Exact Eigenvalues Research Articles

Advancements to generalized equivalence theory for preserving cross-sections of whole core simulations

The conventional two-step approach, without the use of an equivalence method, can introduce significant error for the simulation of non-LWRs (Light Water Reactors), especially fast reactors, due to the large leakage and high anisotropic neutron distribution. To improve upon the accuracy of the two-step approach, a 3D whole core model is simulated with a transport method to generate region-wise homogenized cross-sections (XS). These XS can then be used in a diffusion whole core solver during the second step to extend the application to cycle and transient analysis. Discontinuity factors (DFs) are then introduced to improve the accuracy during the simulation of the second step. With properly generated DFs from Generalized Equivalence Theory (GET), the region-averaged solutions from the first transport step can then be reproduced by the second step diffusion solver. The Monte Carlo method was selected to perform the whole core transport simulation to generate region-wise XS. However, simulating whole core problems with Monte Carlo may result in poor statistics near the peripheral region especially for partial current tallies. This paper introduces an advancement to GET to reproduce region-wise solutions for select regions when the reaction rates and surface currents have good statistics in these regions but poor statistics in other regions. A reference high-fidelity model was constructed using the Serpent 2 Monte Carlo code based on the EBR-II benchmark evaluation and verification was carried out using the TriPEN-4 method in PARCS. The results show that it is possible to reproduce the exact eigenvalue and power distributions of whole core problems in a feasible manner.

Efficient Shift-and-Invert Preconditioning for Multi-GPU Accelerated Density Functional Calculations.

To accelerate the iterative diagonalization of electronic structure calculations, we propose a new inexact shift-and-invert (ISI) preconditioning method. The key idea is to improve shift values in the ISI preconditioning to be closer to the exact eigenvalues, leading to a significant boost in the convergence speed of the iterative diagonalization. Furthermore, we adopted a preconditioned conjugate gradient solver to rapidly evaluate an inversion process. Finally, we accelerated overall processes, including the proposed modification, with state-of-the-art graphical processing units (GPUs) and assessed its parallel efficiency with real-space density functional calculations of 1D, 2D, and 3D periodic systems. Our method attains both fast diagonalization convergence and high multi-GPU parallel efficiency. This is evident from the fact that single-point density functional calculations for hundreds of atom systems can be done in approximately 10 s using 8 GPUs. The proposed method can be generally applied to any electronic structure calculation methods involving large-scale diagonalizations.

Estimating Bethe roots with VQE

Abstract Bethe equations, whose solutions determine exact eigenvalues and eigenstates of corresponding integrable Hamiltonians, are generally hard to solve. We implement a Variational Quantum Eigensolver approach to estimating Bethe roots of the spin-1/2 XXZ quantum spin chain, by using Bethe states as trial states, and treating Bethe roots as variational parameters. In numerical simulations of systems of size up to 6, we obtain estimates for Bethe roots corresponding to both ground states and excited states with up to 5 down-spins, for both the closed and open XXZ chains. This approach is not limited to real Bethe roots.

Global monopole effects on exact s-state solution under trigonometric Pöschl–Teller potential

In this paper, we address into the quantum dynamics of quantum particles interacting with a trigonometric potential within the context of a space-time background featuring a point-like global monopole. Our aim is to derive exact s-states eigenvalue solutions. To achieve this, we first derive the angular and radial wave equations of the Schrödinger equation within this particular geometry, considering Pöschl–Teller potential. While the angular solutions are well-known, the radial wave equation necessitates the utilisation of special functions for its solution. Our analysis demonstrates that the presence of the global monopole induces modifications in both the exact energy eigenvalues and the corresponding wave functions of quantum particles. Consequently, these modifications deviate from the behaviour typically observed in flat space when considering this potential. Afterwards, we compute Shannon entropy for this quantum systems and show the influences of various parameters.

Fourier series method for the stability solution of simply supported thin beams on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models

