Behavior Of Eigenvalues Research Articles

Asymptotic Behavior for Eigenvalues of the Singular Sturm-Liouville Problem with Transmission Conditions

In this study, we consider the singular Sturm-Liouville (S-L) problem with transmission conditions (TC). The eigenvalues of the singular S-L problem with TC are investigated. By defining a new Hilbert space which is related to TC, the self-adjointness of the S-L problem in this associated Hilbert space is proved. And we give the condition for $$\lambda $$ being the eigenvalue of the singular S-L problem. Furthermore, the asymptotic behavior of eigenvalues of the singular S-L problem is described.

Asymptotic Behavior of Eigenvalues of a Boundary Value Problem for a Second-Order Elliptic Differential–Operator Equation with Spectral Parameter in the Equation and a Boundary Condition

In a separable Hilbert space $$H$$, we study the asymptotic behavior of eigenvalues of a boundary value problem for second-order elliptic differential–operator equations for the case in which the spectral parameter occurs in the equation quadratically and one of the boundary conditions is a quadratic trinomial in the same spectral parameter. We derive asymptotic formulas for the eigenvalues of this boundary value problem. An application of the abstract results obtained here to elliptic boundary value problems is indicated.

Eigenvalue variation under moving mixed Dirichlet–Neumann boundary conditions and applications

We deal with the sharp asymptotic behaviour of eigenvalues of elliptic operators with varying mixed Dirichlet–Neumann boundary conditions. In case of simple eigenvalues, we compute explicitly the constant appearing in front of the expansion’s leading term. This allows inferring some remarkable consequences for Aharonov–Bohm eigenvalues when the singular part of the operator has two coalescing poles.

Universal Behavior of the Corners of Orbital Beta Processes

Abstract There is a unique unitarily-invariant ensemble of $N\times N$ Hermitian matrices with a fixed set of real eigenvalues $a_1> \dots > a_N$. The joint eigenvalue distribution of the $(N-1)$ top-left principal submatrices of a random matrix from this ensemble is called the orbital unitary process. There are analogous matrix ensembles of symmetric and quaternionic Hermitian matrices that lead to the orbital orthogonal and symplectic processes, respectively. By extrapolation, on the dimension of the base field, of the explicit density formulas, we define the orbital beta processes. We prove the universal behavior of the virtual eigenvalues of the smallest $m$ principal submatrices, when $m$ is independent of $N$ and the eigenvalues $a_1> \dots > a_N$ grow linearly in $N$ and in such a way that the rescaled empirical measures converge weakly. The limiting object is the Gaussian beta corners process. As a byproduct of our approach, we prove a theorem on the asymptotics of multivariate Bessel functions.

Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere

We consider the eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a spherical domain. Especially, we investigate the case when the domain is a large zonal one and letting the zone larger so that the zone covers the whole sphere as a limit. We discuss the behavior of eigenvalues according to the rate of expansion of the zone.

Extremal eigenvalues of the Dirichlet biharmonic operator on rectangles

We study the behaviour of extremal eigenvalues of the Dirichlet biharmonic operator over rectangles with a given fixed area. We begin by proving that the principal eigenvalue is minimal for a rectangle for which the ratio between the longest and the shortest side lengths does not exceed 1.066459 1.066459 . We then consider the sequence formed by the minimal k k th eigenvalue and show that the corresponding sequence of minimising rectangles converges to the square as k k goes to infinity.

Analysis of the ground-state energy eigenvalues of fractal quantum potentials

Traditional models in solid-state physics study the motion of electrons in a periodic lattice. Recent developments in the physics of graphene allow scientists to construct two-dimensional structures with fractal geometry and conduct experiments on them. Recently, some theoretical approaches have been developed to study the optical and electrical properties of semiconductor layers with self-similar characteristics. This newly emerged direction in solid-state physics inspired us to focus on studying the quantum mechanical properties of fractal potentials. We first introduce sequences of potential wells converging towards different fractal structures. Then, we calculate the ground-state energy eigenvalues of the time-independent Schrödinger equation for these potential functions using two numerical methods, the Numerov and analytical transfer matrix method, to demonstrate the effect of the potential structure morphology and properties on the behavior of the energy eigenvalues. Ground-state energies for the generalized Cantor set, the Smith–Volterra–Cantor set, a multi-level Cantor set and the Weierstrass function will be calculated and compared. We will deal with the question of how the ground state of these potential functions changes as the fractal generator is applied, and we show that properties such as the Lebesgue measure of the Cantor potentials and the Hausdorff dimension of the Weierstrass function strongly control the convergence of energy eigenvalues.

