Eigenvalue Equation Research Articles

Quantum simulation of excited states from parallel contracted quantum eigensolvers

Computing excited-state properties of molecules and solids is considered one of the most important near-term applications of quantum computers. While many of the current excited-state quantum algorithms differ in circuit architecture, specific exploitation of quantum advantage, or result quality, one common feature is their rooting in the Schrödinger equation. However, through contracting (or projecting) the eigenvalue equation, more efficient strategies can be designed for near-term quantum devices. Here we demonstrate that when combined with the Rayleigh–Ritz variational principle for mixed quantum states, the ground-state contracted quantum eigensolver (CQE) can be generalized to compute any number of quantum eigenstates simultaneously. We introduce two excited-state (anti-Hermitian) CQEs that perform the excited-state calculation while inheriting many of the remarkable features of the original ground-state version of the algorithm, such as its scalability. To showcase our approach, we study several model and chemical Hamiltonians and investigate the performance of different implementations.

The complete Kaluza-Klein spectra of N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 and N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 0 M-theory on AdS4× (squashed S7)

The squashed seven-sphere operator spectrum is completed by deriving the spectrum of the spin-3/2 operator. The implications of the results for the AdS4N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 1 supermultiplets obtained from compactification of eleven-dimensional supergravity are analysed. The weak G2 holonomy plays an important role when solving the eigenvalue equations on the squashed sphere. Here, a novel and more universal algebraic approach to the whole eigenvalue problem on coset manifolds is provided. Having obtained full control of all the operator spectra, we can finally determine the irreps D(E0, s) for all supermultiplets in the left-squashed vacuum. This includes an analysis of possible boundary conditions. By performing an orientation flip on the seven-sphere, we also obtain the full spectrum for the non-supersymmetric right-squashed compactification which is of interest in the swampland context and in particular for the AdS swampland conjecture. Here, a number of boundary condition ambiguities arise making the analysis of dual marginal operators somewhat involved. This work is a direct continuation of [1] and [2].

A two-direction cross-iteration solution for in-plane vibrations of thin-walled composite plates with a simplified explicit vibration mechanical model

For the fine and high reliability design of advanced precision engineering structures, this work presents an analytically cross-iteration method for in-plane vibrations analysis of thin-walled composite plates, the single-layer and laminated structures both involved. The governing equations and corresponding boundary conditions are derived adopting Rayleigh’s principle, in which the governing partial differential equations are transferred into ordinary differential equations. The most important advantage of this method lies in the fact that the explicit eigenvalue equations and mode functions are obtained with only a series of four-by-four matrices by simplifying boundary conditions in the two-direction cross-iteration procedures. Accuracy of the present method is verified by comparison with the previous studies and the finite element method in natural characteristics solutions, and convergence analysis on the single-layer orthotropic plates and composite laminates reveals the high efficiency and stability. Finally, the effects of boundary conditions, ply orientation angles, stacking sequences, aspect ratios and stiffness ratios on the in-plane natural characteristics are investigated. The extensive results presented in this work for the first time can be taken as the benchmarks data compared with other methods.

A regularized (XP)2 model

We investigate a dynamic model described by the classical Hamiltonian H(x, p) = (x 2 + a 2)(p 2 + a 2), where a 2 > 0, in classical, semi-classical, and quantum mechanics, respectively. In the high-energy E limit, the phase path resembles that of the XP2 model. However, the non-zero value of a acts as a regulator, removing the singularities that appear in the region where x, p ∼ 0, resulting in a discrete spectrum characterized by a logarithmic increase in state density. Classical solutions are described by elliptic functions, with the period being determined by elliptic integrals. In semi-classical approximation, we speculate that the asymptotic Riemann-Siegel formula may be interpreted as summing over contributions from multiply phase paths. We present three different forms of quantized Hamiltonians, and reformulate them into the standard Schrödinger equation with cosh2x -like potentials. Numerical evaluations of the spectra for these forms are carried out and reveal minor differences in energy levels. Among them, one interesting form possesses Hamiltonian in the Schrödinger equation that is identical to its classical version. In such scenarios, the eigenvalue equations can be expressed as the vanishing of the Mathieu functions’ value at i ∞ points, and furthermore, the Mathieu functions can be represented as the wave functions.

Symplectic solutions of the plane annular sectors in micropolar elasticity

The symplectic approach is utilized to derive the solutions of the plane annular sector in micropolar elasticity. According to the Hellinger-Reissner variational principle, the Hamiltonian canonical equations are obtained for the plane problem in an annular sectorial region. Through the method of separation of variables, we obtain an eigenvalue equation of the Hamiltonian operator matrix which is a system of linear differential equations with variable coefficients. By the method of series solutions, its general solutions can be expressed as the linear combination of Bessel functions and power functions. The corresponding eigensolutions for three types of homogeneous boundary conditions are derived. According to the adjoint symplectic orthogonality of the eigensolutions, the solution of the micropolar plane annular sector is expressed as a series expansion of eigensolutions. Numerical analyses are presented to demonstrate the validity of the proposed method and confirm its ability to accurately capture the size effect.