The analysis of stability problems of beams on two-parameter foundations (Bo2PFs) is an important part of their design for compressive loads. This work presents novel first principles derivation of the governing differential equations of elastic stability (GDES) of thin beams resting on two-parameter elastic foundations of the Pasternak, Filonenko-Borodich, Hetenyi or Vlasov models. The requirements of translational and rotational equilibrium of all the applied, reactive and internal forces on an infinitesimal segment of the Bo2PF and the laws of infinitesimal calculus were used to formulate the GDES as a fourth order ordinary differential equation (ODE) in terms of the transverse displacement function ux. The GDES is non-homogeneous in the presence of applied transverse load qx but homogeneous when qx vanishes. This study presents the Fourier series method (FSM) for solving the governing differential equation of stability (GDES) for the case of Dirichlet boundary conditions. The FSM has the advantage of amenability to differentiation, and integration due to the orthogonality properties of the sinusoidal functions. Implementation of the FSM by assuming the unknown function in the GDES as a Fourier series of infinite terms and the exploitation of orthogonalization simplifies the problem to an algebraic eigenvalue problem which is the characteristic buckling equation. The exact eigenvalues are found by algebraic solution of the buckling equation. The exact eigenvalues were used to find the exact buckling loads and the exact buckling load coefficients. The critical buckling load was found to correspond to the first buckling mode (n= 1), and is identical with previous solutions in the literature. Numerical calculations for the critical buckling load parameters Kcr were presented for the Bo2PF problem for values of the dimensionless foundation parameters k-1= 0, k-2= 0; k-1= 100, k-2= 0; k-1= 0, k-2= 1; k-1= 100, k-2= 100; k-1= 0, k-2= 2.5; k-1= 100, k-2= 2.5. The present solutions were compared with previous solutions for Kcr in the literature. The comparison shows that the present FSM results are identical with previous results obtained using various other methods such as Recursive Differentiation Method, Finite Element Method, Generalized Integral Transform Method (GITM) and Stodola-Vianello Iteration Method. The study has illustrated the effectiveness of the FSM for solving Bo2PFs.

Gaussian-process-regression-based method for the localization of exceptional points in complex resonance spectra

Resonances in open quantum systems depending on at least two controllable parameters can show the phenomenon of exceptional points (EPs), where not only the eigenvalues but also the eigenvectors of two or more resonances coalesce. Their exact localization in the parameter space is challenging, in particular in systems, where the computation of the quantum spectra and resonances is numerically very expensive. We introduce an efficient machine learning algorithm to find EPs based on Gaussian process regression (GPR). The GPR-model is trained with an initial set of eigenvalue pairs belonging to an EP and used for a first estimation of the EP position via a numerically cheap root search. The estimate is then improved iteratively by adding selected exact eigenvalue pairs as training points to the GPR-model. The GPR-based method is developed and tested on a simple low-dimensional matrix model and then applied to a challenging real physical system, viz., the localization of EPs in the resonance spectra of excitons in cuprous oxide in external electric and magnetic fields. The precise computation of EPs, by taking into account the complete valence band structure and central-cell corrections of the crystal, can be the basis for the experimental observation of EPs in this system.

Exact Solution of the Bose-Hubbard Model with Unidirectional Hopping.

A one-dimensional Bose-Hubbard model with unidirectional hopping is shown to be exactly solvable. Applying the algebraic Bethe ansatz method, we prove the integrability of the model and derive the Bethe ansatz equations. The exact eigenvalue spectrum can be obtained by solving these equations. The distribution of Bethe roots reveals the presence of a superfluid-Mott insulator transition at the ground state, and the critical point is determined. By adjusting the boundary parameter, we demonstrate the existence of a non-Hermitian skin effect even in the presence of interaction, but it is completely suppressed for the Mott insulator state in the thermodynamical limit. Our result represents a new class of exactly solvable non-Hermitian many-body systems, which has no Hermitian correspondence and can be used as a benchmark for various numerical techniques developed for non-Hermitian many-body systems.

Asymptotic formulas of the eigenvalues for the linearization of a one-dimensional sinh-Poisson equation

We are concerned with a Neumann problem of a one-dimensional sinh-Poisson equation u′′+λsinhu=0for0<x<1,u′(0)=u′(1)=0,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} {\\left\\{ \\begin{array}{ll} u''+\\lambda \\sinh u=0 &{} \ ext {for}\\ 0<x<1,\\\\ u'(0)=u'(1)=0, \\end{array}\\right. } \\end{aligned}$$\\end{document}where lambda >0 is a parameter. A complete bifurcation diagram of this problem is obtained. We also consider the linearized eigenvalue problem at every nontrivial solution u. We derive exact expressions of all the eigenvalues and eigenfunctions, using Jacobi elliptic functions and complete elliptic integrals. Then, we also derive asymptotic formulas of eigenvalues as lambda rightarrow 0. Exact eigenvalues and eigenfunctions for a Dirichlet problem are presented without proof. The main technical tool is an ODE technique.

Topological effects on non-relativistic eigenvalue solutions under AB-flux field with pseudoharmonic- and Mie-type potentials

In this paper, we investigate the quantum dynamics of a non-relativistic particle confined by the Aharonov–Bohm quantum flux field with pseudoharmonic-type potential in the background of topological defect produced by a point-like global monopole. We solve the radial Schrödinger equation analytically and determine the exact eigenvalue solution of the quantum system. Afterwards, we consider a Mie-type potential in the quantum system and solve the radial equation analytically and obtain the eigenvalue solution. We analyze the effects of the topological defect and the quantum flux with these potentials on the energy eigenvalue and wave function of the non-relativistic particles. In fact, it is shown that the energy levels and wave functions are influenced by the topological defect shifted the result compared to the flat space results. In addition, the quantum flux field also shifted the eigenvalue solutions and an analogue of the Aharonov–Bohm effect for bound-states is observed. Finally, we utilize these eigenvalue solutions to some known diatomic molecular potential models and presented the energy eigenvalue and wave function.

