FIG 19 - uploaded by Eugene Bogomolny
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In this figure the isolated periodic orbit connects the two open circles. A nearby path from r to r is represented. There is an inversion along the periodic orbit for the transverse coordinate y see the text. The diffractive periodic orbit goes from one open circle to the apex at r 0 and to the other open circle. It has a length L d L2 2 /L.
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We derive contributions to the trace formula for the spectral density accounting for the role of diffractive orbits in two-dimensional polygonal billiards. In polygons, diffraction typically occurs at the boundary of a family of trajectories. In this case the first diffractive correction to the contribution of the family to the periodic orbit expan...
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Citations
... On the other hand, as we shall see later in this paper, we are able to recover the product of the sine of fluxes using the leading singularities of closed diffractive geodesics. Bogomolny-Pavloff-Schmit [BPS00] studied the diffractive singularities of rectangular billiards with one solenoid using the geometric theory of diffraction. They were only able to deal with the diffractive singularities of one solenoid in a rectangular domain. ...
We study leading order singularities of the wave trace of the Aharonov–Bohm Hamiltonian on R2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf {R}^2$$\end{document} with multiple solenoids under a generic assumption that no three solenoids are collinear. Then we apply our formula to get a lower bound of scattering resonances in a logarithmic neighborhood near the real axis.
... On the other hand, as we shall see later in this paper, we are able to recover the product of the sine of fluxes using the leading singularities of closed diffractive geodesics. Bogomolny-Pavloff-Schmit [BPS00] studied the diffractive singularities of rectangular billiards with one solenoid using the geometric theory of diffraction. They were only able to deal with the diffractive singularities of one solenoid in a rectangular domain. ...
We study leading order singularities of the wave trace of the Aharonov--Bohm Hamiltonian on $\mathbf{R}^2$ with multiple solenoids under a generic assumption that no three solenoids are collinear. Then we apply our formula to get a lower bound of scattering resonances in a logarithmic neighborhood near the real axis.
... Difficult problems appear when inside these intermediate regions there are other points of singular diffractions which is inevitable for plane polygonal billiards. For a finite number of singular diffraction vertices it is possible to develop the uniform approximations which give a good description of the multiple singular diffraction in the semiclassical limit ¥ k (see, e.g., [21] and references therein). For an infinite number of singular diffractions the situation is less clear. ...
... The simplest method consists in the construction of the Kirchhoff-type approximation to this problem. It has been done in [21] and briefly reviewed in section 2.1. It is known that the condition of applicability of the Kirchhoff approximation is not easy to be rigorously established. ...
... In this approximation the role of wedges is reduced to the restriction of integration domains to half-lines ( ) ¥ 0, (cf figure 5). This problem has been investigated in [21] where it has been proved that the contribution to the trace formula from such trajectories for a finite number (n) of wedges (i.e. ( ) + n 1 -fold integral over all x j at figure 5) can be calculated analytically and the result is ...
Polygonal billiards constitute a special class of models. Though they have zero Lyapunov exponent their classical and quantum properties are involved due to scattering on singular vertices with angles ≠ π / n with integer n . It is demonstrated that in the semiclassical limit the multiple singular scattering on such vertices, when optical boundaries of many scatters overlap, leads to the vanishing of quantum wave functions along straight lines built by these scatters. This phenomenon has an especially important consequence for polygonal billiards where periodic orbits (when they exist) form pencils of parallel rays restricted from the both sides by singular vertices. Due to the singular scattering on boundary vertices, waves propagated inside a periodic orbit pencil tend in the semiclassical limit to zero along pencil boundaries thus forming weakly interacting quasi-modes. Contrary to scars in chaotic systems, the discussed quasi-modes in polygonal billiards become almost exact for high-excited states and for brevity they are designated as superscars. Many pictures of eigenfunctions for a triangular billiard and a barrier billiard which have clear superscar structures are presented in the paper. Special attention is given to the development of quantitative methods of detecting and analysing such superscars. In particular, it is demonstrated that the overlap between superscar waves associated with a fixed periodic orbit and eigenfunctions of a barrier billiard is distributed according to the Breit-Wigner distribution typical for weakly interacting quasi-modes. For special sub-class of rational polygonal billiards called Veech polygons where all periodic orbits can be calculated analytically it is argued and checked numerically that their eigenfunctions are fractal in the Fourier space.
... However in the presence of singularities such as wedges or corners such expansions have to be modified. This reflects the fact that there exist so-called diffractive orbits which obey the law of classical mechanics everywhere except on the singularities of the potential (see for instance [52]). In our simple one-dimensional setting, there are only two classical paths connecting y and x in a given time t, namely the direct path and the one which is reflected at the origin. ...
