Real eigenvalues determined by recursion of eigenstates

Quantum physics is primarily concerned with real eigenvalues,stemming from the unitarity of time evolutions.With the introduction of PT symmetry,a widely accepted consensus is that,even if the Hamiltonian of the system is not Hermitian,the eigenvalues can still be purely real under specific symmetry.Hence,great enthusiasm has been devoted to exploring the eigenvalue problem of non-Hermitian systems.In this work,from a distinct perspective,we demonstrate that real eigenvalues can also emerge under the appropriate recursive condition of eigenstates.Consequently,our findings provide another path to extract the real energy spectrum of non-Hermitian systems,which guarantees the conservation of probability and stimulates future experimental observations.

Efficient method to calculate the eigenvalues of the Zakharov–Shabat system

A numerical method is proposed to calculate the eigenvalues of the Zakharov–Shabat system based on Chebyshev polynomials. A mapping in the form of tanh(ax) is constructed according to the asymptotic of the potential function for the Zakharov–Shabat eigenvalue problem. The mapping can distribute Chebyshev nodes very well considering the gradient for the potential function. Using Chebyshev polynomials, tanh(ax) mapping, and Chebyshev nodes, the Zakharov–Shabat eigenvalue problem is transformed into a matrix eigenvalue problem. This method has good convergence for the Satsuma–Yajima potential and the convergence rate is faster than the Fourier collocation method. This method is not only suitable for simple potential functions but also converges quickly for a complex Y-shape potential. It can also be further extended to other linear eigenvalue problems.

On the eigenvalues and eigendisplacement of the critical mode in horizontally layered media

Wave propagation in horizontally layered media is a classical problem in seismic-wave theory.In semi-infinite space,a nondispersive Rayleigh wave mode exists,and the eigendisplacement decays exponentially with depth.In a layered model with increasing layer velocity,the phase velocity of the Rayleigh wave varies between the S-wave velocity of the bottom half-space and that of the classical Rayleigh wave propagated in a supposed half-space formed by the parameters of the top layer.If the phase velocity is the same as the P-or S-wave velocity of the layer,which is called the critical mode or critical phase velocity of surface waves,the general solution of the wave equation is not a homogeneous(expressed by trigonometric functions)or inhomogeneous(expressed by exponential functions)plane wave,but one whose amplitude changes linearly with depth(expressed by a linear function).Theories based on a general solution containing only trigonometric or exponential functions do not apply to the critical mode,owing to the singularity at the critical phase velocity.In this study,based on the classical framework of generalized reflection and transmission coefficients,the propagation of surface waves in horizontally layered media was studied by introducing a solution for the linear function at the critical phase velocity.Therefore,the eigenvalues and eigenfunctions of the critical mode can be calculated by solving a singular problem.The eigendisplacement characteristics associated with the critical phase velocity were investigated for different layered models.In contrast to the normal mode,the eigendisplacement associated with the critical phase velocity exhibits different characteristics.If the phase velocity is equal to the S-wave velocity in the bottom half-space,the eigendisplacement remains constant with increasing depth.

Petrochemical eigenvalues and diagrams for the identification of metamorphic rocks'protolith,taking the host rocks of Dashuigou tellurium deposit in China as an example