Numerical solution of an extra-wide angle parabolic equation through diagonalization of a 1-D indefinite Schrödinger operator with a piecewise constant potential

We present a numerical method for computing the solution of a partial differential equation (PDE) for modeling acoustic pressure, known as an extra-wide angle parabolic equation, that features the square root of a differential operator. The differential operator is the negative of an indefinite Schrödinger operator with a piecewise constant potential. This work primarily deals with the 3-piece case; however, a generalization is made the case of an arbitrary number of pieces. Through restriction to a judiciously chosen lower-dimensional subspace, approximate eigenfunctions are used to obtain estimates for the eigenvalues of the operator. Then, the estimated eigenvalues are used as initial guesses for the Secant Method to find the exact eigenvalues, up to roundoff error. An eigenfunction expansion of the solution is then constructed. The computational expense of obtaining each eigenpair is independent of the grid size. The accuracy, efficiency, and scalability of this method is shown through numerical experiments and comparisons with other methods.

Analytical Solutions of the Molecular Kratzer-Feus potential by means of the Nikiforov-Uvarov Method

The analytical methods for solving Schrödinger equation are essential and effective tools with which we can investigate the spectroscopic properties, the electronic structure, and the energetic properties of the diatomic molecules (DMs). Accordingly, in this work, we used the Nikiforov-Uvarov (NU) method to solve the three-dimensional nonrelativistic Schrödinger equation with the molecular Kratzer-Feus (KF) potential and obtain the exact analytical bound state energy eigenvalues as well as their corresponding normalized eigenfunctions. The effective KF diatomic molecular potential well is investigated and represented graphically for several well-known DMs. The bound state energy levels are tabulated numerically for arbitrary values of the vibrational and rotational quantum numbers. The results obtained in this work are found to be in excellent agreement with the already-existing results in the literature.

임의 형상 음향 공동의 전 주파수 대역 고유치 추출을 위한 개선된 NDIF법 개발: 제1부(이론 정립)

A modified NDIF method for highly accurate eigenvalue analysis of arbitrarily shaped two-dimensional acoustic cavities is proposed in the paper. The original NDIF method has the weak point of not extracting lower-order eigenvalues when the number of nodes increases to obtain more accurate eigenvalues. It turns out in the study that this weak point is because the system matrix of the acoustic cavity of interest becomes ill-conditioned and doesn’t have full rank. The paper proposes a practical way to extract eigenvalues and mode shapes of acoustic cavities in the entire frequency range by means of calculating accurately the determinant of the system matrix ill-conditioned in the low frequency range. It is confirmed through the example study for the circular acoustic cavity having an exact solution that eigenvalues and mode shapes obtained by the proposed method using only 32 nodes agree exactly with the exact solution although eigenvalues obtained by ANSYS using 2840 nodes have some errors with respect to the exact eigenvalues.

Noise induced bistability in a fluctuating environment

Bistability in some systems does not occur at the mean-field level but is induced by noise. In this work, we consider a system which exhibits bistability induced by noise but in the presence of environmental fluctuations. The system can transition between two environmental states at a specified rate. It has absorbing boundaries in one state and can switch between two metastable states in the other. At small values of a system parameter (ϵ), the model exhibits noise-induced switching behaviour and the probability distribution is bimodal which becomes unimodal at large values of ϵ. We calculate the exact eigenvalues and steady-state solutions of the system. We further extend the investigation by calculating the mean switching time between the two metastable states in the small parameter limit and compare its accuracy with simulation results.

Generalized fractional of the extended Nikiforov–Uvarov method for heavy tetraquark masses spectra

In this paper, the new analytic–exact energy eigenvalues and eigenfunctions are obtained in the fractional forms by using the extended Nikiforov–Uvarov approach in which the heavy diquark systems are described. We have recalculated the mass spectra and fractional radial wave of heavy tetraquarks. The mass spectra are compared to the experimental data. The present results show a good agreement with the experimental data and are improved in comparison with other studies. We conclude that the fractional models play a good role in the heavy tetraquark masses.

