Exponentially small asymptotic formulas for the length spectrum in some billiard tables

Let $q \ge 3$ be a period. There are at least two $(1,q)$-periodic trajectories inside any smooth strictly convex billiard table, and all of them have the same length when the table is an ellipse or a circle. We quantify the chaotic dynamics of axisymmetric billiard tables close to their borders by studying the asymptotic behavior of the differences of the lengths of their axisymmetric $(1,q)$-periodic trajectories as $q \to +\infty$. Based on numerical experiments, we conjecture that, if the billiard table is a generic axisymmetric analytic strictly convex curve, then these differences behave asymptotically like an exponentially small factor $q^{-3} e^{-r q}$ times either a constant or an oscillating function, and the exponent $r$ is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the $(1,q)$-periodic trajectories. Our experiments are restricted to some perturbed ellipses and circles, which allows us to compare the numerical results with some analytical predictions obtained by Melnikov methods and also to detect some non-generic behaviors due to the presence of extra symmetries. Our computations require a multiple-precision arithmetic and have been programmed in PARI/GP.Comment: 21 pages, 37 figure

On the length and area spectrum of analytic convex domains

Area-preserving twist maps have at least two different (p, q)-periodic orbits and every (p, q)-periodic orbit has its (p, q)-periodic action for suitable couples (p, q). We establish an exponentially small upper bound for the differences of (p, q)-periodic actions when the map is analytic on a (m, n)-resonant rotational invariant curve (resonant RIC) and p/q is 'sufficiently close' to m/n. The exponent in this upper bound is closely related to the analyticity strip width of a suitable angular variable. The result is obtained in two steps. First, we prove a Neishtadt-like theorem, in which the n-th power of the twist map is written as an integrable twist map plus an exponentially small remainder on the distance to the RIC. Second, we apply the MacKay-Meiss-Percival action principle. We apply our exponentially small upper bound to several billiard problems. The resonant RIC is a boundary of the phase space in almost all of them. For instance, we show that the lengths (respectively, areas) of all the (1, q)-periodic billiard (respectively, dual billiard) trajectories inside (respectively, outside) analytic strictly convex domains are exponentially close in the period q. This improves some classical results of Marvizi, Melrose, Colin de Verdiere, Tabachnikov, and others about the smooth case.Peer ReviewedPostprint (author's final draft

Magnetic edge states

Magnetic edge states are responsible for various phenomena of magneto-transport. Their importance is due to the fact that, unlike the bulk of the eigenstates in a magnetic system, they carry electric current along the boundary of a confined domain. Edge states can exist both as interior (quantum dot) and exterior (anti-dot) states. In the present report we develop a consistent and practical spectral theory for the edge states encountered in magnetic billiards. It provides an objective definition for the notion of edge states, is applicable for interior and exterior problems, facilitates efficient quantization schemes, and forms a convenient starting point for both the semiclassical description and the statistical analysis. After elaborating these topics we use the semiclassical spectral theory to uncover nontrivial spectral correlations between the interior and the exterior edge states. We show that they are the quantum manifestation of a classical duality between the trajectories in an interior and an exterior magnetic billiard.Comment: 170 pages, 48 figures (high quality version available at http://www.klaus-hornberger.de

On the length and area spectrum of analytic convex domains

Area-preserving twist maps have at least two different $(p,q)$-periodic orbits and every $(p,q)$-periodic orbit has its $(p,q)$-periodic action for suitable couples $(p,q)$. We establish an exponentially small upper bound for the differences of $(p,q)$-periodic actions when the map is analytic on a $(m,n)$-resonant rotational invariant curve (resonant RIC) and $p/q$ is "sufficiently close" to $m/n$. The exponent in this upper bound is closely related to the analyticity strip width of a suitable angular variable. The result is obtained in two steps. First, we prove a Neishtadt-like theorem, in which the $n$-th power of the twist map is written as an integrable twist map plus an exponentially small remainder on the distance to the RIC. Second, we apply the MacKay-Meiss-Percival action principle. We apply our exponentially small upper bound to several billiard problems. The resonant RIC is a boundary of the phase space in almost all of them. For instance, we show that the lengths (respectively, areas) of all the $(1,q)$-periodic billiard (respectively, dual billiard) trajectories inside (respectively, outside) analytic strictly convex domains are exponentially close in the period $q$. This improves some classical results of Marvizi, Melrose, Colin de Verdi\`ere, Tabachnikov, and others about the smooth case