The Dashuigou tellurium deposit is the world’s only known independent tellurium deposit.By restoring metamorphic rocks’protolith,we seek to understand not only the development and evolution trajectory of the region but also the origin of the relevant deposits.While there are many ways to restore metamorphic rocks’protolith,we take the host metamorphic rocks of Dashuigou tellurium deposit and leverage various petrochemical eigenvalues and related diagrams previously proposed to reveal the deposit’s host metamorphic rocks’protolith.The petrochemical eigenvalues include molecular number,Niggli’s value,REE parity ratio,CaO/Al_(2)O_(3)ratio,Fe^(3+) /(Fe^(3+) -+Fe^(2+) )ratio,chondrite-normalized REE value,logarithmic REE value,various REE eigenvalues including scandium,Eu/Sm ratio,total REE amount,light and heavy REEs,δEu,Eu anomaly,Sm/Nd ratio,and silicon isotope δ^(30) SiNBS-29‰,etc.The petrochemical plots include ACMs,100 mg-c-(al+alk),SiO_(2)-(Na_(2)O+K_(2)O),(al+fm)-(c+alk)versus Si,FeO+Fe_(2)O^(3+) TiO)-Al_(2)O_(3)-MgO,c-mg,Al_(2)O_(3)-(Na_(2)O+K_(2)O),chondrite-normalized REE model,La/Yb-REE,and Sm/Nd ratio,etc.On the basis of these comprehensive analyses,the following conclusions are drawn,starting from the many mantle-derived types of basalt developed in the study area of different geological ages,combined with the previously published research results on the deposit s fluid inclusions and sulfur and lead isotopes.The deposit is formed by mantle degassing in the form of a mantle plume in the late Yanshanian orogeny.The degassed fluids are rich in nano-sc ale substances including Fe,Te,S,As,Bi,Au,Se,H_(2),CO_(2),N_(2),H_(2)O,and CH_(4),which are enriched by nano-effect,and then rise to a certain part of the crust in the form of mantle plume along the lithospheric fault to form the deposit.The ultimate power for tellurium mineralization was from H_(2)flow with high energy,which was produced through radiation from the melted iron of the Earth’s outer core.The H,flow results in the Earth’s degassing,as well as the mantle and crust’s uplift.

Time-domain dynamic constitutive model suitable for mucky soil site seismic response

Soil nonlinear behavior displays noticeable effects on the site seismic response.This study proposes a new functional expression of the skeleton curve to replace the hyperbolic skeleton curve.By integrating shear modulus and combining the dynamic skeleton curve and the damping degradation coefficient,the constitutive equation of the logarithmic dynamic skeleton can be obtained,which considers the damping effect in a soil dynamics problem.Based on the finite difference method and the multi-transmitting boundary condition,a 1D site seismic response analysis program called Soilresp1D has been developed herein and used to analyze the time-domain seismic response in three types of sites.At the same time,this study also provides numerical simulation results based on the hyperbolic constitutive model and the equivalent linear method.The results verify the rationality of the new soil dynamic constitutive model.It can analyze the mucky soil site nonlinear seismic response,reflecting the deformation characteristics and damping effect of the silty soil.The hysteresis loop area is more extensive,and the residual strain is evident.

LARGE DEVIATIONS FOR TOP EIGENVALUES OFβ-JACOBI ENSEMBLES AT SCALING TEMPERATURES

Letλ=(λ_(1),…,λ_(n))beβ-Jacobi ensembles with parameters p_(1),p_(2),n andβ,withβvarying with n.Set■.Suppose that■and 0≤σγ<1.We offer the large deviation for p_(1)+p_(2)/p_(1)■λ_(i)whenγ>0 via the large deviation of the corresponding empirical measure and via a direct estimate,respectively,whenγ=0.

Estimates on the eigenvalues of complex nonlocal Sturm-Liouville problems

The present paper deals with the eigenvalues of complex nonlocal Sturm-Liouville boundary value problems.The bounds of the real and imaginary parts of eigenvalues for the nonlocal Sturm-Liouville differential equation involving complex nonlocal potential terms associated with nonlocal boundary conditions are obtained in terms of the integrable conditions of coefficients and the real part of the eigenvalues.

