Scarring in open chaotic systems: The local density of states

Chaotic Hamiltonians are known to follow Random Matrix Theory (RMT) ensembles in the apparent randomness of their spectra and wavefunction statistics. Deviations form RMT also do occur, however, due to system-specific properties, or as quantum signatures of classical chaos. Scarring, for instance, is the enhancement of wavefunction intensity near classical periodic orbits, and it can be characterized by a local density of states (or local spectrum) that clearly deviates from RMT expectations, by exhibiting a peaked envelope, which has been described semiclassically. Here, the system is connected to an opening, the local density of states is introduced for the resulting non-Hermitian chaotic Hamiltonian, and estimated a priori in terms of the Green's function of the closed system and the open channels. The predictions obtained are tested on quantum maps coupled both to a single-channel opening and to a Fresnel-type continuous opening. The main outcome is that strong coupling to the opening gradually suppresses the energy dependence of the local density of states due to scarring, and restores RMT behavior.Comment: 9 pages, 3 figure

Neighborhoods of periodic orbits and the stationary distribution of a noisy chaotic system

The finest state space resolution that can be achieved in a physical dynamical system is limited by the presence of noise. In the weak-noise approximation the neighborhoods of deterministic periodic orbits can be computed as distributions stationary under the action of a local Fokker-Planck operator and its adjoint. We derive explicit formulae for widths of these distributions in the case of chaotic dynamics, when the periodic orbits are hyperbolic. The resulting neighborhoods form a basis for functions on the attractor. The global stationary distribution, needed for calculation of long-time expectation values of observables, can be expressed in this basis.Comment: 6 pages, 3 figure

Knowing when to stop: how noise frees us from determinism

Deterministic chaotic dynamics presumes that the state space can be partitioned arbitrarily finely. In a physical system, the inevitable presence of some noise sets a finite limit to the finest possible resolution that can be attained. Much previous research deals with what this attainable resolution might be, all of it based on global averages over a stochastic flow. We show how to compute the locally optimal partition, for a given dynamical system and given noise, in terms of local eigenfunctions of the Fokker-Planck operator and its adjoint. We first analyze the interplay of the deterministic dynamics with the noise in the neighborhood of a periodic orbit of a map, by using a discretized version of Fokker-Planck formalism. Then we propose a method to determine the 'optimal resolution' of the state space, based on solving Fokker-Planck's equation locally, on sets of unstable periodic orbits of the deterministic system. We test our hypothesis on unimodal maps.Comment: 45 pages, 11 figure

Escape-rate response to noise of all amplitudes in leaky chaos

We study the effect of homogeneous noise on the escape rate of strongly chaotic area-preserving maps with a small opening. While in the noiseless dynamics the escape rate analytically depends on the instability of the shortest periodic orbit inside the hole, adding noise overall enhances escape, which, however, exhibits a non-trivial response to the noise amplitude, featuring an initial plateau and a successive rapid growth up to a saturation value. Numerical analysis is performed on cat maps with a hole, and the salient traits of the response to noise of the escape rate are reproduced analytically by an approximate model.Comment: 25 pages, 20 figure

Statistics of Chaotic Resonances in an Optical Microcavity

Distributions of eigenmodes are widely concerned in both bounded and open systems. In the realm of chaos, counting resonances can characterize the underlying dynamics (regular vs. chaotic), and is often instrumental to identify classical-to-quantum correspondence. Here, we study, both theoretically and experimentally, the statistics of chaotic resonances in an optical microcavity with a mixed phase space of both regular and chaotic dynamics. Information on the number of chaotic modes is extracted by counting regular modes, which couple to the former via dynamical tunneling. The experimental data are in agreement with a known semiclassical prediction for the dependence of the number of chaotic resonances on the number of open channels, while they deviate significantly from a purely random-matrix-theory-based treatment, in general. We ascribe this result to the ballistic decay of the rays, which occurs within Ehrenfest time, and importantly, within the timescale of transient chaos. The present approach may provide a general tool for the statistical analysis of chaotic resonances in open systems.Comment: 5 pages, 5 figures, and a supplemental informatio

Substernal oxyphil parathyroid adenoma producing PTHrP with hypercalcemia and normal PTH level

<p>Abstract</p> <p>Background</p> <p>Parathyroid adenoma is the most common cause of primary hyperparathyroidism. Preoperative serum calcium and intact-parathyroid hormone levels are the most useful diagnostic parameters that allow differentiating primary hyperparathyroidism from non-parathyroid-dependent hypercalcemia. Parathyroidectomy is the definitive treatment for primary hyperparathyroidism. Approximately 5% of patients who underwent parathyroidectomy present with persistent or recurrent hyperparathyroidism due to ectopic localization of the adenoma. Functioning oxyphil parathyroid adenoma is an uncommon histological form, seldom causing primary hyperparathyroidism. Parathyroid adenoma with hypercalcemia exhibiting normal parathyroid hormone level is rare. An incidence of 5% to 33% has been documented in the literature; no etiologic explanation has been given. In 1987, parathyroid-hormone-related peptide was isolated as a causative factor of humeral hypercalcemia of malignancy. The presence of parathyroid-hormone-related peptide in parathyroid tissue under normal and pathological conditions has been described in the literature; however, its role in causing hyperparathyroidism has not yet been defined.</p> <p>Case presentation</p> <p>We present a case of persistent hypercalcemia with a normal level of intact-parathyroid hormone due to a substernal parathyroid adenoma, treated with radioguided parathyroidectomy. The final histological diagnosis was oxyphil adenoma, positive for parathyroid-hormone-related peptide antigens.</p> <p>Conclusion</p> <p>In clinical practice, this atypical biochemical presentation of primary hyperparathyroidism should be considered in the differential diagnosis of hypercalcemia. The parathyroid-hormone-related peptide should be considered not only in the presence of malignancy.</p