Behavior Of Eigenvalues Research Articles

Asymptotic behavior of eigenvalues of fourth-order differential operators with spectral parameter in the boundary conditions

Asymptotic behavior of eigenvalues of fourth-order differential operators with spectral parameter in the boundary conditions

A numerical study of the generalized Steklov problem in planar domains

Abstract We numerically investigate the generalized Steklov problem for the modified Helmholtz equation and focus on the relation between its spectrum and the geometric structure of the domain. We address three distinct aspects: (i) the asymptotic behavior of eigenvalues for polygonal domains; (ii) the dependence of the integrals of eigenfunctions on domain symmetries; and (iii) the localization and exponential decay of Steklov eigenfunctions away from the boundary for smooth shapes and in the presence of corners. For this purpose, we implemented two complementary numerical methods to compute the eigenvalues and eigenfunctions of the associated Dirichlet-to-Neumann operator for planar bounded domains. We also discuss applications of the obtained results in the theory of diffusion-controlled reactions and formulate conjectures with relevance in spectral geometry.

Match-based solution of general parametric eigenvalue problems

We describe a novel algorithm for solving general parametric (nonlinear) eigenvalue problems. Our method has two steps: first, high-accuracy solutions of non-parametric versions of the problem are gathered at some values of the parameters; these are then combined to obtain global approximations of the parametric eigenvalues. To gather the non-parametric data, we use non-intrusive contour-integration-based methods, which, however, cannot track eigenvalues that migrate into/out of the contour as the parameter changes. Special strategies are described for performing the combination-over-parameter step despite having only partial information on such migrating eigenvalues. Moreover, we dedicate a special focus to the approximation of eigenvalues that undergo bifurcations. Finally, we propose an adaptive strategy that allows one to effectively apply our method even without any a priori information on the behavior of the sought-after eigenvalues. Numerical tests are performed, showing that our algorithm can achieve remarkably high approximation accuracy.

Deriving Characteristic Mode Eigenvalue Behavior Using Subduction of Group Representations

Deriving Characteristic Mode Eigenvalue Behavior Using Subduction of Group Representations

Moment analysis on channel eigenvalue for DAS‐to‐DAS system

AbstractIn this paper, we study on channel eigenvalue distribution of MIMO system, namely, DAS‐to‐DAS system, of which transmit and receive nodes are geometrically distributed in unplanned manner. It offers key to performance upper bound analysis of collaborative transmission in unplanned network and multi‐user transmission of distributed antenna system. Previous studies on stochastic geometry have analyzed performance of single typical receiver, while transmitters are geometrically distributed. And random matrix theory has unveiled probabilistic behavior of MIMO channel eigenvalues, while transmit and receive antennas are collocated. Our study integrates knowledge from the two different research fields to extend the applicability to analysis on channel eigenvalue distribution of DAS‐to‐DAS system. As a preliminary research on such channel eigenvalue distribution, its first few moments are derived in analytic form. And its coherency of the analysis results with Monte‐Carlo simulation results has examined.

Spectral Properties of Mimetic Operators for Robust Fluid–Structure Interaction in the Design of Aircraft Wings

This paper presents a comprehensive study on the spectral properties of mimetic finite-difference operators and their application in the robust fluid–structure interaction (FSI) analysis of aircraft wings under uncertain operating conditions. By delving into the eigenvalue behavior of mimetic Laplacian operators and extending the analysis to stochastic settings, we develop a novel stochastic mimetic framework tailored for addressing uncertainties inherent in the fluid dynamics and structural mechanics of aircraft wings. The framework integrates random matrix theory with mimetic discretization methods, enabling the incorporation of uncertainties in fluid properties, structural parameters, and coupling conditions at the fluid–structure interface. Through spectral and localization analysis of the coupled stochastic mimetic operator, we assess the system’s stability, sensitivity to perturbations, and computational efficiency. Our results highlight the potential of the stochastic mimetic approach for enhancing reliability and robustness in the design of aircraft wings, paving the way for optimization algorithms that integrate uncertainties directly into the design process. Our findings reveal a significant impact of stochastic perturbations on the spectral radius and eigenfunction localization, indicating heightened system sensitivity. The introduction of randomized singular value decomposition (RSVD) within our framework not only enhances computational efficiency but also preserves accuracy in low-rank approximations, which is critical for handling large-scale systems. Moreover, Monte Carlo simulations validate the robustness of our stochastic mimetic framework, showcasing its efficacy in capturing the nuanced dynamics of FSI under uncertainty. This study contributes to the fields of numerical methods and aerospace engineering by offering a rigorous and scalable approach for conducting uncertainty-aware FSI analysis, which is crucial for the development of safer and more efficient aircraft.

