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Jan 2012

Volume 53, Issue 1, Articles (01xxxx)

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J. Math. Phys. 53, 013302 (2012); http://dx.doi.org/10.1063/1.3677978 (24 pages)

Vita Borovyk and Robert Sims
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back to top Quantum Mechanics (General and Nonrelativistic)

The gap equation for spin-polarized fermions

Abraham Freiji, Christian Hainzl, and Robert Seiringer

J. Math. Phys. 53, 012101 (2012); http://dx.doi.org/10.1063/1.3670747 (19 pages)

Online Publication Date: 4 January 2012

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We study the BCS gap equation for a Fermi gas with unequal population of spin-up and spin-down states. For cosh (δμ/T) ⩽ 2, with T the temperature and δμ the chemical potential difference, the question of existence of non-trivial solutions can be reduced to spectral properties of a linear operator, similar to the unpolarized case studied previously in [Frank, R. L., Hainzl, C., Naboko, S., and Seiringer, R., J., Geom. Anal. 17, 559–567 (2007)10.1007/BF02937429; Hainzl, C., Hamza, E., Seiringer, R., and Solovej, J. P., Commun., Math. Phys. 281, 349–367 (2008)10.1007/s00220-008-0489-2; and Hainzl, C. and Seiringer, R., Phys. Rev. B 77, 184517-110 435 (2008)]10.1103/PhysRevB.77.184517. For cosh (δμ/T) > 2 the phase diagram is more complicated, however. We derive upper and lower bounds for the critical temperature, and study their behavior in the small coupling limit.
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05.30.Fk Fermion systems and electron gas
05.70.-a Thermodynamics

Quasi-coherent states for harmonic oscillator with time-dependent parameters

Nuri Ünal

J. Math. Phys. 53, 012102 (2012); http://dx.doi.org/10.1063/1.3676072 (8 pages)

Online Publication Date: 6 January 2012

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In this study, we discuss the harmonic oscillator with the time-dependent frequency, ω(t), and the mass, M(t), by generalizing the holomorphic coordinates for the harmonic oscillator. In general cases, we solve the Schrödinger equation by reducing it into the Riccati equation and discuss the uncertainties for the quasi-coherent states of the time-dependent harmonic oscillator. In special cases, we find the following results: First, for a time-dependent harmonic oscillator, if [ω(t)M(t)] is constant, then the coherent states will evolve as the coherent states. Second, for the driven harmonic oscillator, the coherent states will evolve as the coherent states with new eigenvalues. Third, we derive quasi-coherent states for the Caldirola–Kanai Hamiltonian and show that the product of uncertainties, ΔxΔp, is larger than minimum value; however, it is constant. We also discuss the classical equations of motion for the system.
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03.65.Ge Solutions of wave equations: bound states
02.10.Ud Linear algebra

Phase coherent states with circular Jacobi polynomials for the pseudoharmonic oscillator

Zouhaïr Mouayn

J. Math. Phys. 53, 012103 (2012); http://dx.doi.org/10.1063/1.3675914 (11 pages)

Online Publication Date: 10 January 2012

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We construct a class of generalized phase coherent states ∣eiθ; γ, α, ɛ >, indexed by points eiθ of the unit circle and depending on three positive parameters γ, α, and ɛ, by replacing the labeling coefficient zn/math of the canonical coherent states by circular Jacobi polynomials gnγ(eiθ) with parameter γ ⩾ 0. The special case of γ = 0 corresponds to the well-known phase coherent states. The constructed states are superposition of eigenstates of a one-parameter pseudoharmonic oscillator depending on α and constitute a resolution of the identity of the state Hilbert space at the limit ɛ → 0+. Closed form for their wavefunctions are obtained in the case α = γ + 1 and their associated coherent states transforms are defined.
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03.65.Ge Solutions of wave equations: bound states
03.65.Aa Quantum systems with finite Hilbert space
02.10.Yn Matrix theory