Robust estimation for image noise based on eigenvalue distributions of large sample covariance matrices

In this paper, we propose a novel algorithm to estimate Gaussian noise levels for captured natural images by rigorously analyzing the limiting distributions of the eigenvalue spectrum of a large covariance matrix with Gaussian samples. In order to select a relatively homogeneous region that best represents the noise, the corresponding image patches are first rearranged to construct a high-dimensional noise covariance matrix. And then, an optimal criterion for classifying homogeneous regions is derived based on the statistical relationship between the largest and the second largest eigenvalues of a sample covariance matrix. Moreover, we further explore the reasons for the bias of the maximum likelihood estimator of the noise variance both in high-dimensional settings and finite samples. According to random matrix theory, we clarify the asymptotic properties of the trace of a sample covariance matrix to measure the error bounds of estimation and then propose a new bias-corrected estimator. To this end, an effective estimation method for the noise level is devised based on the boundness and asymptotic behavior of pure noise eigenvalues of the selected patches. The estimation performance of our method has been guaranteed both theoretically and empirically. Experimental results have demonstrated that our approach can reliably infer true noise variance and is superior to the competing methods in terms of both estimation accuracy and robustness.

Running scales in causal dynamical triangulations

The spectrum of the Laplace-Beltrami operator, computed on the spatial slices of Causal Dynamical Triangulations, is a powerful probe of the geometrical properties of the configurations sampled in the various phases of the lattice theory. We study the behavior of the lowest eigenvalues of the spectrum and show that this can provide information about the running of length scales as a function of the bare parameters of the theory, hence about the critical behavior around possible second order transition points in the CDT phase diagram, where a continuum limit could be defined.

Superstatistics and canonical quantization of the damped harmonic oscillator

In this paper, we investigate the behavior of the energy eigenvalues of the Schrödinger equation by using the canonical quantization method. We obtain the Hamiltonian of the Schrödinger equation by the Lagrangian in terms of the new coordinates. Then we calculate the partition function by the eigenvalues and the thermodynamic properties of the system in the superstatistics formalism for the modified Dirac delta and the Gamma distributions. All results in the limiting cases satisfy that of the harmonic oscillator. Furthermore, the effects of the all parameters in the problem of energy eigenvalues and thermodynamic properties are calculated and shown graphically.

On the Outlying Eigenvalues of a Polynomial in Large Independent Random Matrices

Abstract Given a selfadjoint polynomial $P(X,Y)$ in two noncommuting selfadjoint indeterminates, we investigate the asymptotic eigenvalue behavior of the random matrix $P(A_N,B_N)$, where $A_N$ and $B_N$ are independent Hermitian random matrices and the distribution of $B_N$ is invariant under conjugation by unitary operators. We assume that the empirical eigenvalue distributions of $A_N$ and $B_N$ converge almost surely to deterministic probability measures $\mu$ and $\nu$, respectively. In addition, the eigenvalues of $A_N$ and $B_N$ are assumed to converge uniformly almost surely to the support of $\mu$ and $\nu ,$ respectively, except for a fixed finite number of fixed eigenvalues (spikes) of $A_N$. It is known that almost surely the empirical distribution of the eigenvalues of $P(A_N,B_N)$ converges to a certain deterministic probability measure $\eta \ (\textrm{sometimes denoted}\ P^\square(\mu,\nu))$ and, when there are no spikes, the eigenvalues of $P(A_N,B_N)$ converge uniformly almost surely to the support of $\eta$. When spikes are present, we show that the eigenvalues of $P(A_N,B_N)$ still converge uniformly to the support of $\eta$, with the possible exception of certain isolated outliers whose location can be determined in terms of $\mu ,\nu ,P$, and the spikes of $A_N$. We establish a similar result when $B_N$ is replaced by a Wigner matrix. The relation between outliers and spikes is described using the operator-valued subordination functions of free probability theory. These results extend known facts from the special case in which $P(X,Y)=X+Y$.

Any l-state solutions of the Schrodinger equation interacting with Hellmann–Kratzer potential model

The approximate analytical solutions of the radial Schrodinger equation have been obtained with a newly proposed potential called Hellmann–Kratzer potential. The potential is a superposition of Hellmann potential and modified Kratzer potential. The Hellmann–Kratzer potential actually comprises of three different potentials which include Yukawa potential, Coulomb potential and Kratzer potential. The aim of combining these potentials is to have a wide application. The energy eigenvalue and the corresponding wave function are calculated in a closed and compact form using the Nikiforov–Uvarov method. The energy equation for some potentials such as Kratzer, Hellmann, Yukawa and Coulomb potentials has also been obtained by varying some potential parameters. Our results excellently agree with the already existing literature. Some numerical results have been computed. We have plotted the behaviour of the energy eigenvalues with different potential parameters and also reported on the numerical result.

Nonlinear time domain and stability analysis of beams under partially distributed follower force

• A geometrically exact beam formulation is used to obtain the stability and nonlinear time domain analysis. • A wide range of beams with different flexibility ratios are utilized in this investigation. • The behavior of the eigenvalues at pre- and post-instability as well as the frequency and time domain results are obtained. • The stability and nonlinear behavior are highly affected by partially follower forces length and position. This paper aims to investigate linear and nonlinear behavior of beams subjected to externally applied partially distributed follower forces. In this investigation, the nonlinear composite beam theory of Hodges is used. The system of nonlinear equations is linearized about the equilibrium, or rest structure state, and the linear system is solved numerically. The effects of follower force position on the behavior of eigenvalues at pre- and post-instability are reported. Additionally, the contours of critical follower force are obtained by changing the position of follower force in span-wise and chord-wise directions. The effects of different parameters such as the length, and position of follower force and the ratios of stiffnesses on the critical follower force as well as the nonlinear limit cycle oscillation (LCO) are reported. The obtained results indicate that the length and the position of the partially distributed follower forces considerably affect the stability of the beam.