Behavior of large eigenvalues for the two-photon asymmetric quantum Rabi model

The asymptotic behavior of large eigenvalues is studied for the two-photon quantum Rabi model with a finite bias. It is proved that the spectrum of this Hamiltonian model consists of two eigenvalue sequences { E n + } n = 0 ∞ \lbrace E_n^+\rbrace _{n=0}^{\infty } , { E n − } n = 0 ∞ \lbrace E_n^-\rbrace _{n=0}^{\infty } , and their large n n asymptotic behavior with error term O ⁡ ( n − 1 / 2 ) \operatorname {O}(n^{-1/2}) is described. The principal tool is the method of near-similarity of operators introduced by G. V. Rozenblum and developed in works of J. Janas, S. Naboko, and E. A. Yanovich (Tur).

Sharp spectral gap estimates for higher-order operators on Cartan–Hadamard manifolds

The goal of this paper is to provide sharp spectral gap estimates for problems involving higher-order operators (including both the clamped and buckling plate problems) on Cartan–Hadamard manifolds. The proofs are symmetrization-free — thus no sharp isoperimetric inequality is needed — based on two general, yet elementary functional inequalities. The spectral gap estimate for clamped plates solves a sharp asymptotic problem from [Q.-M. Cheng and H. Yang, Universal inequalities for eigenvalues of a clamped plate problem on a hyperbolic space, Proc. Amer. Math. Soc. 139(2) (2011) 461–471] concerning the behavior of higher-order eigenvalues on hyperbolic spaces, and answers a question raised in [A. Kristály, Fundamental tones of clamped plates in nonpositively curved spaces, Adv. Math. 367(39) (2020) 107113] on the validity of such sharp estimates in high-dimensional Cartan–Hadamard manifolds. As a byproduct of the general functional inequalities, various Rellich inequalities are established in the same geometric setting.

Inverse nodal problems for singular diffusion equation

In this study, some properties of the pencils of singular Sturm–Liouville operators are investigated. Firstly, the behaviors of eigenvalues and eigenfunctions is learned, then for each discontinuity point a solution of the inverse problem is given to determine the potential function and parameters , and with the help of a dense set of nodes. And finally, a constructive method is given for solving the given inverse problem.

Hydrodynamic stability of Poiseuille flow in a bidisperse porous medium with slip effect

This study focuses on examining the hydrodynamic stability of an incompressible fluid flowing through a bidisperse porous medium. Specifically, the impact of slip boundary conditions on instability is investigated. The study looks at a scenario in which the Darcy theory is used for micropores and the Brinkman theory is used for macropores. An incompressible fluid is located within an unbound channel with a constant pressure gradient along its length in the system under investigation. The fluid flows laminarly along the pressure gradient, resulting in a stable parabolic velocity distribution that does not alter over time. Based on our observations, it appears that increasing the values of the slip parameter, permeability ratio, porous parameter, interaction parameter and Darcy Reynolds number leads to an improvement in the stability of the system. The spectrum behavior of eigenvalues in the Orr–Sommerfeld problem for Poiseuille flow exhibits significant sensitivity and is influenced by multiple factors, encompassing both the mathematical attributes of the problem and the specific numerical techniques utilized for approximation.

INVERSE NODAL PROBLEMS FOR PENCILS OF SINGULAR STURM-LIOUVILLE OPERATORS

In this study, some properties of the pencils of singular Sturm-Liouville operators are investigated. Firstly, the behaviors of eigenvalues and eigenfunctions is learned, then for each discontinuity point

Behaviour of large eigenvalues for the asymmetric quantum Rabi model

We prove that the spectrum of the asymmetric quantum Rabi model consists of two eigenvalue sequences ( E m + ) m = 0 ∞ , ( E m − ) m = 0 ∞ , satisfying a two-term asymptotic formula with error estimate of the form O ( m − 1 / 4 ), when m tends to infinity.

On a family of finitely many point interaction Hamiltonians free of ultraviolet pathologies

Abstract We re-investigate the entire family of many center point interaction Hamiltonians. Under the assumption of exchange symmetry with respect to the point positions, we show that a large sub-family of point interaction Hamiltonian operators does not become either singular or trivial when the positions of two or more scattering centers tend to coincide. In this sense, they appear to be renormalized by default as opposed to the point interaction Hamiltonians usually considered in the literature. Functions in their domains satisfy regularized boundary conditions which turn out to be very similar to the ones proposed recently in many-body quantum mechanics to define three-particle system Hamiltonians with contact interactions bounded from below. In the two-center case, we study the behavior of the negative eigenvalues as a function of the center distance. The result is used to analyze a formal Born–Oppenheimer approximation of a three-particle system with two heavy bosons and one light particle. We demonstrate that this simplified model describes a stable system (no ‘fall to the center’ problem is present). Furthermore, in the unitary limit, the energy spectrum is characterized by an infinite sequence of negative energy eigenvalues accumulating at zero according to the geometrical Efimov law.