Compound transfer matrices: Constructive and destructive interference

Petarpa Boonserm and Matt Visser

J. Math. Phys. 53, 012104 (2012); http://dx.doi.org/10.1063/1.3676070 (13 pages)

Online Publication Date: 17 January 2012

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Scattering from a compound barrier, one composed of a number of distinct non-overlapping sub-barriers, has a number of interesting and subtle mathematical features. If one is scattering classical particles, where the wave aspects of the particle can be ignored, the transmission probability of the compound barrier is simply given by the product of the transmission probabilities of the individual sub-barriers. In contrast, if one is scattering waves (whether we are dealing with either purely classical waves or quantum Schrodinger wavefunctions) each sub-barrier contributes phase information (as well as a transmission probability), and these phases can lead to either constructive or destructive interference, with the transmission probability oscillating between nontrivial upper and lower bounds. In this article, we shall study these upper and lower bounds in some detail, and also derive bounds on the closely related process of quantum excitation (particle production) via parametric resonance.
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03.65.Ge Solutions of wave equations: bound states
02.10.Yn Matrix theory
02.50.Cw Probability theory

Generalization of H-pseudoalgebraic structures

Qinxiu Sun

J. Math. Phys. 53, 012105 (2012); http://dx.doi.org/10.1063/1.3665708 (18 pages)

Online Publication Date: 18 January 2012

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The goal of this paper is to study Hom-H-pseudoalgebraic structures generalizing H-pseudoalgebras of associative and Lie type. We provide examples of the new structures and present some properties and construction theorems. Similar to the case of H-pseudoalgebras, the annihilator algebras are also discussed. We also describe the equivalent definitions of Hom-H-pseudoalgebra. Finally, we consider the cohomology theory of Hom-H-pseudoalgebra with coefficients in an arbitrary module. Furthermore, the connection between the cohomology of Hom-Lie H-pseudoalgebra and its corresponding annihilator algebra is depicted.
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02.20.Sv Lie algebras of Lie groups
02.10.Ud Linear algebra

Notes on the Riccati operator equation in open quantum systems

Bartłomiej Gardas and Zbigniew Puchała

J. Math. Phys. 53, 012106 (2012); http://dx.doi.org/10.1063/1.3676309 (4 pages)

Online Publication Date: 18 January 2012

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A recent problem [B. Gardas, J. Math. Phys. 52, 042104 (2011)] concerning an antilinear solution of the Riccati equation is solved. We also exemplify that a simplification of the Riccati equation, even under reasonable assumptions, can lead to a not equivalent equation.
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03.65.Ta Foundations of quantum mechanics; measurement theory
02.30.Hq Ordinary differential equations

Coherences of accelerated detectors and the local character of the Unruh effect

Charis Anastopoulos and Ntina Savvidou

J. Math. Phys. 53, 012107 (2012); http://dx.doi.org/10.1063/1.3679554 (20 pages)

Online Publication Date: 30 January 2012

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We study the locality of the acceleration temperature in the Unruh effect. To this end, we develop a new formalism for the modeling of macroscopic irreversible detectors. In particular, the formalism allows for the derivation of the higher-order coherence functions, analogous to the ones employed in quantum optics, that encode temporal fluctuations and correlations in particle detection. We derive a causal and approximately local-in-time expression for an Unruh-DeWitt detector moving in a general path in Minkowski spacetime. Moreover, we derive the second-order coherence function for uniformly accelerated Unruh-DeWitt detectors. We find that the fluctuations in detection time for a single Unruh-DeWitt detector are thermal. However, the correlations in detection time between two Unruh-DeWitt detectors with the same acceleration but separated by a finite distance are not thermal. This result suggests that the Unruh effect is fundamentally local, in the sense that the notion of acceleration temperature applies only to the properties of local field observables.
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03.65.Pm Relativistic wave equations
11.10.-z Field theory

The trace formula for a point scatterer on a compact hyperbolic surface

Henrik Ueberschär

J. Math. Phys. 53, 012108 (2012); http://dx.doi.org/10.1063/1.3679761 (24 pages)

Online Publication Date: 30 January 2012

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An exact trace formula for the perturbation of the Laplacian by a Dirac delta potential on a compact hyperbolic Riemann surface is derived. The formula can be considered an analogue of the Selberg trace formula. The difference of perturbed and unperturbed trace is expressed as an identity term plus a sum over combinations of diffractive orbits which visit the position of the potential.
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03.65.Pm Relativistic wave equations
05.45.-a Nonlinear dynamics and chaos
02.30.Sa Functional analysis

PT symmetric, Hermitian and P-self-adjoint operators related to potentials in PT quantum mechanics