Highly Accurate Golden Section Search Algorithms and Fictitious Time Integration Method for Solving Nonlinear Eigenvalue Problems

This study sets up two new merit functions,which are minimized for the detection of real eigenvalue and complex eigenvalue to address nonlinear eigenvalue problems.For each eigen-parameter the vector variable is solved from a nonhomogeneous linear system obtained by reducing the number of eigen-equation one less,where one of the nonzero components of the eigenvector is normalized to the unit and moves the column containing that component to the right-hand side as a nonzero input vector.1D and 2D golden section search algorithms are employed to minimize the merit functions to locate real and complex eigenvalues.Simultaneously,the real and complex eigenvectors can be computed very accurately.A simpler approach to the nonlinear eigenvalue problems is proposed,which implements a normalization condition for the uniqueness of the eigenvector into the eigenequation directly.The real eigenvalues can be computed by the fictitious time integration method(FTIM),which saves computational costs compared to the one-dimensional golden section search algorithm(1D GSSA).The simpler method is also combined with the Newton iterationmethod,which is convergent very fast.All the proposed methods are easily programmed to compute the eigenvalue and eigenvector with high accuracy and efficiency.

A Simple Embedding Method for the Laplace-Beltrami Eigenvalue Problem onImplicit Surfaces

We propose a simple embedding method for computing the eigenvalues and eigenfunctions of the Laplace-Beltrami operator on implicit surfaces.The approach follows an embedding approach for solving the surface eikonal equation.We replace the differential operator on the interface with a typical Cartesian differential operator in the surface neighborhood.Our proposed algorithm is easy to implement and efficient.We will give some two-and three-dimensional numerical examples to demonstrate the effectiveness of our proposed approach.

Reciprocal Distance Laplacian Eigenvalue Distribution Based on Graph Parameters

Let G be a connected graph of order n and m_(RD)^(L)_(G)I denote the number of reciprocal distance Laplacian eigenvaluesof G in an interval I.For a given interval I,we mainly present several bounds on m_(RD)^(L)_(G)I in terms of various structuralparameters of the graph G,including vertex-connectivity,independence number and pendant vertices.

THE INTERIOR TRANSMISSION EIGENVALUE PROBLEM FOR AN ANISOTROPIC MEDIUM BY A PARTIALLY COATED BOUNDARY

We consider the interior transmission eigenvalue problem corresponding to the scattering for an anisotropic medium of the scalar Helmholtz equation in the case where the boundary?Ωis split into two disjoint parts and possesses different transmission conditions.Using the variational method,we obtain the well posedness of the interior transmission problem,which plays an important role in the proof of the discreteness of eigenvalues.Then we achieve the existence of an infinite discrete set of transmission eigenvalues provided that n≡1,where a fourth order differential operator is applied.In the case of n■1,we show the discreteness of the transmission eigenvalues under restrictive assumptions by the analytic Fredholm theory and the T-coercive method.

Time-Domain Analysis of Body Freedom Flutter Based on 6DOF Equation

The reduced weight and improved efficiency of modern aeronautical structures result in a decreasing separation of frequency ranges of rigid and elastic modes.Particularly,a high-aspect-ratio flexible flying wing is prone to body freedomflutter(BFF),which is a result of coupling of the rigid body short-periodmodewith 1st wing bendingmode.Accurate prediction of the BFF characteristics is helpful to reflect the attitude changes of the vehicle intuitively and design the active flutter suppression control law.Instead of using the rigid body mode,this work simulates the rigid bodymotion of the model by using the six-degree-of-freedom(6DOF)equation.A dynamicmesh generation strategy particularly suitable for BFF simulation of free flying aircraft is developed.An accurate Computational Fluid Dynamics/Computational Structural Dynamics/six-degree-of-freedom equation(CFD/CSD/6DOF)-based BFF prediction method is proposed.Firstly,the time-domain CFD/CSD method is used to calculate the static equilibrium state of the model.Based on this state,the CFD/CSD/6DOF equation is solved in time domain to evaluate the structural response of themodel.Then combinedwith the variable stiffnessmethod,the critical flutter point of the model is obtained.This method is applied to the BFF calculation of a flyingwing model.The calculation results of the BFF characteristics of the model agree well with those fromthe modalmethod andNastran software.Finally,the method is used to analyze the influence factors of BFF.The analysis results show that the flutter speed can be improved by either releasing plunge constraint or moving the center ofmass forward or increasing the pitch inertia.