Euclidean time method in generalized eigenvalue equation

Euclidean time method in generalized eigenvalue equation

Quasi-static testing based structural modal parameter estimation

ABSTRACT Structural modal identification is generally implemented within a dynamic framework, requiring multidisciplinary knowledge and adequate operational experience. This preliminary study attempts to explore an easy-to-handle quasi-static alternative method for dynamics-based modal identification of beam-type structures. Acceleration time-history data are replaced by quasi-static deflections induced by slow moving loads. A concept of quasi-static deflection influence surface is proposed based on the theory of influence lines and the principle of virtual work. Its analytical expression is simplified into a deflection matrix by choosing specific points on the surface according to deflection measurement coordinates. The matrix form is divided by external loads to obtain a generalised quasi-static flexibility matrix, which is used to replace the inversion of the global stiffness matrix in the modal eigenvalue equation. The lumped-mass method is also employed to establish the mass matrix. Subsequently, the eigenvalue equation is analytically solved seeking for modal frequencies and mode shapeswithout performing modal tests. The feasibility of the proposed method has been successfully verified against three experimental examples including a continuous box girder with variable cross sections. It was observed that the modal frequencies and mode shapes estimated by the proposed method were very close to those given by the dynamically modal tests.

Effects of anisotropy on the electronic structure and nonlinear optical rectification in a quantum dot with hydrogen molecular ion

In this study, the effects of anisotropy and quantum dot size on energy levels and nonlinear optical rectification of a D_{2}^{ + } molecular complex confined to a two-dimensional quantum dot are investigated. The energy eigenvalues and corresponding eigenfunctions of the resulting eigenvalue equation are obtained using the adiabatic approach and the two-dimensional diagonalization method. We conclude that the evolution of the energy spectrum depends predominantly on the anisotropy-induced change of geometric confinement size. In addition, it has been determined that the peak position of the nonlinear optical rectification shifts to lower or higher energies as the asymmetry of the system increases, depending on whether the QD is in the oblate or prolate case. This feature can be considered an advantage in tuning the nonlinear optical properties of systems based on two-dimensional semiconductor heterostructures.

Integral Operator-Based Characteristic Mode Theory for Conducting, Material, and Lossy Structures

Integral operator-based theory of characteristic modes (CMs) for conducting, material, and lossy structures is reviewed. CMs are defined as solutions of a generalized eigenvalue equation (GEE). These GEEs are presented for various surface and volume integral operators and material structures. Interpretation of the characteristic eigenvalues in terms of electromagnetic power is studied based on the Mie expansion and integral operator formalism. Orthogonality and diagonalizing properties of the CMs are summarized. Challenges related to dielectric-magnetic bodies, lossy structures and spurious modes are discussed, as well as differences between the surface and volume operator approaches.

Comment on: “Some aspects of the interaction of a magnetic quadrupole moment with an electric field in a rotating frame” [J. Math. Phys. 58, 102103 (2017)

We discuss results obtained recently for a quantum-mechanical model given by a neutral particle with a magnetic quadrupole moment in a radial electric field and a scalar potential proportional to the radial variable in cylindrical coordinates that also includes the noninertial effects of a rotating reference frame. We show that the conjectured allowed values of the cyclotron frequency are a mere artifact of the truncation of the power series used to solve the radial eigenvalue equation. Our analysis proves that the analytical expression for the eigenvalues is incorrect.

Bound-state energy spectrum and thermochemical functions of the deformed Schiöberg oscillator

In this study, a diatomic molecule interacting potential such as the deformed Schiöberg oscillator (DSO) have been applied to diatomic systems. By solving the Schrödinger equation with the DSO, analytical equations for energy eigenvalues, molar entropy, molar enthalpy, molar Gibbs free energy and constant pressure molar heat capacity are obtained. The obtained equations were used to analyze the physical properties of diatomic molecules. With the aid of the DSO, the percentage average absolute deviation (PAAD) of computed data from the experimental data of the 7Li2 (2 3Πg), NaBr (X 1Σ+), KBr (X 1Σ+) and KRb (B 1Π) molecules are 1.3319%, 0.2108%, 0.2359% and 0.8841%, respectively. The PAAD values obtained by employing the equations of molar entropy, scaled molar enthalpy, scaled molar Gibbs free energy and isobaric molar heat capacity are 1.2919%, 1.5639%, 1.5957% and 2.4041%, respectively, from the experimental data of the KBr (X 1Σ+) molecule. The results for the potential energies, bound-state energy spectra, and thermodynamic functions are in good agreement with the literature on diatomic molecules.