Tomas Ya. Azizov and Carsten Trunk

J. Math. Phys. 53, 012109 (2012); http://dx.doi.org/10.1063/1.3677368 (18 pages)

Online Publication Date: 31 January 2012

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In the recent years, a generalization H = p2 + x2(ix)ε of the harmonic oscillator using a complex deformation was investigated, where ε is a real parameter. Here, we will consider the most simple case: ε even and x real. We will give a complete characterization of three different classes of operators associated with the differential expression H: The class of all self-adjoint (Hermitian) operators, the class of all PT symmetric operators, and the class of all P-self-adjoint operators. Surprisingly, some of the PT symmetric operators associated to this expression have no resolvent set.
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03.65.Ge Solutions of wave equations: bound states
back to top Quantum Information and Computation

Generalized channels: Channels for convex subsets of the state space

Anna Jenčová

J. Math. Phys. 53, 012201 (2012); http://dx.doi.org/10.1063/1.3676294 (23 pages)

Online Publication Date: 25 January 2012

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Let K be a convex subset of the state space of a finite-dimensional C*-algebra. We study the properties of channels on K, which are defined as affine maps from K into the state space of another algebra, extending to completely positive maps on the subspace generated by K. We show that each such map is the restriction of a completely positive map on the whole algebra, called a generalized channel. We characterize the set of generalized channels and also the equivalence classes of generalized channels having the same value on K. Moreover, if K contains the tracial state, the set of generalized channels forms again a convex subset of a multipartite state space, this leads to a definition of a generalized supermap, which is a generalized channel with respect to this subset. We prove a decomposition theorem for generalized supermaps and describe the equivalence classes. The set of generalized supermaps having the same value on equivalent generalized channels is also characterized. Special cases include quantum combs and process positive operator valued measures (POVMs).
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03.67.Hk Quantum communication
02.10.Ab Logic and set theory
03.65.Fd Algebraic methods

Entropic uncertainty relations and the stabilizer formalism

Sönke Niekamp, Matthias Kleinmann, and Otfried Gühne

J. Math. Phys. 53, 012202 (2012); http://dx.doi.org/10.1063/1.3678200 (13 pages)

Online Publication Date: 27 January 2012

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Entropic uncertainty relations express the quantum mechanical uncertainty principle by quantifying uncertainty in terms of entropy. Central questions include the derivation of lower bounds on the total uncertainty for given observables, the characterization of observables that allow strong uncertainty relations, and the construction of such relations for the case of several observables. We demonstrate how the stabilizer formalism can be applied to these questions. We show that the Maassen–Uffink entropic uncertainty relation is tight for the measurement in any pair of stabilizer bases. We compare the relative strengths of variance-based and various entropic uncertainty relations for dichotomic anticommuting observables.
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03.65.Ta Foundations of quantum mechanics; measurement theory
05.70.Ce Thermodynamic functions and equations of state
back to top Relativistic Quantum Mechanics, Field Theory, Brane Theory (Including Strings)

Superalgebra realization of the 3-algebras in N = 6,8 Chern-Simons-matter theories

Fa-Min Chen

J. Math. Phys. 53, 012301 (2012); http://dx.doi.org/10.1063/1.3674989 (17 pages)

Online Publication Date: 6 January 2012

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We use superalgebras to realize the 3-algebras used to construct N = 6,8 Chern-Simons-matter (CSM) theories. We demonstrate that the superalgebra realization of the 3-algebras provides a unified framework for classifying the gauge groups of the N ≥ 5 theories based on 3-algebras. Using this realization, we rederive the ordinary Lie algebra construction of the general N = 6 CSM theory from its 3-algebra counterpart and reproduce all known examples as well. In particular, we explicitly construct the Nambu 3-bracket in terms of a double graded commutator of PSU(2|2). The N = 8 theory of Bagger, Lambert and Gustavsson (BLG) with SO(4) gauge group is constructed by using several different ways. A quantization scheme for the 3-brackets is proposed by promoting the double graded commutators as quantum mechanical double graded commutators.
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11.15.Yc Chern-Simons gauge theory
02.20.Sv Lie algebras of Lie groups
03.65.Fd Algebraic methods
11.30.Ly Other internal and higher symmetries

