Eigenvalue density of the non-Hermitian Wilson Dirac operator

We find the lattice spacing dependence of the eigenvalue density of the non-Hermitian Wilson Dirac operator in the ? domain. The starting point is the joint probability density of the corresponding random matrix theory. In addition to the density of the complex eigenvalues we also obtain the density of the real eigenvalues separately for positive and negative chiralities as well as an explicit analytical expression for the number of additional real modes.     

A study of the generalized density matrix in the SU(3) model of Elliott for an arbitrary oscillator n-shell

The SU(3) model of Elliott for an arbitrary oscillator n-shell is considered. Exact solutions corresponding to the low-lying collective SU(3) multiplets are obtained. These multiplets exhibiting distinct collective properties are used as a “collective band”. All the matrix elements from the one-particle density matrix operator inside a given band are investigated. The so-formed matrix R, i.e. the generalized density matrix (GDM), is diagonalized and an explicit expression for the eigenvalues of the GDM in the case of low-lying multiplets is found. The GDM diagonal form contains a number of vanishing eigenvalues (the number of such “zero” eigenvalues is equal to that of the occupied orbits in the oscillator n-shell). Most of the remaining eigenvalues are close to unity. The asymptotic behaviour of the nonzero eigenvalues is analyzed in the limit of large nucleon number and the accuracy of the normalization condition R²= R is estimated.     

Spectrum Density Factor of Photons and Its Application in the Casimir Force

The observables of continuous eigenvalues are de?ned in an in?nite-dimensional ket space. The complete set of such eigenstates demands a spectrum density factor, for example, for the photons in the free space and electrons in the vacuum. From the derivation of the Casimir force without an arti?cial regulator we determine the explicit expression of the spectrum density factor for the photon ?eld to be an exponential function. The undetermined constant in the function is ?xed by the experimental data for the Lamb shift. With that, we predict that there exists a correction to the Casimir force.     

Statistical analysis of the transmission based on the DMPK equation: an application to Pb nano-contacts

The density of the transmission eigenvalues of Pb nano-contacts has been estimated recently in mechanically controllable break-junction experiments. Motivated by these experimental analyses, here we study the evolution of the density of the transmission eigenvalues with the disorder strength and the number of channels supported by the ballistic constriction of a quantum point contact in the framework of the Dorokhov-Mello-Pereyra-Kumar equation. We find that the transmission density evolves rapidly into the density in the diffusive metallic regime as the number of channels N_c of the constriction increase. Therefore, the transmission density distribution for a few N_c channels comes close to the known bimodal density distribution in the metallic limit. This is in agreement with the experimental statistical-studies in Pb nano-contacts. For the two analyzed cases, we show that the experimental densities are seen to be well described by the corresponding theoretical results.     

Nonclassical degrees of freedom in the Riemann Hamiltonian

The Hilbert-Pólya conjecture states that the imaginary parts of the zeros of the Riemann zeta function are eigenvalues of a quantum Hamiltonian. If so, conjectures by Katz and Sarnak put this Hamiltonian in the Altland-Zirnbauer universality class?C. This implies that the system must have a nonclassical two-valued degree of freedom. In such a system, the dominant primitive periodic orbits contribute to the density of states with a phase factor of -1. This resolves a previously mysterious sign problem with the oscillatory contributions to the density of the Riemann zeros.     

The Spectral Density of the Scattering Matrix for High Energies

We determine the density of eigenvalues of the scattering matrix of the Schr?dinger operator with a short range potential in the high energy asymptotic regime. We give an explicit formula for this density in terms of the X-ray transform of the potential.     

Eigenvalues of the Zakharov-Shabat scattering problem for real symmetric pulses

The classical problem of determining the solitons generated from symmetric real initial conditions in the nonlinear Schr?dinger equation is revisited. The corresponding Zakharov-Shabat scattering problem is solved for real and symmetric double-humped rectangular initial pulse forms. It is found that such real symmetric pulses may generate eigenvalues with nonzero real parts corresponding to separating soliton pulse pairs. Moreover, it is found that the classical formula relating the number of eigenvalues to the area of the pulse is not always correct.     