INEQUALITIES OF EIGENVALUES FOR BI-KOHN LAPLACIAN ON HEISENBERG GROUP

In this article, we consider the eigenvalue problem for the bi-Kohn Laplacian and obtain universal bounds on the （k ＋ 1）-th eigenvalue in terms of the first k eigenvalues independent of the domains.

ASYMPTOTIC BEHAVIOUR OF EIGENVALUES FOR THE DISCONTINUOUS BOUNDARY-VALUE PROBLEM WITH FUNCTIONAL-TRANSMISSION CONDITIONS

In this study, the boundary-value problem with eigenvalue parameter generated by the differential equation with discontinuous coefficients and boundary conditions which contains not only endpoints of the considered interval, but also point of discontinuity and linear functionals is investigated. So, the problem is not pure boundary-value. The authors single out a class of linear functionals and find simple algebraic conditions on coefficients, which garantee the existence of infinit number eigenvalues. Also the asymptotic formulas for eigenvalues are found.

A SIMPLIFIED BRAUER'S THEOREM ON MATRIX EIGENVALUES

Let A= (aij)∈Cn×n and ri = ∑ j≠i｜aij｜. Suppose thatforeach row of A there isatleastonenonzero off-diagonalentry. Itis proved thatalleigenvalues ofAarecontained in Ω～= ∪aij≠0,i≠j｛z∈C:｜z- aii｜｜z- ajj｜≤rirj｝. The resultre- duces the num berofovals in originalBrauer'stheorem in m any cases. Eigenval- ues(and associated eigenvectors) thatlocate in theboundary ofΩ～ arediscussed.

UNIVERSAL BOUNDS FOR EIGENVALUES OF LAPLACIAN OPERATOR OF ANY ORDER

Let Ω be a connected bounded domain in R^n. Denote by λi the i-th eigenvalue of the Lapla^ian operator with any order p：{u=Эn→^-Эu=…=Эn→p-1^-Эp-1u=0 on ЭΩ （-△）pu=λu in Ω.In this article, we give some expressions for upper bound of the （k ＋ 1）-th eigenvalue ）λk＋l in terms of the first k eigenvalues.

One Computational Method of the Eigenvalues of the Horizontal Vibration Problem of Beam

This paper considers one computational method of the eigenvalues approximate value of the horizontal vibration problem of beam. The proof of our main result is based on the variational formula. First of all, Cauchy inequality is used to obtain a basic inequality. Secondly, the functions of basis are made by Galerkin method, and the error estimates of eignevalues are obtained by Cauchy inequality. At last, the computational method of the approximate value of the eigenvalues turns out immediately, and acc...

ESTIMATES FOR EIGENVALUES OF FOURTH-ORDER WEIGHTED POLYNOMIAL OPERATOR

In this paper,we investigate the Dirichlet eigenvalue problem of fourth-order weighted polynomial operator △2u-a△u＋bu=Λρu,inΩRn,u｜Ω=uvΩ=0,where the constants a,b≥0.We obtain some estimates for the upper bounds of the （k＋1）-th eigenvalueΛ_k＋1 in terms of the first k eigenvalues.Moreover,these results contain some results for the biharmonic operator.

ESTIMATES ON EIGENVALUES FOR THE BIHARMONIC OPERATOR ON A BOUNDED DOMAIN IN H^n(-1)

In this paper, we consider eigenvalues of the Dirichlet biharmonic operator on a bounded domain in a hyperbolic space. We obtain universal bounds on the （k ＋ 1）th eigenvalue in terms of the first kth eigenvalues independent of the domains.

An Efficient Technique for Finding the Eigenvalues of Fourth-Order Sturm-Liouville Problems

In this work, we present a computational method for solving eigenvalue problems of fourth-order ordinary differential equations which based on the use of Chebychev method. The efficiency of the method is demonstrated by three numerical examples. Comparison results with others will be presented.