Cohomology of line bundles: Applications

Ralph Blumenhagen, Benjamin Jurke, Thorsten Rahn, and Helmut Roschy

J. Math. Phys. 53, 012302 (2012); http://dx.doi.org/10.1063/1.3677646 (38 pages)

Online Publication Date: 26 January 2012

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Massless modes of both heterotic and Type II string compactifications on compact manifolds are determined by vector bundle valued cohomology classes. Various applications of our recent algorithm for the computation of line bundle valued cohomology classes over toric varieties are presented. For the heterotic string, the prime examples are so-called monad constructions on Calabi-Yau manifolds. In the context of Type II orientifolds, one often needs to compute cohomology for line bundles on finite group action coset spaces, necessitating us to generalize our algorithm to this case. Moreover, we exemplify that the different terms in Batyrev's formula and its generalizations can be given a one-to-one cohomological interpretation. Furthermore, we derive a combinatorial closed form expression for two Hodge numbers of a codimension two Calabi-Yau fourfold.
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11.25.-w Strings and branes
04.20.Gz Spacetime topology, causal structure, spinor structure
back to top General Relativity and Gravitation

Static solutions from the point of view of comparison geometry

Martin Reiris

J. Math. Phys. 53, 012501 (2012); http://dx.doi.org/10.1063/1.3668045 (31 pages)

Online Publication Date: 5 January 2012

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We analyze (the harmonic map representation of) static solutions of the Einstein equations in dimension three from the point of view of comparison geometry. We find simple monotonic quantities capturing sharply the influence of the Lapse function on the focussing of geodesics. This allows, in particular, a sharp estimation of the Laplacian of the distance function to a given (hyper)-surface. We apply the technique to asymptotically flat solutions with regular and connected horizons and, after a detailed analysis of the distance function to the horizon, we recover the Penrose inequality and the uniqueness of the Schwarzschild solution. The proof of this last result does not require proving conformal flatness at any intermediate step.
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04.20.-q Classical general relativity
02.30.Jr Partial differential equations
02.40.Hw Classical differential geometry

Relative entropy as a measure of inhomogeneity in general relativity

Nikolas Akerblom and Gunther Cornelissen

J. Math. Phys. 53, 012502 (2012); http://dx.doi.org/10.1063/1.3675440 (10 pages)

Online Publication Date: 10 January 2012

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We introduce the notion of relative volume entropy for two spacetimes with preferred compact spacelike foliations. This is accomplished by applying the notion of Kullback-Leibler divergence to the volume elements induced on spacelike slices. The resulting quantity gives a lower bound on the number of bits which are necessary to describe one metric given the other. For illustration, we study some examples, in particular gravitational waves, and conclude that the relative volume entropy is a suitable device for quantitative comparison of the inhomogeneity of two spacetimes.
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05.70.Ce Thermodynamic functions and equations of state
04.30.Nk Wave propagation and interactions
04.20.Gz Spacetime topology, causal structure, spinor structure
04.20.Jb Exact solutions

Spinor representation for loop quantum gravity

Etera Livine and Johannes Tambornino

J. Math. Phys. 53, 012503 (2012); http://dx.doi.org/10.1063/1.3675465 (33 pages) | Cited 1 time

Online Publication Date: 20 January 2012

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We perform a quantization of the loop gravity phase space purely in terms of spinorial variables, which have recently been shown to provide a direct link between spin network states and simplicial geometries. The natural Hilbert space to represent these spinors is the Bargmann space of holomorphic square-integrable functions over complex numbers. We show the unitary equivalence between the resulting generalized Bargmann space and the standard loop quantum gravity Hilbert space by explicitly constructing the unitary map. The latter maps SU(2)-holonomies, when written as a function of spinors, to their holomorphic part. We analyze the properties of this map in detail. We show that the subspace of gauge invariant states can be characterized particularly easy in this representation of loop gravity. Furthermore, this map provides a tool to efficiently calculate physical quantities since integrals over the group are exchanged for straightforward integrals over the complex plane.
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04.60.Pp Loop quantum gravity, quantum geometry, spin foams
03.65.Ta Foundations of quantum mechanics; measurement theory
02.20.Sv Lie algebras of Lie groups

Inversion of a general hyperelliptic integral and particle motion in Hořava–Lifshitz black hole space-times

Victor Enolski, Betti Hartmann, Valeria Kagramanova, Jutta Kunz, Claus Lämmerzahl, and Parinya Sirimachan