SPECTRAL PROPERTIES AND INVERSE NODAL PROBLEMS FOR SINGULAR DIFFUSION EQUATION

In this study, some properties for the pencils of singular Sturm-Liouville operators are investigated. Firstly, the behaviors of eigenvalues were learned, then the solutions of the inverse problem were given to determine the potential function and parameters of the boundary condition with the help of a dense set of nodal points and lastly we obtain a constructive solution to the inverse problems of this class.

Asymptotic behavior for eigenvalues and eigenfunctions associated to Stokes operator in the presence of a rapidly oscillating boundary

In this paper, we analyze rigorously the asymptotic behaviors for perturbation of eigenvalues and eigenfunctions associated to the Stokes eigenvalue problem in the presence of a rapidly oscillating boundary. The aim is to construct asymptotic approximations, as , of the eigenvalues and corresponding eigenfunctions for the case where the eigenvalue of the reference problem is simple or multiple. Taking advantage of small oscillations, we use the method of matching of asymptotic expansions to construct the associated leading terms. We believe that our results are ambitious tools for determining shape and/or size of the small perturbed part of the domain by taking eigenvalue measurements.

A note on the prediction error of principal component regression in high dimensions

We analyze the prediction error of principal component regression (PCR) and prove high probability bounds for the corresponding squared risk conditional on the design. Our first main result shows that PCR performs comparably to the oracle method obtained by replacing empirical principal components by their population counterparts, provided that an effective rank condition holds. On the other hand, if the latter condition is violated, then empirical eigenvalues start to have a significant upward bias, resulting in a self-induced regularization of PCR. Our approach relies on the behavior of empirical eigenvalues, empirical eigenvectors and the excess risk of principal component analysis in high-dimensional regimes.

Inverse nodal problems for Sturm–Liouville operators on quantum star graphs

In this work, we study the nodal inverse problem for Sturm–Liouville operators on quantum star graphs. We show that the zeros of eigenfunctions characterize the potential, and we present a simple algorithm that enables us to approximate it. The method does not rely on a priori information about the asymptotic behavior of eigenvalues. We manage to reduce the problem to a family of constant coefficient eigenvalue problems that can be solved explicitly and contain enough information to recover the potential. We also consider the multiplicity of eigenvalues and their vanishing properties, which only hold generically.

Sturm-Liouville Problem with Mixed Boundary Conditions for a Differential Equation with a Fractional Derivative and Its Application in Viscoelasticity Models

In this study, we obtained a system of eigenfunctions and eigenvalues for the mixed homogeneous Sturm-Liouville problem of a second-order differential equation containing a fractional derivative operator. The fractional differentiation operator was considered according to two definitions: Gerasimov-Caputo and Riemann-Liouville-Visualizations of the system of eigenfunctions, the biorthogonal system, and the distribution of eigenvalues on the real axis were presented. The numerical behavior of eigenvalues was studied depending on the order of the fractional derivative for both definitions of the fractional derivative.

Principal spectral theory for nonlocal systems and applications to stem cell regeneration models

We first study the eigenvalue problem for a system of coupled integral operators and investigate existence and monotonicity of the principal eigenvalue and asymptotic behavior of the principal eigenpair. As applications of the spectral theory, we investigate a single genotype stem cell regeneration model with epigenetic transitions, and multiple genotype stem cell regeneration models with epigenetic transitions, with or without gene mutations. These models give rise to three classes of nonlinear integro-differential evolution equations. We consider existence, uniqueness and multiplicity of positive steady states to these equations, as well as the long time behavior of time-dependent solutions. Various explicit formulae for threshold values for tissue development, degeneration and abnormal growth are obtained.

Robust PCA for high‐dimensional data based on characteristic transformation

SummaryIn this paper, we propose a novel robust principal component analysis (PCA) for high‐dimensional data in the presence of various heterogeneities, in particular strong tailing and outliers. A transformation motivated by the characteristic function is constructed to improve the robustness of the classical PCA. The suggested method has the distinct advantage of dealing with heavy‐tail‐distributed data, whose covariances may be non‐existent (positively infinite, for instance), in addition to the usual outliers. The proposed approach is also a case of kernel principal component analysis (KPCA) and employs the robust and non‐linear properties via a bounded and non‐linear kernel function. The merits of the new method are illustrated by some statistical properties, including the upper bound of the excess error and the behaviour of the large eigenvalues under a spiked covariance model. Additionally, using a variety of simulations, we demonstrate the benefits of our approach over the classical PCA. Finally, using data on protein expression in mice of various genotypes in a biological study, we apply the novel robust PCA to categorise the mice and find that our approach is more effective at identifying abnormal mice than the classical PCA.