Maximal Convex Combinations of Sequential Steklov Eigenvalues

In this paper, we study a shape optimization problem in two dimensions where the objective function is the convex combination of two sequential Steklov eigenvalues of a domain with a fixed area constraint. We show the existence of the optimal domain and the nondecreasing, Lipschitz continuity, and convexity of the optimal objective function with respect to the convex combination constant. On one-parameter family of rectangular domains, asymptotic behaviors of lower eigenvalues are found. For general shapes, numerical approaches are used to find optimal shapes. The range of the first two Steklov eigenvalues are discussed for several one-parameter families of shapes including Cassini oval shapes and Hippopede shapes.

The limiting behaviour of solutions to a family of eigenvalue problems

ABSTRACTWe investigate the asymptotic behaviour of first eigenvalues and their corresponding eigenfunctions for a family of eigenvalue problems involving Finsler p-Laplace type operators and nonlocal terms. We prove that there exists a subsequence of eigenfunctions which converges uniformly to a continuous function which is the viscosity solution of a limiting equation that is identified.

On a New Characterization of Some Class Nonlinear Eigenvalue Problem

A normal mode analysis of a vibrating mechanical or electrical system gives rise to an eigenvalue problem. Faber made a fairly complete study of the existence and asymptotic behavior of eigenvalues and eigenfunctions, Green’s function, and expansion properties. We will investigate a new characterization of some class nonlinear eigenvalue problem.

Exact eigenvalues and eigenfunctions for a one-dimensional Gel’fand problem

It is known that every positive solution of a one-dimensional Gel’fand problem can be written explicitly. In this paper, we give exact expressions of all the eigenvalues and eigenfunctions of the linearized eigenvalue problem at each solution. We study asymptotic behaviors of eigenvalues and eigenfunctions as the L∞-norm of the solution goes to infinity. We also study the problem u″ + λe−u = 0 and the associated linearized problem.

On the distribution of real eigenvalues in linear viscoelastic oscillators

SummaryIn this paper, a linear viscoelastic system is considered where the viscoelastic force depends on the past history of motion via a convolution integral over an exponentially decaying kernel function. The free‐motion equation of this nonviscous system yields a nonlinear eigenvalue problem that has a certain number of real eigenvalues corresponding to the nonoscillatory nature. The quality of the current numerical methods for deriving those eigenvalues is directly related to damping properties of the viscoelastic system. The main contribution of this paper is to explore the structure of the set of nonviscous eigenvalues of the system while the damping coefficient matrices are rank deficient and the damping level is changing. This problem will be investigated in the cases of low and high levels of damping, and a theorem that summarizes the possible distribution of real eigenvalues will be proved. Moreover, upper and lower bounds are provided for some of the eigenvalues regarding the damping properties of the system. Some physically realistic examples are provided, which give us insight into the behavior of the real eigenvalues while the damping level is changing.

On the Behavior of Clamped Plates under Large Compression

We determine the asymptotic behavior of eigenvalues of clamped plates under large compression by relating this problem to eigenvalues of the Laplacian with Robin boundary conditions. Using the method of fundamental solutions, we then carry out a numerical study of the extremal domains for the first eigenvalue, from which we see that these depend on the value of the compression, and start developing a boundary structure as this parameter is increased. The corresponding number of nodal domains of the first eigenfunction of the extremal domain also increases with the compression.

Effects of Asymmetry on Thermoacoustic Modes in Annular Combustors: A Higher-Order Perturbation Study

Gas-turbine combustion chambers typically consist of nominally identical sectors arranged in a rotationally symmetric pattern. However, in practice, the geometry is not perfectly symmetric. This may be due to design decisions, such as placing dampers in an azimuthally nonuniform fashion, or to uncertainties in the design parameters, which break the rotational symmetry of the combustion chamber. The question is whether these deviations from symmetry have impact on the thermoacoustic-stability calculation. The paper addresses this question by proposing a fast adjoint-based perturbation method. This method can be integrated into numerical frameworks that are industrial standard such as lumped-network models, Helmholtz and linearized Euler equations. The thermoacoustic stability of asymmetric combustion chambers is investigated by perturbing rotationally symmetric combustor models. The approach proposed in this paper is applied to a realistic three-dimensional combustion chamber model with an experimentally measured flame transfer function (FTF). The model equations are solved with a Helmholtz solver. Results for modes of zeroth, first, and second azimuthal order are presented and compared to exact solutions of the problem. A focus of the discussion is set on the loss of mode-degeneracy due to symmetry breaking and the capability of the perturbation theory to accurately predict it. In particular, an “inclination rule” that explains the behavior of degenerate eigenvalues at first order is proven.