J. Math. Phys. 53, 012504 (2012); http://dx.doi.org/10.1063/1.3677831 (35 pages)

Online Publication Date: 31 January 2012

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The description of many dynamical problems such as the particle motion in higher dimensional spherically and axially symmetric space-times is reduced to the inversion of hyperelliptic integrals of all three kinds. The result of the inversion is defined locally, using the algebro-geometric techniques of the standard Jacobi inversion problem and the foregoing restriction to the θ-divisor. For a representation of the hyperelliptic functions the Klein–Weierstraß multi-variable σ-function is introduced. It is shown that all parameters needed for the calculations such as period matrices and abelian images of branch points can be expressed in terms of the periods of holomorphic differentials and θ-constants. The cases of genus two, three, and four are considered in detail. The method is exemplified by the particle motion associated with genus one elliptic and genus three hyperelliptic curves. Applications are for instance solutions to the geodesic equations in the space-times of static, spherically symmetric Hořava–Lifshitz black holes.
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04.70.-s Physics of black holes
02.40.Hw Classical differential geometry
02.10.-v Logic, set theory, and algebra
02.30.Rz Integral equations
04.20.Gz Spacetime topology, causal structure, spinor structure
back to top Dynamical Systems

Global attractors for weighted p-Laplacian equations with boundary degeneracy

Shan Ma and Hongtao Li

J. Math. Phys. 53, 012701 (2012); http://dx.doi.org/10.1063/1.3675441 (8 pages)

Online Publication Date: 5 January 2012

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Using a new a priori estimate method which is introduced by Zhong et al. [J. Differ. Equations 223(2), 367 (2006)]10.1016/j.jde.2005.06.008, we establish the existence of a global attractor in L2(Ω) and Lq(Ω)(q ⩾ 2), respectively, for weighted p-Laplacian equations with boundary degeneracy and without any restriction on the growing order of the nonlinearity.
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05.45.-a Nonlinear dynamics and chaos
02.60.Lj Ordinary and partial differential equations; boundary value problems

Periodic orbits and non-integrability in a cosmological scalar field

Jaume Llibre and Claudio Vidal

J. Math. Phys. 53, 012702 (2012); http://dx.doi.org/10.1063/1.3675493 (14 pages)

Online Publication Date: 10 January 2012

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We apply the averaging theory of first order to study the periodic orbits of Hamiltonian systems describing a universe filled with a scalar field which possesses three parameters. The main results are the following. First, we provide sufficient conditions on the parameters of these cosmological model, which guarantee that at any positive or negative Hamiltonian level, the Hamiltonian system has periodic orbits, the number of such periodic orbits and their stability change with the values of the parameters. These periodic orbits live in the whole phase space in a continuous family of periodic orbits parameterized by the Hamiltonian level. Second, under convenient assumptions we show the non-integrability of these cosmological systems in the sense of Liouville–Arnol'd, proving that there cannot exist any second first integral of class C1. It is important to mention that the tools (i.e., the averaging theory for studying the existence of periodic orbits and their kind of stability, and the multipliers of these periodic orbits for studying the integrability of the Hamiltonian system) used here for proving our results on the cosmological scalar field can be applied to Hamiltonian systems with an arbitrary number of degrees of freedom.
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98.80.Jk Mathematical and relativistic aspects of cosmology
04.20.Jb Exact solutions
02.30.Sa Functional analysis
back to top Classical Mechanics and Classical Fields

Classical ladder operators, polynomial Poisson algebras, and classification of superintegrable systems

Ian Marquette

J. Math. Phys. 53, 012901 (2012); http://dx.doi.org/10.1063/1.3676075 (12 pages)

Online Publication Date: 6 January 2012

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We recall results concerning one-dimensional classical and quantum systems with ladder operators. We obtain the most general one-dimensional classical systems, respectively, with a third and a fourth-order ladder operators satisfying polynomial Heisenberg algebras. These systems are written in terms of the solutions of quartic and quintic equations. They are the classical equivalent of quantum systems involving the fourth and fifth Painlevé transcendents. We use these results to present two new families of superintegrable systems and examples of trajectories that are deformation of Lissajous's figures.
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03.65.Ge Solutions of wave equations: bound states
02.10.Yn Matrix theory

Limitations on cloning in classical mechanics

Aaron Fenyes

J. Math. Phys. 53, 012902 (2012); http://dx.doi.org/10.1063/1.3676295 (9 pages)

Online Publication Date: 12 January 2012

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In this paper, we show that a result precisely analogous to the traditional quantum no-cloning theorem holds in classical mechanics. This classical no-cloning theorem does not prohibit classical cloning, we argue, because it is based on a too-restrictive definition of cloning. Using a less popular, more inclusive definition of cloning, we give examples of classical cloning processes. We also prove that a cloning machine must be at least as complicated as the object it is supposed to clone.
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45.05.+x General theory of classical mechanics of discrete systems
03.65.Ta Foundations of quantum mechanics; measurement theory
back to top Fluids

