Scale anomaly and quantum chaos in billiards with pointlike scatterers.

We argue that the random-matrix like energy spectra found in pseudointegrable billiards with pointlike scatterers are related to the quantum violation of scale invariance of classical analogue system. It is shown that the behavior of the running coupling constant explains the key characteristics of the level statistics of pseudointegrable billiards. 5.45.+b, 3.65.Db, 11.10.Gh Typeset using REVTEX 1 The concepts of scale anomaly and the asymptotic freedom are among the key features of the gauge field theories which describe the interaction of the elementary particles. It is less widely recognized, however, that the scale anomaly can be found in a vastly simpler setting of one particle quantum mechanics. A particle scattered off a pointlike scatterer in two spacial dimension is known to have energy dependent s-wave phase shift defying the scale invariance of its classical analogue [1]. There exists a sister problem of particle motion confined in a hard-wall boundary with a pointlike scatterer inside. When the shape of the boundary is a rectangle, the problem belongs to a larger category of systems known as pseudointegrable billiards [2–5]. This system is known for puzzling statistical properties of its energy eigenvalues [3,5]. It is shown through numerical experiments that the level statistics of the pseudointegrable billiard resembles to that of random-matrix ensembles [6] which is generally associated with chaotic dynamics [7]. This is in seeming contradiction with the absence of chaotic dynamics in classical analogue system. Further, when the levels are collected at higher energy region, the level statistics moves toward the Poisson distribution which characterizes the integrable classical dynamics. Also, the system tends to show more of Wigner-like statistics when the genus of the billiard is increased, that is, in the present context, when the number of the singular scatterers is increased. These facts have never received sufficient explanations, in spite of several attempted studies based on the semiclassical periodic orbit quantization theory [8,9]. In this paper, we argue that the behavior of spectra of the pseudointegrable billiard with pointlike scatterers is a direct result of the scale anomaly of the system. Specifically, the dependence of the level statistics on the energy and the number of the scatterers is shown to be well explained by the high energy behavior of the effective coupling strength of the pointlike scatterer. We consider a quantum particle of unit mass moving freely inside a boundary B in two spacial dimension on which its wave functions are assumed to vanish. We denote the eigenvalue and eigenfunction of the system as εn and φn, namely