We study the statistics of last-passage time for linear diffusions. First we present an elementary derivation of the Laplace transform of the probability density of the last-passage time, thus recovering known results from the mathematical literature. We then illustrate them on several explicit examples. In a second step we study the spectral properties of the Schrödinger operator associated to such diffusions in an even potential U(x) = U(- x) , unveiling the role played by the so-called Weyl coefficient. Indeed, in this case, our approach allows us to relate the last-passage times for dual diffusions (i.e., diffusions driven by opposite force fields) and to obtain new explicit formulae for the mean last-passage time. We further show that, for such even potentials, the small time t expansion of the mean last-passage time on the interval [0, t] involves the Korteveg–de Vries invariants, which are well known in the theory of Schrödinger operators. Finally, we apply these results to study the emptying time of a one-dimensional box, of size L, containing N independent Brownian particles subjected to a constant drift. In the scaling limit where both N→ ∞ and L→ ∞, keeping the density ρ= N/ L fixed, we show that the limiting density of the emptying time is given by a Gumbel distribution. Our analysis provides a new example of the applications of extreme value statistics to out-of-equilibrium systems. © 2020, Springer Science+Business Media, LLC, part of Springer Nature.
... For finite number of singular diffraction vertices it is possible to develop uniform approximations which give good description of multiple singular diffraction in the semiclassical limit k → ∞ (see e.g. [21] and references therein). For infinite number of singular diffractions the situation is less clear. ...
... The simplest method consists in the construction of the Kirchhoff-type approximation to this problem. It has been done in Ref. [21] and briefly reviewed in Section II A. It is known that the condition of applicability of the Kirchhoff approximation is not easy to be rigorously established. To get more precise information of this process, the exact solution for the scattering on staggered periodic set of half-planes as indicated at Fig. 6 a) derived by Carlson and Heins in 1947 [22] and analysed in the semiclassical limit in [23] is discussed in Section II B. Section II C is devoted to numerical investigation of wave propagation inside periodic array of slits depicted at Fig. 6 b). ...
... Fig. 5). This problem has been investigated in Ref. [21] where it has been proved that the contribution to the trace formula from such trajectories (i.e. (n + 1)-fold integral over all x j at Fig. 5) can be calculated analytically even for a finite number (n) of wedges and the result is ...
Polygonal billiards constitute a special class of models. Though they have zero Lyapunov exponent their classical and quantum properties are involved due to scattering on singular vertices. It is demonstrated that in the semiclassical limit multiple singular scattering on such vertices when optical boundaries of many scatters overlap leads to vanishing of quantum wave functions along straight lines built by these scatters. This phenomenon has an especially important consequence for polygonal billiards where periodic orbits (when they exist) form pencils of parallel rays restricted from the both sides by singular vertices. Due to singular scattering on boundary vertices, waves propagated inside periodic orbit pencils in the semiclassical limit tend to zero along pencil boundaries thus forming weakly interacting quasi-modes. Contrary to scars in chaotic systems the discussed quasi-modes in polygonal billiards become almost exact for high-excited states and for brevity they are designated as superscars. Many pictures of eigenfunctions for a triangular billiard and a barrier billiard which have clear superscar structures are presented in the paper. Special attention is given to the development of quantitative methods of detecting and analysing such superscars. In particular, it is demonstrated that the overlap between superscar waves associated with a fixed periodic orbit and eigenfunctions of a barrier billiard is distributed according to the Breit-Wigner distribution typical for weakly interacting quasi-modes (or doorway states). For special sub-class of rational polygonal billiards called Veech polygons where all periodic orbits can be calculated analytically it is argued and checked numerically that their eigenfunctions are fractal in the Fourier space.
... Or, there is the uniform formula for the far wavefield on the underlying on the helicoid surface (Richens 1982, Berry 2010. These latter, and also a uniform small wavelength formula in Sieber (1999) (see also (Bogomolny et al 2000)), that can be reinterpreted as a far distance formula, are the closest treatments to the one presented here. All of them contain the most important ingredient; a Fresnel integral (the complex error function). ...
A uniform asymptotic formula for the Aharonov–Bohm wavefield (that of a plane quantum wave scattered by a thin straight solenoid) far away from the solenoid is obtained in a direct way. Actually quite good accuracy is achieved even down to one wavelength away. The error is numerically of order radius^(−3/2) for all values of polar angle, including directly forwards. Several previous formulas, uniform and otherwise, for the far field limit exist in the literature. All contain the essential ingredient: the Fresnel integral (complex error function), but ordinarily the error in these formulas is of order radius^(−1/2) in the forwards direction where the Fresnel contribution is most important.
... We first use the method of reflections to express the Dirichlet and Neumann wave kernels in terms of the wave kernel of the double of the trapezoid, which can be realized as a Euclidean surface with conical singularities (ESCS). We then obtain new wave trace invariants using two results of Hillairet [31] for ESCSs (see also [5] for results in the physics literature). The first is a parametrix construction of the wave propagator near diffractive geodesics as a Fourier Integral Operator (FIO). ...
... There has also been a lot of research on the contribution of diffractive geodesics to the wave trace: Friedlander [23], Durso [19], Wunsch [59], Hillairet [30,31], Ford and Wunsch [21], and more recently Ford et al. [20]. In the physics literature, we also note the works of Bogomolny et al. [5] and Pavloff and Schmit [49]. A standard technique used in studying the wave trace on polygonal tables is to double the polygon along its edges to obtain a compact Euclidean surface with conical singularities, or ESCS, as commonly abbreviated in the literature. ...