Particle occupation factors without large number approximations

New derivations of particle occupation factors that are based on mean values and do not require large number approximations (LNA) are provided for fermions, bosons, and Boltzmann particles. The derivations are closely related to traditional combinatorial approaches, so the physical content of the latter is preserved without recourse to most probable values or LNA. The approach is based on the use of eigenvalues of the density matrix as probabilities entering into the Shannon entropy. The degeneracies of the eigenvalues of the density matrix can be identified with Boltzmann enumeration factors. Problems associated with steepest descent procedures, limit theorem techniques, and the Boltzmann thermodynamic probability are avoided. In addition, since the approach is based on the eigenvalues of the density matrix, the concept of ensembles is not required.     

Efficient computation of the spectrum of viscoelastic flows

The understanding of viscoelastic flows in many situations requires not only the steady state solution of the governing equations, but also its sensitivity to small perturbations. Linear stability analysis leads to a generalized eigenvalue problem (GEVP), whose numerical analysis may be challenging, even for Newtonian liquids, because the incompressibility constraint creates singularities that lead to non-physical eigenvalues at infinity. For viscoelastic flows, the difficulties increase due to the presence of continuous spectrum, related to the constitutive equations.The Couette flow of upper convected Maxwell (UCM) liquids has been used as a case study of the stability of viscoelastic flows. The spectrum consists of two discrete eigenvalues and a continuous segment with real part equal to ?1/We (We is the Weissenberg number). Most of the approximations in the literature were obtained using spectral expansions. The eigenvalues close to the continuous part of the spectrum show very slow convergence.In this work, the linear stability of Couette flow of a UCM liquid is studied using a finite element method. A new procedure to eliminate the eigenvalues at infinity from the GEVP is proposed. The procedure takes advantage of the structure of the matrices involved and avoids the computational overhead of the usual mapping techniques. The GEVP is transformed into a non-degenerate GEVP of dimension five times smaller. The computed eigenfunctions related to the continuous spectrum are in good agreement with the analytic solutions obtained by Graham [M.D. Graham, Effect of axial flow on viscoelastic Taylor–Couette instability, J. Fluid Mech. 360 (1998) 341].     

Time reversal for a single spherical scatterer.

We show that the time reversal operator for a planar time reversal mirror (TRM) can have up to four distinct eigenvalues with a small spherical acoustic scatterer. Each eigenstate represents a resonance between the TRM and an induced scattering moment of the sphere. Their amplitude distributions on the TRM are orthogonal superpositions of the radiation patterns from a monopole and up to three orthogonal dipoles. The induced monopole moment is associated with the compressibility contrast between the sphere and the medium, while the dipole moments are associated with density contrast. The number of eigenstates is related to the number of orthogonal orientations of each induced multipole. For hard spheres (glass, metals) the contribution of the monopole moment to the eigenvalues is much greater than that of the dipole moments, leading to a single dominant eigenvalue. The other eigenvalues are much smaller, making it unlikely multiple eigenvalues could have been observed in previous experiments using hard materials. However, for soft materials such as wood, plastic, or air bubbles the eigenvalues are comparable in magnitude and should be observable. The presence of multiple eigenstates breaks the one-to-one correspondence between eigenstates and distinguishable scatterers discussed previously by Prada and Fink [Wave Motion 20, 151-163 (1994)]. However, eigenfunctions from separate scatterers would have different phases for their eigenfunctions, potentially restoring the ability to distinguish separate scatterers. Since relative magnitudes of the eigenvalues for a single scatterer are governed by the ratio of the compressibility contrast to the density contrast, measurement of the eigenvalue spectrum would provide information on the composition of the scatterer.     

A novel quasi-exactly solvable spin chain with nearest-neighbors interactions

In this paper we study a novel spin chain with nearest-neighbors interactions depending on the sites coordinates, which in some sense is intermediate between the Heisenberg chain and the spin chains of Haldane–Shastry type. We show that when the number of spins is sufficiently large both the density of sites and the strength of the interaction between consecutive spins follow the Gaussian law. We develop an extension of the standard freezing trick argument that enables us to exactly compute a certain number of eigenvalues and their corresponding eigenfunctions. The eigenvalues thus computed are all integers, and in fact our numerical studies evidence that these are the only integer eigenvalues of the chain under consideration. This fact suggests that this chain can be regarded as a finite-dimensional analog of the class of quasi-exactly solvable Schrödinger operators, which has been extensively studied in the last two decades. We have applied the method of moments to study some statistical properties of the chain's spectrum, showing in particular that the density of eigenvalues follows a Wigner-like law. Finally, we emphasize that, unlike the original freezing trick, the extension thereof developed in this paper can be applied to spin chains whose associated dynamical spin model is only quasi-exactly solvable.     