Singular phenomena in the length spectrum of analytic convex curves

Consider the billiard map defined inside an analytic closed strictly convex curve Q. Given q>2 and 0<p<q relatively prime integers, there exist at least two (p,q)-periodic trajectories inside Q. The main goal of this thesis is to study the maximal difference of lengths among (p,q)-periodic trajectories on the billiard, D(p,q). The quantity D(p,q) gives some dynamical and geometrical information. First, it characterizes part of the length spectrum of Q and so it relates to Kac's question, "Can one hear the shape of a drum?''. Second, D(p,q) is an upper bound of Mather's DW(p/q) and so it quantifies the chaotic dynamics of the billiard table. We first focus on the study of the maximal difference of lengths among (1,q)-periodic orbits. These orbits approach the boundary of the billiard table as q tends to infinity. The study of D(1,q) is twofold. On the one hand, we obtain an exponentially small upper bound in the period q for D(1,q). The result is obtained on the general framework of the maximal difference of (p,q)-periodic actions among (p,q)-periodic orbits on analytic exact twist maps. Precisely, we establish an exponentially small upper bound for differences of (p,q)-periodic actions when the map is analytic on a (m,n)-resonant rotational invariant curve and p/q is ``sufficiently close'' to m/n. The exponent in the upper bound is closely related to the analyticity strip width of a suitable angular variable. The result is obtained in two steps. First, we prove a Neishtadt-like theorem. Second, we apply the MacKay-Meiss-Percival action principle. This result implies that the lengths of all the (1,q)-periodic billiard trajectories inside analytic strictly convex domains are exponentially close in the period q, which improves the classical result of Marvizi and Melrose about the smooth case. But it also has several other applications in both classical and dual billiards. For instance, we show that the areas of the (1,q)-periodic dual billiard trajectories outside Q are exponentially close in the period q. This result improves Tabachnikov's classical result about the smooth case. On the other hand, we discuss some exponentially small asymptotic formulas for D(1,q) when the billiard table is a generic axisymmetric analytic strictly convex curve. In this context, we conjecture that the differences behave asymptotically like an exponentially small factor q^(-3)*exp(-rq) times either a constant or an oscillatory function. Also, the exponent r is half of the radius of convergence of the Borel transform of the well-known asymptotic series for the lengths of the (1,q)-periodic trajectories. This conjecture is strongly supported by numerical experiments. Our computations require a multiple-precision arithmetic and we have used PARI/GP. The experiments are restricted to some perturbed ellipses and circles, which allow us to compare the numerical results with some analytical Melnikov predictions and also to detect some non-generic behaviors due to the presence of extra symmetries. The asymptotic formulas we obtain resemble the ones obtained for the splitting of separatrices on many analytic maps, where the behavior of the splitting size is of order h(-m)*exp(-r/h). In such cases, the parameter h>0 is small and continuous so the formulas are exponentially small in 1/h instead. The exponent r has been proved to be (or is strongly numerically supported, depending on the map studied) 2pi times the distance to the real axis of the set of complex singularities of the homoclinic solution of a limit Hamiltonian flow. We propose and study an equivalent limit problem in the billiard setting. Next, we give some insight on how D(p,q) behaves when (p,q)-periodic orbits do not tend to the boundary of Q but to other regions of the phase space. Namely, we consider the cases of p/q tends to an irrational number or to P/Q. The study of D(p,q) in these cases consists of a phenomenological study based on some numerical results.Considereu l'aplicació billard definida dins d'una corba tancada, analítica i estrictament convexa Q. Per q>2 i 00 és petit i contínu i les fórmules són exponencialment petites en 1/h. S'ha demostrat (o està recolzat fortament per experiments numèriques) que l'exponent r és 2Pi vegades la distància a l'eix real del conjunt de singularitats complexes de la solució homoclínica del flux d'un Hamiltonià límit. Proposem i estudiem un equivalent a problema límit per l'aplicació billar. A continuació, comentem com es comporta D(p,q) per òrbites (p,q)-periòdiques que tendeixen a regions de l'espai de fases diferents de la frontera de Q. En concret, considerem els casos de p/q tendint a un nombre irracional o a P/Q. L'estudi de D(p,q) en aquests casos es basa en un estudi numèric dels fenòmens

Marked Length Spectral determination of analytic chaotic billiards with axial symmetries

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 non-escaping orbits is conjugated to a subshift, which provides a natural labeling of periodic orbits. We show that under suitable symmetry and genericity assumptions, the Marked Length Spectrum determines the geometry of the billiard table.Comment: 57 pages, 8 figure

Elliptic Quantum Billiard

The exact and semiclassical quantum mechanics of the elliptic billiard is investigated. The classical system is integrable and exhibits a separatrix, dividing the phasespace into regions of oscillatory and rotational motion. The classical separability carries over to quantum mechanics, and the Schr\"odinger equation is shown to be equivalent to the spheroidal wave equation. The quantum eigenvalues show a clear pattern when transformed into the classical action space. The implication of the separatrix on the wave functions is illustrated. A uniform WKB quantization taking into account complex orbits is shown to be adequate for the semiclassical quantization in the presence of a separatrix. The pattern of states in classical action space is nicely explained by this quantization procedure. We extract an effective Maslov phase varying smoothly on the energy surface, which is used to modify the Berry-Tabor trace formula, resulting in a summation over non-periodic orbits. This modified trace formula produces the correct number of states, even close to the separatrix. The Fourier transform of the density of states is explained in terms of classical orbits, and the amplitude and form of the different kinds of peaks is analytically calculated.Comment: 33 pages, Latex2e, 19 figures,macros: epsfig, amssymb, amstext, submitted to Annals of Physic

Inverse spectral problem for analytic plane domains I: Balian-Bloch trace formula

We give a rigorous version of the classical Balian-Bloch trace formula, a semiclassical expansion around a periodic reflecting ray of the (regularized) resolvent of the Dirichlet Laplacian on a bounded smooth plane domain. It is equivalent to the Poisson relation (or wave trace formula) between spectrum and closed geodesics. We view it primarily as a computational device for explicitly calculating wave trace invariants. Its effectiveness will be illustrated in subsquent articles in the series in which concrete inverse spectral results are proved.Comment: First in a series on the inverse spectral problem for analytic plane domains. 53 pages, 1 figure. Added some reference

Approximating invariant densities of metastable systems

We consider a piecewise smooth expanding map of the interval possessing two invariant subsets of positive Lebesgue measure and exactly two ergodic absolutely continuous invariant probability measures (ACIMs). When this system is perturbed slightly to make the invariant sets merge, we describe how the unique ACIM of the perturbed map can be approximated by a convex combination of the two initial ergodic ACIMs.Comment: 19 pages, 6 figure