We show that non-obtuse trapezoids with identical Neumann spectra are congruent up to rigid motions of the plane. The proof is based on heat trace invariants and some new wave trace invariants associated to certain diffractive billiard trajectories. We use the method of reflections to express the Dirichlet and Neumann wave kernels in terms of the wave kernel of the double polygon. Using Hillairet’s trace formulas for isolated diffractive geodesics and one-parameter families of regular geodesics with geometrically diffractive boundaries for Euclidean surfaces with conical singularities (Hillairet in J Funct Anal 226(1):48–89, 2005), we obtain the new wave trace invariants for trapezoids. To handle the reflected term, we use another result of Hillairet (J Funct Anal 226(1):48–89, 2005), which gives a Fourier integral operator representation for the Keller and Friedlander parametrix (Keller in Proc Symp Appl Math 8:27–52, 1958; Friedlander in Math Proc Camb Philos Soc 90(2):335–341, 1981) of the wave propagator near regular diffractive geodesics. The reason we can only treat the Neumann case is that the wave trace is “more singular” for the Neumann case compared to the Dirichlet case. This is a new observation which is of independent interest.
... For instance, Wunsch showed in [Wun02] that singularities may appear at length of periodic diffractive orbits. For some periodic diffractive geodesics, the leading singularity is then computed in [Hil05] in the Euclidean case and in [FW] in a more general case (see also [BPS00] for related results from a physics perspective). Both these results are built upon a precise description of the wave propagator that is microlocalized in the vicinity of given periodic (possibly diffractive) geodesic. ...
... It is interesting to note that a α,±,1 (q 1 , q 2 , s = 0, ω) is actually a regularization of the symbol on the diffracted front. The latter blows up when approaching the intersection and this formula gives an effective way of regularizing the contribution of a diffractive geodesic when the diffraction angle approaches ±π (compare with the approach of [BPS00]). ...
We investigate the singularities of the trace of the half-wave group,
$\mathrm{Tr} \, e^{-it\sqrt\Delta}$, on Euclidean surfaces with conical
singularities $(X,g)$. We compute the leading-order singularity associated to
periodic orbits with successive degenerate diffractions. This result extends
the previous work of the third author \cite{Hil} and the two-dimensional case
of the work of the first author and Wunsch \cite{ForWun} as well as the seminal
result of Duistermaat and Guillemin \cite{DuiGui} in the smooth setting. As an
intermediate step, we identify the wave propagators on $X$ as singular Fourier
integral operators associated to intersecting Lagrangian submanifolds,
originally developed by Melrose and Uhlmann \cite{MelUhl}.
... On the other hand, a wave impinging on such a corner is diffracted into various directions. This leads to contributions from diffractive POs for example in the trace formulas for closed resonators [58,59]. Another important point is that the dynamics of wave systems is less prone to geometrical perturbations than classical systems. ...
We investigated experimentally the ray-wave correspondence in organic
microlasers of various triangular shapes. Triangular billiards are of interest
since they are the simplest cases of polygonal billiards and the existence and
properties of periodic orbits in general triangles are not yet fully
understood. The microlasers with symmetric shapes that were investigated
exhibited states localized on simple periodic orbits, and almost all of their
lasing characteristics like spectra and far-field distributions could be well
explained by simple ray-optical calculations. Furthermore, asymmetric triangles
that do not feature simple periodic orbits were studied. The modes of these
microlasers were not localized on classical periodic orbits as for the
symmetric triangles, but instead seem to be related to diffractive orbits.
... Le cas des billards métalliques ayant la forme d'un triangle rationnel (la signification de rationnel est donnée à la section suivante) a souvent été étudié grâce à l'approche des superscars [77] et souvent dans le but de quantifier l'influence de la diffraction par les coins sur les spectres de résonances des résonateurs [78,79,80]. ...
We investigate in this thesis fundamental and applied properties of solid-state laser micro-sources made of organic materials, with possible applications to information and biosensing technologies. We explored three-dimensional features pertaining both to fabrication and characterization of such lasers. Regarding the fabrication, we extended the geometry of organic microlasers, previously restricted to quasi-two-dimensional (2D) as from thin film patterning, onto full 3D structures such as cubes. Deep UV lithography and direct laser writing with two-photon-polymerization processes have been adapted in order to fabricate customized shapes, which incorporate a laser dye. To study the highly anisotropic emission from these lasers, we conceived a new set-up, called solid angle scanner (SAS), allowing for high angular accuracy detection of the emission from a micro-laser in all directions in space. When applied to 2D micro-lasers, SAS measurements allowed us to observe that they emit mainly out of their plane. We developed a model to account for this effect and infer predictions. Moreover, various shapes of 2D micro-lasers have been investigated, through the angular and spectral features of their emission, with experiments satisfactorily connected to a semi-classical theoretical approach of periodic orbits. We paid special attention to triangular shapes, for which a diffractive orbit was observed, opening the way to the study of diffraction by a dielectric corner. We also propose an explanation for the directionality of the emission by square micro-lasers. Finally, 3D characterizations of solid state 3D organic micro-lasers are presented for the first time to our knowledge.