Monte Carlo Calculation Method for Reactor Period Utilizing the Differential Operator Sampling Technique

The inverse reactor period α is a fundamental mode eigenvalue of the α-mode nonlinear Boltzmann eigenvalue equation that considers delayed neutron contributions. Thus far, several Monte Carlo methods, including the α-k, weight balancing, and transition rate matrix methods, have been developed to calculate α. This study presents a new Monte Carlo method for predicting α by using the derivatives of the k-eigenvalue with respect to α. Formulae are derived to calculate the first and second derivatives using the differential operator sampling method. The key feature of the new proposed method is its ability to estimate the uncertainty of the predicted α by considering the uncertainty of the k-eigenvalue and its derivatives with respect to α.

Two string theory flavours of generalised Eisenstein series

Generalised Eisenstein series are non-holomorphic modular invariant functions of a complex variable, τ, subject to a particular inhomogeneous Laplace eigenvalue equation on the hyperbolic upper-half τ-plane. Two infinite classes of such functions arise quite naturally within different string theory contexts. A first class can be found by studying the coefficients of the effective action for the low-energy expansion of type IIB superstring theory, and relatedly in the analysis of certain integrated four-point functions of stress tensor multiplet operators in N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 4 supersymmetric Yang-Mills theory. A second class of such objects is known to contain all two-loop modular graph functions, which are fundamental building blocks in the low-energy expansion of closed-string scattering amplitudes at genus one. In this work, we present a Poincaré series approach that unifies both classes of generalised Eisenstein series and manifests certain algebraic and differential relations amongst them. We then combine this technique with spectral methods for automorphic forms to find general and non-perturbative expansions at the cusp τ → i∞. Finally, we find intriguing connections between the asymptotic expansion of these modular functions as τ → 0 and the non-trivial zeros of the Riemann zeta function.

Comment on “Harmonic oscillator problem in the background of a topologically charged Ellis-Bronnikov–type wormhole” by Ahmed Faizuddin

Abstract We analyse some results appeared in a paper published recently in this journal and show that the author obtained wrong solutions for the two eigenvalue equations studied. In particular, he obtained bound states for a problem that does not support them. The mistake can be traced back to a failed truncation of the power-series ansatz proposed as a possible solution. We also show that the problem is conditionally solvable and one can obtain analytical solutions only for particular values of the model parameters.

Solution by the self-adjoint operator method for a model of a co/counter-current moving bed heat exchanger with heat losses

A solution by the self-adjoint operator method (SAO) (still not widely used in engineering) is presented for a lumped parameter model (LM) of a co/counter-current, indirect heated, diluted, tubular, vertical moving bed heat exchanger (MBHE), with heat loss to surroundings. The self-adjoint formulation of the problem and several mathematical techniques enabled the development of an ingenious methodology capable of solving the difficult problem of establishing the character of the eigenvalues of the MBHE operator - only partially known to date - for co/counter flow.For co-current flow, it is demonstrated that the self-adjoint solution corresponds to a solution from the literature - obtained by the Laplace transform - but with the character of the roots not yet established. Therefore, the work aims to, through analysis, study the character of the eigenvalues of the MBHE operator to generalize results from the literature, allowing the use of the solution in a comprehensive parametric range, as well as for different flow arrangements, discussed within the scope of case studies. From the analysis of the eigenvalue equation, five thermal behaviors of the MBHE could be identified. The model was also solved numerically, and the results show excellent agreement in both calculation methods. The obtained analytical solution reveals a possible inconsistency in a literature solution of the analogous heat transfer problem in a triple tube heat exchanger (the) with thermal losses at the outer surface. The study generalizes previous results and allows its extension, by analogy, to the solution for the TTHE with thermal losses at the outer surface.

Response of gravitationally coupled gaseous and stellar components to asymmetric warp in disc galaxies

ABSTRACT The outer disc region of most spiral galaxies (approximately 50 per cent of all disc galaxies) shows warping above the galactic mid-plane and is primarily asymmetric by nature. In this work, we explore analytically the effect of the gas component on asymmetric warps in a realistic self-gravitating collision-less disc residing in a cold oblate dark matter halo’s potential field. We consider the disc to be composed of gravitationally coupled stars and gas components. The quadratic eigenvalue equation describing the shape and frequency of the bending mode is formulated and solved numerically. Two stable ground-state bending modes m = 0 and m = 1, representing the U-shape and the mostly observed S-shaped warp in the galactic disc are superimposed linearly to generate and examine the asymmetric warps in the disc. The resulting asymmetry in warp is measured by asymmetric index (Aasym) by varying physical parameters such as the mass of the gas components and the halo flattening parameter. It is shown that the gas fraction in the disc has a negligible contribution to the generation of asymmetric warp in the disc. The disc residing in a spherical dark matter halo is found to be more asymmetry than that in the counterpart oblate halo.