Non-uniform dependence for a modified Camassa-Holm system

Guangying Lv and Mingxin Wang

J. Math. Phys. 53, 013101 (2012); http://dx.doi.org/10.1063/1.3675900 (21 pages)

Online Publication Date: 5 January 2012

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This paper is concerned with the non-uniform dependence on initial data for a modified Camassa-Holm system. We prove that the solution map of the Cauchy problem of the Camassa-Holm system is not uniformly continuous in Hs(math), s > 1. Moreover, we obtain the similar result for the initial boundary value problem for the Camassa-Holm system.
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02.60.Lj Ordinary and partial differential equations; boundary value problems
02.30.Jr Partial differential equations

The Biot-Savart operator and electrodynamics on subdomains of the three-sphere

Jason Parsley

J. Math. Phys. 53, 013102 (2012); http://dx.doi.org/10.1063/1.3673788 (23 pages)

Online Publication Date: 13 January 2012

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We study steady-state magnetic fields in the geometric setting of positive curvature on subdomains of the three-dimensional sphere. By generalizing the Biot-Savart law to an integral operator BS acting on all vector fields, we show that electrodynamics in such a setting behaves rather similarly to Euclidean electrodynamics. For instance, for current J and magnetic field BS(J), we show that Maxwell's equations naturally hold. In all instances, the formulas we give are geometrically meaningful: they are preserved by orientation-preserving isometries of the three-sphere. This article describes several properties of BS: we show it is self-adjoint, bounded, and extends to a compact operator on a Hilbert space. For vector fields that act like currents, we prove the curl operator is a left inverse to BS; thus, the Biot-Savart operator is important in the study of curl eigenvalues, with applications to energy-minimization problems in geometry and physics. We conclude with two examples, which indicate our bounds are typically within an order of magnitude of being sharp.
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03.50.De Classical electromagnetism, Maxwell equations
02.10.Ud Linear algebra

Vortex dynamics in math4

Banavara N. Shashikanth

J. Math. Phys. 53, 013103 (2012); http://dx.doi.org/10.1063/1.3673800 (21 pages)

Online Publication Date: 13 January 2012

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The vortex dynamics of Euler's equations for a constant density fluid flow in math4 is studied. Most of the paper focuses on singular Dirac delta distributions of the vorticity two-form ω in math4. These distributions are supported on two-dimensional surfaces termed membranes and are the analogs of vortex filaments in math3 and point vortices in math2. The self-induced velocity field of a membrane is shown to be unbounded and is regularized using a local induction approximation. The regularized self-induced velocity field is then shown to be proportional to the mean curvature vector field of the membrane but rotated by 90° in the plane of normals. Next, the Hamiltonian membrane model is presented. The symplectic structure for this model is derived from a general formula for vorticity distributions due to Marsden and Weinstein [“Coadjoint orbits, vortices and Clebsch variables for incompressible fluids,” Physica D 7, 305–323 (1983)10.1016/0167-2789(83)90134-3]. Finally, the dynamics of the four-form ω ∧ ω is examined. It is shown that Ertel's vorticity theorem in math3, for the constant density case, can be viewed as a special case of the dynamics of this four-form.
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47.32.C- Vortex dynamics
back to top Statistical Physics

Circular law and arc law for truncation of random unitary matrix

Zhishan Dong, Tiefeng Jiang, and Danning Li

J. Math. Phys. 53, 013301 (2012); http://dx.doi.org/10.1063/1.3672885 (14 pages)

Online Publication Date: 20 January 2012

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Let V be the m × m upper-left corner of an n × n Haar-invariant unitary matrix. Let λ1, …, λm be the eigenvalues of V. We prove that the empirical distribution of a normalization of λ1, …, λm goes to the circular law, that is, the uniform distribution on {z∈math; |z| ≤ 1} as m → ∞ with m/n → 0. We also prove that the empirical distribution of λ1, …, λm goes to the arc law, that is, the uniform distribution on {z∈math; |z| = 1} as m/n → 1. These explain two observations by Życzkowski and Sommers (2000).
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05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion
02.10.Ud Linear algebra
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