Electron energy states and miniband parameters in a class of non-uniform quantum well and superlattice structures

A simple method to compute the carrier energy states, miniband parameters and dispersion characteristics for single and multiple quantum well and superlattice structures is presented. The method utilizes the continuity of the envelope function across the heterojunctions according to the boundary conditions that both the wavefunction ψ and the particle current density $\text{ψ′}\text{m}{}^{\mathrm{★}}$ be continuous at each interface. The nonuniform potential distribution encountered in doped or compositionally graded materials is approximated by piecewise constant potential functions. In addition to being conceptually simple, the method is readily adopted to fairly complex structures where other more sophisticated methods such as LCAO, reduced Hamiltonian and tight binding theories may become unfeasible or unmanageable. It is shown that for an arbitrary stepped potential variation, the eigenvalues (or the energy states) of quantum wells or a finite number of coupled quantum wells can be found by utilizing a transverse resonance method which is readily implemented on a digital computer for the computation of these eigenvalues. For the case of periodic superlattices, the miniband parameters and the dispersion characteristics are computed from a suitably defined transmission matrix associated with a unit cell of the superlattice which may itself consist of multiple layers. Typical results for the computed parameters for several wells and simple, biperiodic, binary and polytype superlattices consisting of various Al_xGa_1?xAs and In_xGa_1?xAs alloys are presented.     

Eigenvalue problem for arbitrary linear combinations of a boson annihilation and creation operator

The eigenvalue problem for arbitrary linear combinations kα + μα^? of a boson annihilation operator α and a boson creation operator α^? is solved. It is shown that these operators possess nondegenerate eigenstates to arbitrary complex eigenvalues. The expansion of these eigenstates into the basic set of number states | n >, (n = 0, 1, 2, …), is found. The eigenstates are normalizable and are therefore states of a Hilbert space for | ζ | < 1 with ζ ? μ/k and represent in this case squeezed coherent states of minimal uncertainty product. They can be considered as states of a rigged Hilbert space for | ζ | ? 1. A completeness relation for these states is derived that generalizes the completeness relation for the coherent states | α 〉. Furthermore, it is shown that there exists a dual orthogonality in the entire set of these states and a connected dual completeness of the eigenstates on widely arbitrary paths over the complex plane of eigenvalues. This duality goes over into a selfduality of the eigenstates of the hermitian operators kα + k* α^? to real eigenvalues. The usually as nonexistent considered eigenstates of the boson creation operator α^? are obtained by a limiting procedure. They belong to the most singular case among the considered general class of eigenstates with ζ ? μ/k as a parameter.     

Automatic determination of the number of targets present when using the time reversal operator

Acoustical time reversal mirrors have been shown to provide a highly accurate means of studying and focusing on acoustical sources. The DORT method is a derivation of the time reversal process, which allows for focusing on multiple targets. An important step in this process is the determination of the number of targets or sources present. This is achieved by examining the eigenvalues of the time reversal operator (TRO). The number of significant eigenvalues is then chosen as the number of sources present. However, as mentioned in [N. Mordant, C. Prada, and M. Fink, J. Acoust. Soc. Am. 105, 2634-2642 (1999) and C. Prada, M. Tanter, and M. Fink, in Proceedings of the IEEE Symposium, 1997, pp. 679-683], factors such as low signal to noise ratio (SNR), small data sample, array configuration and the target location may result in the eigenvalues corresponding to the targets no longer being distinguishable from the background noise eigenvalues. This paper proposes a robust method of automatically determining the number of targets even in the presence of a small number of snapshots. For white Gaussian noise, the profile of the ordered eigenvalues is seen to fit an exponential law. The observed eigenvalues are then compared to this model and a mismatch is detected between the observed profile and the noise-only model. The index of the mismatch gives the number of scatterers present.     

