain may be arbitrarily smooth in a neighborhood of some point in the length spectrum. In other words, the Poisson relation, which asserts that the singular support of the wave trace is contained in the closure of $\pm$ the length spectrum, can almost be made into a strict inclusion. To do so, we construct large families of domains for which there exist multiple periodic billiard orbits having the same length but different Maslov indices. Using the microlocal Balian-Bloch-Zelditch parametrix for wave invariants developed in our previous paper, we solve a large system of equations for the boundary curvature jets, which leads to the required cancellations. We call such periodic orbits silent, since they are undetectable from the ostensibly audible wave trace. Such cancellations show that there are potential limitations in using the wave trace for inverse spectral problems and more fundamentally, that the Laplace spectrum and length spectrum are inherently different mathematical objects, at least insofar as the wave trace is concerned.

math.SP↗

arXiv2024

Balian-Bloch Wave Invariants for Nearly Degenerate Orbits

This paper is part I of a series in which we aim to show that the singular support of the wave trace and the length spectrum of a smooth, strictly convex, and bounded planar billiard table are generally distinct objects. We derive an asymptotic trace formula for the regularized resolvent which is dual to the wave trace and contains the same information. To do this, we consider a class of periodic orbits which have nearly degenerate Poincar\'e maps and study their leading order behavior as the deformation parameter goes to zero, generating large coefficients in the wave trace. We also keep careful track of the Maslov indices, which will allow us to match contributions of opposite signs in our subsequent paper. Each cancellation of coefficients in the resolvent trace corresponds to making the wave trace one degree smoother. The resolvent based approach is due to Balian and Bloch and was significantly expanded upon by Zelditch in a foundational series of papers [Zel09], [Zel04a], [Zel04c] and [Zel00].

math.SP↗

arXiv2023

Compactness of Marked Length Isospectral Sets of Birkhoff Billiard Tables

We prove that equivalence classes of marked length isospectral Birkhoff billiard tables are compact in the $C^\infty$ topology, analogous to the Laplace spectral results of Melrose, Osgood, Phillips and Sarnak. To do so, we derive a hierarchical structure for the integral invariants of Marvizi and Melrose, or equivalently the coefficients of a caustic length-Lazutkin parameter expansion, which are in turn algebraically equivalent to the Taylor coefficients of Mather's $\beta$ function (also called the mean minimal action). Under a generically satisfied noncoincidence condition, these are also Laplace spectral invariants and can be used to hear the shape of certain drumheads. As a byproduct, we obtain an independent proof of the compactness of Laplace isospectral sets for strictly convex planar billiard tables. The proof of the structure theorem uses an interpolating Hamiltonian for nearly glancing billiard orbits and some analytic number theory to compute its Taylor coefficients.

math.DS↗

arXiv2022

The Wave Trace and Birkhoff Billiards

The purpose of this article is to develop a Hadamard-Riesz type parametrix for the wave propagator in bounded planar domains with smooth, strictly convex boundary. This parametrix then allows us to rederive an oscillatory integral representation for the wave trace appearing in \cite{MaMe82} and compute its principal symbol explicitly in terms of geometric data associated to the billiard map. This results in new formulas for the wave invariants. The order of the principal symbol, which appears to be inconsistent in the works of \cite{MaMe82} and \cite{Popov1994}, is also corrected. In those papers, the principal symbol was never actually computed and to our knowledge, this paper contains the first explicit formulas for the principal symbol of the wave trace. The wave trace formulas we provide are localized near both simple lengths corresponding to nondegenerate periodic orbits and degenerate lengths associated to one parameter families of periodic orbits tangent to a single rational caustic. Existence of a Hadamard-Riesz type parametrix with explicit symbol and phase calculations in the interior appears to be new in the literature, with the exception of the author's previous work \cite{Vig18} in the special case of elliptical domains. This allows us to circumvent the symbol calculus in \cite{DuGu75} and \cite{HeZe12} when computing trace formulas, which are instead derived from integrating our explicit parametrix over the diagonal.

math.AP↗

arXiv2020

Robin Spectral Rigidity of the Ellipse

In this paper, we investigate $C^1$ isospectral deformations of the ellipse with Robin boundary conditions, allowing both the Robin function and domain to deform simultaneously. We prove that if the deformations preserve the reflectional symmetries of the ellipse, then the first variation of both the domain and Robin function must vanish. Reparameterizing allows us to show that such smooth deformations must be flat at $ε= 0$. In particular, there exist no such analytic isospectral deformations. The key ingredients are a version of Hadamard's variational formula for variable Robin boundary conditions and an oscillatory integral representation of the wave trace variation which uses action angle coordinates for the billiard map. For the latter, we in fact construct an explicit parametrix for the wave propagator in the interior, microlocally near orbits of rotation number $1/j$.

math.AP↗