Position-dependent mass Schrödinger particles in space-like screw dislocation: associated degeneracies and magnetic and Aharonov–Bohm flux fields effects

We consider non-relativistic position-dependent mass (PDM) Schödinger particles moving in an elastic medium with space-like screw dislocation. Within the cylindrical coordinates $$\left( r,\phi ,z\right) $$ , we study and report the effects of screw dislocation as well as PDM settings on the energy levels of some PDM-quantum mechanical systems. In so doing, we use a power-law type positive-valued dimensionless scalar multiplier $$ f(r)=\lambda r^{\sigma }$$ (which manifestly introduces the metaphoric notion of PDM-Schödinger particles). Next, we subject such PDM particles to magnetic and Aharonov–Bohm flux fields. We report exact or conditionally exact eigenvalues and eigenfunctions for $$V(r)=0$$ and $$V(r) =a+b\,r+c\,r^{2}$$ .

Spectral Determinant of the Two‐Photon Quantum Rabi Model

AbstractThe various generalized spectral determinants (G‐functions) of the two‐photon quantum Rabi model are analyzed with emphasis on the qualitative aspects of the regular spectrum. Whereas all of them yield at least a subset of the exact regular eigenvalues, only theG‐function proposed by Chen et al. in 2012 exhibits an explicitly known pole structure which dictates the approach to the collapse point. This function is derived rigorously employing the ‐symmetry of the model and shown that its zeros correspond to the complete regular spectrum.

Comment on “Confined Klein–Gordon oscillators in Minkowski spacetime and a pseudo-Minkowski spacetime with a space-like dislocation: PDM KG-oscillators, isospectrality and invariance”

We argue that the analytical energies obtained recently in Mustafa (2022) from a radial eigenvalue equation are incorrect because the truncation method used by the author forces a relationship between two model parameters. This relationship (overlooked by the author) depends on the truncation number and as a result the exact analytical energies do not correspond to a single model but are solutions to different physical models. The reason is that the radial eigenvalue equation is not exactly solvable but conditionally solvable so that one can only obtain some particular exact solutions. We also show that the exact eigenvalues obtained for a particular simpler case are also incorrect because the author overlooked that the shift of the radial coordinate leads to a change in the left boundary condition.

An Improved Contour-Integral Algorithm for Calculating Critical Eigenvalues of Power Systems Based on Accurate Number Counting

This paper proposes an improved contour-integral algorithm based on exact number counting inside the interesting region to calculate critical eigenvalues of large-scale power systems. Firstly, the difference between the eigenvalues inside and outside the region of the non-hermitian matrix is derived using trapezoidal integral. Based on this difference, a small matrix can be skillfully constructed with reasonable parameters so that the exact number of eigenvalues is precisely equal to the eigenvalue number of the small matrix whose real parts are larger than 0.5. Further, rational parameters can be adaptively adjusted to obtain exact results exploiting the minimum of eigenvalues of the constructed matrix. The exact eigenvalue number can help set the contour-integral algorithm's initial subspace. We automatically expand the space by comparing the results of each iteration with the exact number to ensure that all eigenvalues are obtained. Numerical experiments demonstrate the effectiveness and reliability of the precise number counting algorithm and the improved contour integral algorithm.

Finiteness of entanglement entropy in collective field theory

We explore the question of finiteness of the entanglement entropy in gravitational theories whose emergent space is the target space of a holographic dual. In the well studied duality of two-dimensional non-critical string theory and c = 1 matrix model, this question has been studied earlier using fermionic many-body theory in the space of eigenvalues. The entanglement entropy of a subregion of the eigenvalue space, which is the target space entanglement in the matrix model, is finite, with the scale being provided by the local Fermi momentum. The Fermi momentum is, however, a position dependent string coupling, as is clear in the collective field theory formulation. This suggests that the finiteness is a non-perturbative effect. We provide evidence for this expectation by an explicit calculation in the collective field theory of matrix quantum mechanics with vanishing potential. The leading term in the cumulant expansion of the entanglement entropy is calculated using exact eigenstates and eigenvalues of the collective Hamiltonian, yielding a finite result, in precise agreement with the fermion answer. Treating the theory perturbatively, we show that each term in the perturbation expansion is UV divergent. However the series can be resummed, yielding the exact finite result. Our results indicate that the finiteness of the entanglement entropy for higher dimensional string theories is non-perturbative as well, with the scale provided by Newton’s constant.

Comment on “PDM Klein–Gordon oscillators in cosmic string spacetime in magnetic and Aharonov–Bohm flux fields within the Kaluza–Klein theory”

We show that the spectra of some models with supposed physical utility derived recently are not correct. The eigenvalues to the radial differential equations do not satisfy well-known theorems. The origin of the problem is that those equations are not exactly solvable but conditionally solvable. Present analysis shows that the author overlooked a second condition that provides a relationship between the model parameters. Consequently, all the conclusions derived from those supposedly exact eigenvalues cannot be correct.