Quantum and Arithmetical Chaos

Bogomolny, E.

doi:10.1007/3-540-31347-8_1

Cited by 17 publications

(31 citation statements)

References 45 publications

(102 reference statements)

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“…where the functions u ± (τ) are periodic with period T γ and u ± are independent eigenvectors of the monodromy matrix M. An expression of the monodromy matrix in terms of second derivatives of the action function S can be derived (Bogomolny 2006). The action function S is defined by a trajectory from the position q i to q f for a given energy or frequency ω (Gutzwiller 1990).…”

Section: Solutions Of the Parabolic Wave Equationmentioning

confidence: 99%

Regular oscillation sub-spectrum of rapidly rotating stars

Pasek

Lignières

Georgeot

et al. 2012

A&A

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Aims. We present an asymptotic theory that describes regular frequency spacings of pressure modes in rapidly rotating stars. Methods. We use an asymptotic method based on an approximate solution of the pressure wave equation constructed from a stable periodic solution of the ray limit. The approximate solution has a Gaussian envelope around the stable ray, and its quantization yields the frequency spectrum. Results. We construct semi-analytical formulas for regular frequency spacings and mode spatial distributions of a subclass of pressure modes in rapidly rotating stars. The results of these formulas are in good agreement with numerical data for oscillations in polytropic stellar models. The regular frequency spacings depend explicitly on internal properties of the star, and their computation for different rotation rates gives new insights on the evolution of mode frequencies with rotation.

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Section: Solutions Of the Parabolic Wave Equationmentioning

confidence: 99%

Regular oscillation sub-spectrum of rapidly rotating stars

Pasek

Lignières

Georgeot

et al. 2012

A&A

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“…For arithmetic surfaces, although the distribution is very irregular, the average over intervals [l, l + l] is at least ce l/2 /l [7,8]. Numerical results also suggest such exponential growth (with a smaller exponent) for certain non-arithmetic surfaces associated to Hecke triangle groups [6,9]. In contrast, one sees that metric graphs give rise to much greater multiplicities.…”

Section: Theorem 1 Under Assumptions 1 and 2mentioning

confidence: 73%

“…It is widely conjectured that typical chaotic systems satisfy Random Matrix statistics. However, numerical results for arithmetic surfaces indicate Poissonian statistics [6] and it is believed that this may be connected to the rather high multiplicities in the length spectrum. In contrast, under certain conditions, families of progressively larger (quantum) metric graphs exhibit Random Matrix statistics in the limit.…”

Section: Theorem 1 Under Assumptions 1 and 2mentioning

confidence: 96%

Degeneracy in the length spectrum for metric graphs

Sharp

2010

Geom Dedicata

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In this note we show that the length spectrum for metric graphs exhibits a very high degree of degeneracy. More precisely, we obtain an asymptotic for the number of pairs of closed geodesics (or closed cycles) with the same metric length. Mathematics Subject Classification IntroductionLet G = (V, E) be a finite (connected) graph with vertices V and edges E. We write E o for the set of oriented edges; for e ∈ E o ,ē ∈ E o denotes the same unoriented edge with orientation reversed. (In the physics literature, the vertices are referred to as nodes and the edges as bonds.) The degree deg(v) of a vertex v is the number of outgoing oriented edges. A path in G is a sequence of successive oriented edges and is called non-backtracking if an oriented edge e is not immediately followed byē. A path is called closed if it returns to its starting point. A closed geodesic on G is a closed non-backtracking path, modulo the obvious cyclic permutation and we denote the (countable) set of closed geodesics by C. For γ ∈ C, we write |γ | for the number of edges it traverses.We make G into a metric graph by giving each edge e ∈ E a length l(e) > 0 and thus identifying it with the interval [0, l(e)]. (We may also regard l as a function on E o satisfying l(ē) = l(e).) For γ ∈ C, we define the metric length l(γ ) to be the sum of the lengths of the edges it traverses.We impose two assumptions on G and l:Assumption 1 Each vertex has degree at least 3.

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“…This statement is made precise by the Quantum Unique Ergodicity (QUE) conjecture, which posits that hyperbolic surfaces have no quantum scars and that all eigenstates are ergodic [64]. Perhaps ironically, the one set of manifolds for which QUE has been rigorously established is the set of arithmetic billiards [65,66], for which eigenvalue statistics is known to be that of integrable systems [67,68]. This happens because the manifolds in question are quotients of the hyperbolic plane by subgroups of the modular group, leaving a large number of symmetries present in the system -without making the theory integrable.…”

Section: Quantum Mechanics In Two Dimensionsmentioning

confidence: 99%

The ergodicity landscape of quantum theories

Radičević

2018

Int. J. Mod. Phys. A

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This paper is a physicist's review of the major conceptual issues concerning the problem of spectral universality in quantum systems. Here we present a unified, graph-based view of all archetypical models of such universality (billiards, particles in random media, interacting spin or fermion systems). We find phenomenological relations between the onset of ergodicity (Gaussian-random delocalization of eigenstates) and the structure of the appropriate graphs, and we construct a heuristic picture of summing trajectories on graphs that describes why a generic interacting system should be ergodic. We also provide an operator-based discussion of quantum chaos and propose criteria to distinguish bases that can usefully diagnose ergodicity. The result of this analysis is a rough but systematic outline of how ergodicity changes across the space of all theories with a given Hilbert space dimension. As a particular example, we study the SYK model and report on the transition from maximal to partial ergodicity as the disorder strength is decreased.

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Quantum and Arithmetical Chaos

Cited by 17 publications

References 45 publications

Regular oscillation sub-spectrum of rapidly rotating stars

Regular oscillation sub-spectrum of rapidly rotating stars

Degeneracy in the length spectrum for metric graphs

The ergodicity landscape of quantum theories

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