Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajectories

McDonald, Steven W.; Kaufman, Allan N.

doi:10.1103/physrevlett.42.1189

Cited by 597 publications

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“…Spectral fluctuations are a principle example as they gave the first support to one of the main results linking classical chaos and random matrix theory [1], the Bohigas-Giannoni-Schmit conjecture [2,3]. Needless to say, the statistical properties of eigenfunctions are also a subject of paramount interest [4,5,6,7,8,9,10,11,12].For chaotic systems, a widely accepted starting point for the treatment of eigenfunction fluctuations locally, such as the amplitude distribution or the two-point correlation function c(|r − r ′ |) = ψ(r)ψ(r ′ ) of a given eigenfunction ψ, is a modeling in terms of a random superposition of plane waves (RPW) [4,5]. For two-degree-of-freedom systems, c(r) can be understood as being given approximately by a Bessel function.…”

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Residual Coulomb Interaction Fluctuations in Chaotic Systems: The Boundary, Random Plane Waves, and Semiclassical Theory

2008

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New fluctuation properties arise in problems where both spatial integration and energy summation are necessary ingredients. The quintessential example is given by the short-range approximation to the first order ground state contribution of the residual Coulomb interaction. The dominant features come from the region near the boundary where there is an interplay between Friedel oscillations and fluctuations in the eigenstates. Quite naturally, the fluctuation scale is significantly enhanced for Neumann boundary conditions as compared to Dirichlet. Elements missing from random plane wave modeling of chaotic eigenstates lead surprisingly to significant errors, which can be corrected within a purely semiclassical approach. The characterization of quantum systems with any of a variety of underlying classical dynamics, ranging from diffusive to chaotic to regular, has often demonstrated that the study of their statistical properties is of primary importance. Spectral fluctuations are a principle example as they gave the first support to one of the main results linking classical chaos and random matrix theory [1], the Bohigas-Giannoni-Schmit conjecture [2,3]. Needless to say, the statistical properties of eigenfunctions are also a subject of paramount interest [4,5,6,7,8,9,10,11,12].For chaotic systems, a widely accepted starting point for the treatment of eigenfunction fluctuations locally, such as the amplitude distribution or the two-point correlation function c(|r − r ′ |) = ψ(r)ψ(r ′ ) of a given eigenfunction ψ, is a modeling in terms of a random superposition of plane waves (RPW) [4,5]. For two-degree-of-freedom systems, c(r) can be understood as being given approximately by a Bessel function. For distances |r − r ′ | short compared to the system size, and in the absence of effects related to classical dynamics [8,9,13], this is roughly observed in numerical [7,14,15] and experimental [16] studies. Our interest here is in statistical properties of eigenfunctions going beyond local quantities such as c(r).One motivation for the introduction of these new statistical measures is to study the interplay between interferences and interactions in mesoscopic systems. For typical electronic densities, the screening length is close to the Fermi wavelength λ F , and the screened Coulomb interaction can be approximated by the short range expression V sc (r − r ′ ) = (2ν) −1 F a 0 δ(r − r ′ ) with 2ν the mean local density of states, including spin degeneracy, (ν = m/2π 2 for d = 2) and To this level of approximation, the first order ground state energy contribution of the residual interactions can be expressed in the formwith N σ the unperturbed ground state density of particles with spin σ. From this expression, it is seen that the increase of interaction energy associated with the addition of an extra electron is related toand the understanding that E i < E + i < E i+1 . Our goal in this letter is to study the fluctuation properties of the S i , concentrating on the case of two dimensional billiards with either Dirichlet ...

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Residual Coulomb Interaction Fluctuations in Chaotic Systems: The Boundary, Random Plane Waves, and Semiclassical Theory

2008

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“…Historically, testing a quantum system for integrability has proved to be a much harder problem than its classical counterpart. Efforts in this direction have ranged across systematically searching for quantum integrals of motion using Moyal brackets [35,36], constructing hypothetical second invariants [37], finding suitable Lax pairs [38], and perhaps most ubiquitously, examining the statistical properties of the quantised energy spectrum [39]. In particular, soon after Percival's conjecture [40], it was realized that the level-spacing distributions of integrable systems, harmonic oscillators notwithstanding, exhibit Poisson-like behavior [41].…”

Section: Resultsmentioning

confidence: 99%

A difference-equation formalism for the nodal domains of separable billiards

2016

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Recently, the nodal domain counts of planar, integrable billiards with Dirichlet boundary conditions were shown to satisfy certain difference equations in [Ann. Phys. 351, 1 (2014)]. The exact solutions of these equations give the number of domains explicitly. For complete generality, we demonstrate this novel formulation for three additional separable systems and thus extend the statement to all integrable billiards.