Random Matrices with Equispaced External Source

We study Hermitian random matrix models with an external source matrix which has equispaced eigenvalues, and with an external field such that the limiting mean density of eigenvalues is supported on a single interval as the dimension tends to infinity. We obtain strong asymptotics for the multiple orthogonal polynomials associated to these models, and as a consequence for the average characteristic polynomials. One feature of the multiple orthogonal polynomials analyzed in this paper is that the number of orthogonality weights of the polynomials grows with the degree. Nevertheless we are able to characterize them in terms of a pair of 2 × 1 vector-valued Riemann–Hilbert problems, and to perform an asymptotic analysis of the Riemann–Hilbert problems.     

On the computation of a very large number of eigenvalues for selfadjoint elliptic operators by means of multigrid methods

Recent results in the study of quantum manifestations in classical chaos raise the problem of computing a very large number of eigenvalues of selfadjoint elliptic operators. The standard numerical methods for large eigenvalue problems cover the range of applications where a few of the leading eigenvalues are needed. They are not appropriate and generally fail to solve problems involving a number of eigenvalues exceeding a few hundreds. Further, the accurate computation of a large number of eigenvalues leads to much larger problem dimension in comparison with the usual case dealing with only a few eigenvalues. A new method is presented which combines multigrid techniques with the Lanczos process. The resulting scheme requires O(mn) arithmetic operations and O(n) storage requirement, where n is the number of unknowns and m, the number of needed eigenvalues. The discretization of the considered differential operators is realized by means of p-finite elements and is applicable on general geometries. Numerical experiments validate the proposed approach and demonstrate that it allows to tackle problems considered to be beyond the range of standard iterative methods, at least on current workstations. The ability to compute more than 9000 eigenvalues of an operator of dimension exceeding 8 million on a PC shows the potential of this method. Practical applications are found, e.g. in the numerical simulation of quantum billiards.     

Quantum field theoretical description of unstable behavior of trapped Bose-Einstein condensates with complex eigenvalues of Bogoliubov-de Gennes equations

The Bogoliubov-de Gennes equations are used for a number of theoretical works on the trapped Bose-Einstein condensates. These equations are known to give the energies of the quasi-particles when all the eigenvalues are real. We consider the case in which these equations have complex eigenvalues. We give the complete set including those modes whose eigenvalues are complex. The quantum fields which represent neutral atoms are expanded in terms of the complete set. It is shown that the state space is an indefinite metric one and that the free Hamiltonian is not diagonalizable in the conventional bosonic representation. We introduce a criterion to select quantum states describing the metastablity of the condensate, called the physical state conditions. In order to study the instability, we formulate the linear response of the density against the time-dependent external perturbation within the regime of Kubo’s linear response theory. Some states, satisfying all the physical state conditions, give the blow-up and damping behavior of the density distributions corresponding to the complex eigenmodes. It is qualitatively consistent with the result of the recent analyses using the time-dependent Gross-Pitaevskii equation.     

有限尺寸石墨烯的电子态

根据π电子的紧束缚模型, 通过有限系统的Bloch定理方法, 解析计算了有限尺寸石墨烯的电子态和能带. 研究发现, 其电子态有且只有两类, 分别是驻波态和边缘态.驻波态时, 波函数形式是两个方向都是正弦函数; 边缘态时, 波函数形式是Armchair边界的方向是双曲正弦函数, Zigzag边界的方向是正弦函数. 其能带由总碳原子数N个离散的本征值组成, 推导了定量计算边缘态的本征值个数的表达式, 并通过态密度来分析边缘态的存在和与无限大情况的一致性. 所有的分析中数值结果与解析理论都完全一致, 当两个受限方向都变成无限长时, 可以得到与无限大石墨烯相同的结果. 关键词： 紧束缚模型 石墨烯 边缘态 态密度     

Overlap Dirac operator at nonzero chemical potential and random matrix theory

We show how to introduce a quark chemical potential in the overlap Dirac operator. The resulting operator satisfies a Ginsparg-Wilson relation and has exact zero modes. It is no longer gamma5 Hermitian, but its nonreal eigenvalues still occur in pairs. We compute the spectral density of the operator on the lattice and show that, for small eigenvalues, the data agree with analytical predictions of non-Hermitian chiral random matrix theory for both trivial and nontrivial topology. We also explain an observed change in the number of zero modes as a function of chemical potential.