Instability of MHD mixed convection flow of nanofluid in porous channel with thermal radiation, chemical reaction, Dufour and Soret effects

The present paper aims to investigate a stability analysis of the mixed convection of Al2O3/water nanofluid through an anisotropic porous channel heated from below and cooled from above. The effects of the magnetic field, thermal radiation, chemical reaction, mass transfer (Soret effect) and energy transfer (Dufour effect) are taken into consideration. The equations governing the motion of the nanofluid are converted by means of hydrodynamic stability theory into a system of eigenvalue equations which is then solved numerically by the spectral collocation method. The effects of Hartmann number, thermal radiation parameter, volume fraction of nanoparticles, permeability parameter, chemical reaction parameter, Dufour and Soret number are presented and analyzed. The results showed that in presence of nanoparticles the inertia of the working fluid increases, which dampens the disturbances and thus reduces their growth. The Hartman number attenuated the amplitude of the disturbances and thus has a stabilizing effect. The permeability parameter affected the stability of the flow and its growth prevented the redistribution of disturbances within the flow, which reflected its stabilizing character. The numbers of Soret and Dufour showed an important effect on the stability of the convective mixed flow. Their growth increased the transition Reynolds number for which there was a transition to instabilities, which prevented the transition to increasingly turbulent flows. On the other hand, the increase in the chemical reaction parameter decreased the values of transition Reynolds number, thus showing that the chemical reaction maintained the instabilities.

Fast method for the determination of radiation-relevant eigenmodes of underwater structures when using FEM shell elements (“modal reduction”)

A self-developed code based on the BEM for the determination of the backscattered sound pressure level in the far field was extended so that additional FEM shell elements and thus elastic material properties can be considered. For this purpose, a separate FEM equation system is built up and directly integrated into the system of the BEM via corresponding transformation matrices. A common variant for solving the FEM-specific parts is to use a solver based on eigenvalue calculations, which provides the corresponding eigenfrequencies and eigenvectors of the uncoupled FEM equation system for a given upper cutoff frequency. It can be observed that only a part of the determined modes has a considerable sound radiation efficiency and is therefore relevant for the result. An algorithm determines these modes by a fast post-processing routine considering a given percentage, reduces the size of the eigenvalue equation system accordingly, and thus shortens the solution time. Furthermore, it was investigated whether the condition of the entire system of equations is improved by removing the irrelevant modes when using iterative solution methods and whether the number of iterations can thus be reduced. The paper describes the basics of FEM coupling with BEM, presents the reduction algorithm used and shows the results obtained for corresponding test structures depending on the reduction percentage used.

Influences of Different Boundary Conditions and Hygro-Thermal Environment on the Free Vibration Responses of FGM Sandwich Plates Resting on Viscoelastic Foundation

This paper investigates the free vibration of functionally graded material (FGM) sandwich plates supported by different boundary conditions and influenced by a three-parameter viscoelastic foundation and hygro-thermal changes. Three types of FGM sandwich plates are studied and discussed, in which the FGM layers vary according to the power law rule and consist of ceramic and metal materials. An efficient and simple four-variable integral higher-order shear deformation theory (HSDT) is employed to model the analytical solution of the considered problem. Hamilton principle is implemented to obtain the plates’ governing equations and to derive the eigenvalue equation for the free vibration study. The model is verified by comparing numerical results with previous studies on the vibration of exponentially graded plates. New results are presented in this paper showing the influences of different boundary conditions, hygro-thermal changes, viscoelastic parameters, vibration modes, material exponents, and geometric dimensions.

On the Uncertainty Analysis of the Data-Enabled Physics-Informed Neural Network for Solving Neutron Diffusion Eigenvalue Problem

We performed uncertainty analysis and further numerical studies on the data-enabled physics-informed neural network (DEPINN). The purpose of DEPINN is to accurately and efficiently use a small amount of prior data to solve the neutron diffusion eigenvalue equations based on the physics-informed neural network. However, in practical engineering experiments, these prior data are acquired through different kinds of sensors, which are inevitably polluted by noise. Numerical results of three typical benchmark problems show that the classical DEPINN is not so robust with respect to noise. To improve the noise robustness, we propose an interval loss function to deal with the noisy prior data term; the weight of the noisy prior data term is also set to be noise dependent. Numerical results show that the proposed framework effectively enhances the robustness of DEPINN and improves the efficiency of utilizing the noisy prior data and thus promotes the engineering application of DEPINN.