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“…The phenomenon was initially observed by McDonald and Kaufman [44], and posteriorly in the Bunimovich billiard by Heller [45]. Heller called this phenomena a scar.…”

Section: Quantum Diamond-shaped Billiard With Rounded Crown: Level Stmentioning

confidence: 93%

Chaos in the Diamond-Shaped Billiard with Rounded Crown

Salazar

Téllez

Jaramillo

et al. 2015

Rev. Acad. Colomb. Cienc. Ex. Fis. Nat.

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We analyse the classical and quantum behaviour of a particle trapped in a diamond-shaped billiard with rounded crown. We defined this billiard as a half stadium connected with a triangular billiard. A parameter ξ smoothly changes the shape of the billiard from an equilateral triangle (ξ = 1) to a diamond with rounded crown (ξ = 0). The parameter ξ controls the transition between the regular and chaotic regimes. The classical behaviour is regular when the control parameter ξ is one; in contrast, the system is chaotic when ξ = 1 even for values of ξ close to one. Several quantities such as Lyapunov exponent and the entropy of the distribution of the incident angle are used to characterize the chaotic behaviour of the classical system. The average information preserved by the classical trajectories increases rapidly as ξ is decreased from 1 and the Lyapunov exponent remains positive for ξ < 1. The Finite Difference Method was implemented in order to solve the quantum counterpart of the billiard. The energy spectrum and eigenstates were numerically computed for different values of ξ < 1. The spacing distribution between adjacent eigenvalues is analysed as a function of ξ, finding a Poisson and a Gaussian Orthogonal Ensemble (GOE) distribution for regular and chaotic regimes respectively. Several scars and bouncing ball states are shown with their corresponding classical periodic orbits. On the other hand, the results found for the quantum billiard are in agreement with the Bohigas-Giannoni-Schmit conjecture and exhibits the standard features of chaotic billiards such as the scarring of the wavefunction.Keywords: quantum chaos, quantum billiards, random matrices, finite difference method.Caos en el billar de forma de diamante y corona redondeada Resumen Se estudia el comportamiento de una partícula en el interior de un billar triangular donde uno de sus lados toma de medio estadio que se llamó billar diamante con corona redondeada o DSRC por su siglas en inglés. Se definió un parámetro ξ que cambia suavemente la forma la frontera partiendo de un billar triangular ξ = 1 a un billar DSRC ξ = 1. Dicho parámetro controla la transición entre el régimen regular y caótico. Clásicamente, el sistema es regular cuando ξ = 1. Por otro lado, el sistema se torna caótico para ξ = 1 incluyendo valores próximos a 1. Se calcula el coeficiente de Lyapunov y la entropía media de la distribución de losángulos de incidencia para caracterizar el comportamiento caótico del sistema. Se observó un rápido crecimiento de la información de las trayectorias hasta saturar la entropía al cambiar levemente la frontera del billar triangular original. A su vez el coeficiente de Lyapunov se mantuvo positivo durante este proceso una vez que ξ se alejaba de 1. Se implementó el método de diferencias finitas FDM para obtener el espectro y los estados propios de la contraparte cuántica del sistema. La distribución de espaciamiento entre primeros vecinos para varios valores de ξ fue construida numéricamente para diferentes valores de ξ encontrando u...

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Spectrum and Eigenfunctions for a Hamiltonian with Stochastic Trajectories

Cited by 597 publications

References 10 publications

Residual Coulomb Interaction Fluctuations in Chaotic Systems: The Boundary, Random Plane Waves, and Semiclassical Theory

Residual Coulomb Interaction Fluctuations in Chaotic Systems: The Boundary, Random Plane Waves, and Semiclassical Theory

A difference-equation formalism for the nodal domains of separable billiards

Chaos in the Diamond-Shaped Billiard with Rounded Crown

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