Isospectrality in chaotic billiards

Dhar, Abhishek; Rao, D Madhusudhana; N., Udaya Shankar; Sridhar, S.

doi:10.1103/physreve.68.026208

Cited by 26 publications

(25 citation statements)

References 10 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The procedure of designing such isospectral planar domains consists of appropriate cutting the 'drum' into subdomains and rearranging them into a new one with the same spectrum. An experimental confirmation that 'hearing' the shape is impossible was presented by Sridhar and Kudrolli [5] and Dhar et al [6] for a pair of isospectral microwave cavities.…”

Section: Introduction Isospectrality and Isoscatteringmentioning

confidence: 69%

Resonances and poles in isoscattering microwave networks and graphs

et al. 2014

View full text Add to dashboard Cite

Can one hear the shape of a graph? This is a modification of the famous question of Mark Kac "Can one hear the shape of a drum?" which can be asked in the case of scattering systems such In this paper we consider important local characteristics of graphs such as structures of resonances and poles of the determinant of the scattering matrices. Using these characteristics we study experimentally and theoretically properties of graphs and directly confirm that the pair of graphs considered in the cited paper is isoscattering. The experimental results are compared to the theoretical ones for a broad frequency range from 0.01 to 3 GHz. In the numerical calculations of the resonances of the graphs absorption present in the experimental networks is taken into account.

show abstract

Section: Introduction Isospectrality and Isoscatteringmentioning

confidence: 69%

Resonances and poles in isoscattering microwave networks and graphs

et al. 2014

View full text Add to dashboard Cite

show abstract

“…The numerical difficulty is highlighted by the fact that analog computation using microwave cavities is still popular in awkward geometries. 3,14 In contrast, we use boundary-based methods, as explained in Sec. II.…”

Section: Introductionmentioning

confidence: 99%

Quantum mushroom billiards

Barnett

Betcke

2007

Chaos

View full text Add to dashboard Cite

We report the first large-scale statistical study of very high-lying eigenmodes ͑quantum states͒ of the mushroom billiard proposed by L. A. Bunimovich ͓Chaos 11, 802 ͑2001͔͒. The phase space of this mixed system is unusual in that it has a single regular region and a single chaotic region, and no KAM hierarchy. We verify Percival's conjecture to high accuracy ͑1.7%͒. We propose a model for dynamical tunneling and show that it predicts well the chaotic components of predominantly regular modes. Our model explains our observed density of such superpositions dying as E −1/3 ͑E is the eigenvalue͒. We compare eigenvalue spacing distributions against Random Matrix Theory expectations, using 16 000 odd modes ͑an order of magnitude more than any existing study͒. We outline new variants of mesh-free boundary collocation methods which enable us to achieve high accuracy and high mode numbers ͑ϳ10 5 ͒ orders of magnitude faster than with competing methods. © 2007 American Institute of Physics. ͓DOI: 10.1063/1.2816946͔ Quantum chaos is the study of the quantum (wave) properties of Hamiltonian systems whose classical (ray) dynamics is chaotic. Billiards are some of the simplest and most studied examples; physically their wave analogs are vibrating membranes, quantum, electromagnetic, or acoustic cavities. They continue to provide a wealth of theoretical challenges. In particular "mixed" systems, where ray phase space has both regular and chaotic regions (the generic case), are difficult to analyze. Six years ago Bunimovich described 1 a mushroom billiard with simple mixed dynamics free of the usual island hierarchies of Kolmogorov-Arnold-Moser (KAM). He concluded by anticipating the growth of "quantum mushrooms;" it is this gardening task that we achieve here, by developing advanced numerical methods to collect an unprecedented large number n of eigenmodes (much higher than competing numerics 2 or microwave studies 3 ). Since uncertainties scale as n −1Õ2 , a large n is vital for accurate spectral statistics and for studying the semiclassical (high eigenvalue) limit. We address three main issues: (i) The conjecture of Percival 4 that semiclassically modes live exclusively in invariant (regular or chaotic) regions, and occur in proportion to the phase space volumes. (ii) The mechanism for dynamical tunneling, or quantum coupling between classically isolated phase space regions. (iii) The distribution of spacings of nearest-neighbor eigenvalues, about which recent questions have been raised. 3 We show many pictures of modes, including the boundary phase space (the so-called Husimi function).

show abstract

“…6 left, one obtains an example of a pair of chaotic billiards with holes. Similarly Dhar et al (2003) constructed chaotic isospectral billiards based on the same idea: scattering circular disks were added inside the base triangular shape in a way consistent with the unfolding.…”

Section: A Paper-folding Proofmentioning

confidence: 99%

“…Indeed, as shown in Table II, the measured 9-th eigenvalue is very close to its theoretical value E = 5π 2 /d 2 . Later Dhar et al (2003) applied a similar technique to a chaotic isospectral billiard made of the billiard with half-square base tile with scattering circular disks inside, showing experimentally that isospectrality is indeed retained, provided scatterers are added in a way consistent with the unfolding rules.…”

Section: Electromagnetic Waves In Metallic Cavitiesmentioning

confidence: 99%

Hearing shapes of drums: Mathematical and physical aspects of isospectrality

Giraud

Thas

2010

Rev. Mod. Phys.

View full text Add to dashboard Cite

In a celebrated paper "Can one hear the shape of a drum?" M. Kac [Amer. Math. Monthly 73, 1 (1966)] asked his famous question about the existence of nonisometric billiards having the same spectrum of the Laplacian. This question was eventually answered positively in 1992 by the construction of noncongruent planar isospectral pairs. This review highlights mathematical and physical aspects of isospectrality. CONTENTS I. Introduction 2 II. A Pedestrian Proof of Isospectrality 4 A. Paper-folding proof 4 B. Transplantation proof 6 III. Further Examples in Higher Dimensions 8 A. Lattices and flat tori 8 B. Construction of examples 9 C. The four-parameter family of Conway and Sloane 9 D. The eigenvalue spectrum as moduli for flat tori 10

show abstract

Isospectrality in chaotic billiards

Abstract: We consider a modification of isospectral cavities whereby the classical dynamics changes from pseudointegrable to chaotic. We construct an example where we can prove that isospectrality is retained. We then demonstrate this explicitly in microwave resonators.

Cited by 26 publications

References 10 publications

Resonances and poles in isoscattering microwave networks and graphs

Resonances and poles in isoscattering microwave networks and graphs

Quantum mushroom billiards

Hearing shapes of drums: Mathematical and physical aspects of isospectrality

Contact Info

Product

Resources

About