Research
2022
Asymptotic lattices, good labellings, and the rotation number for quantum integrable systems
2022 - with M. Dauge and S. Vũ Ngọc
Discrete and Continuous Dynamical Systems, 42(12): 5683-5735
This article introduces the notion of good labellings for asymptotic lattices in order to study joint spectra of quantum integrable systems from the point of view of inverse spectral theory. As an application, we consider a new spectral quantity for a quantum integrable system, the quantum rotation number. In the case of two degrees of freedom, we obtain a constructive algorithm for the detection of appropriate labellings for joint eigenvalues, which we use to prove that, in the semiclassical limit, the quantum rotation number can be calculated on a joint spectrum in a robust way, and converges to the well-known classical rotation number. The general results are applied to the semitoric case where formulas become particularly natural.
2018
Ergoregions between two ergospheres
2018 - with G. Eskin
Methods of Functional Analysis and Topology, 24, No. 2, 98-106, 83c57
For a stationary spacetime metric, black holes are spatial regions which disturbances may not propagate out of. In our previous work an existence and regularity theorem was proven for black holes in two space dimensions, in the case where the boundary of the ergoregion is a simple closed curve surrounding a singularity. In this paper we study the case of an annular ergoregion, whose boundary has two components.
2016
Stationary Black Hole Metrics in Two Space Dimensions
2016 - with G. Eskin
Inverse Problems, vol.32, 095006
We consider the wave equation for a stationary Lorentzian metric in the case of two space dimensions. Assuming that the metric has a singularity of the appropriate form, surrounded by an ergosphere which is a smooth Jordan curve, we prove the existence of a black hole. It is shown that the boundary of the black hole (the event horizon) is not in general a smooth curve. We demonstrate this in a physical model for acoustic black holes, where the event horizon may have corners.
2015
Spectra for semiclassical operators with periodic bicharacteristics in dimension two
2015 - with M. Hitrik and J. Sjöstrand
International Mathematics Research Notices, vol. 2015, no. 20.
We study the distribution of eigenvalues for selfadjoint h--pseudodifferential operators in dimension two, arising as perturbations of selfadjoint operators with a periodic classical flow. When the strength ε of the perturbation is ≪ h, the spectrum displays a cluster structure, and assuming that ε ≫ h^2 (or sometimes ≫ h^N, for N >1 large), we obtain a complete asymptotic description of the individual eigenvalues inside subclusters, corresponding to the regular values of the leading symbol of the perturbation, averaged along the flow.
2013
Diophantine tori and nonselfadjoint inverse spectral problems
2013 - Mathematics Research Letters, vol 20, no. 2; pp. 255-271.
We study a semiclassical inverse spectral problem based on a spectral asymptotics result of arXiv:math/0502032, which applies to small non-selfadjoint perturbations of selfadjoint h-pseudodifferential operators in dimension 2. The eigenvalues in a suitable complex window have an expansion in terms of a quantum Birkhoff normal form for the operator near several Lagrangian tori which are invariant under the classical dynamics and satisfy a Diophantine condition. In this work we prove that the normal form near a single Diophantine torus is uniquely determined by the associated eigenvalues. We also discuss the normalization procedure and symmetries of the quantum Birkhoff normal form near a Diophantine torus.
2008
Conformal energy, conformal Laplacian, and energy measures on the Sierpinski gasket
2008 - With J. Azzam and R. Strichartz
Transactions of the American Mathematical Society, vol. 360, no. 4; pp. 2089-2130.
On the Sierpinski Gasket (SG) and related fractals, we define a notion of conformal energy Eϕ and conformal Laplacian ∆ϕ for a given conformal factor ϕ, based on the corresponding notions in Riemannian geometry in dimension n != 2. We derive a differential equation that describes the dependence of the effective resistances of Eϕ on ϕ. We show that the spectrum of ∆ϕ (Dirichlet or Neumann) has similar asymptotics compared to the spectrum of the standard Laplacian, and also has similar spectral gaps (provided the function ϕ does not vary too much). We illustrate these results with numerical approximations. We give a linear extension algorithm to compute the energy measures of harmonic functions (with respect to the standard energy), and as an application we show how to compute the Lp dimensions of these measures for integer values of p ≥ 2. We derive analogous linear extension algorithms for energy measures on related fractals.