Negative Eigenvalues Research Articles

Comment on: “Analytical solutions to the Schrödinger equation for a generalized Cornell potential and its applications to diatomic molecules and heavy mesons”

We analyze a recent calculation of the energy levels of some quantum-mechanical models derived from the so-called generalized Cornell potential. We argue that the authors obtained essentially the same spectrum for potentials with considerably different spectra. We show that the authors obtained negative eigenvalues in the case of a positive-definite potential. The reason is due to the approximation used in the solution of the Schrödinger equation. We point out that the omission of the sum over the rotational degrees of freedom in the canonical partition function leads to incomplete expressions for the thermodynamic functions.

Fast convergence of trust-regions for non-isolated minima via analysis of CG on indefinite matrices

AbstractTrust-region methods (TR) can converge quadratically to minima where the Hessian is positive definite. However, if the minima are not isolated, then the Hessian there cannot be positive definite. The weaker Polyak–Łojasiewicz (PŁ) condition is compatible with non-isolated minima, and it is enough for many algorithms to preserve good local behavior. Yet, TR with an exact subproblem solver lacks even basic features such as a capture theorem under PŁ. In practice, a popular inexact subproblem solver is the truncated conjugate gradient method (tCG). Empirically, TR-tCG exhibits superlinear convergence under PŁ. We confirm this theoretically. The main mathematical obstacle is that, under PŁ, at points arbitrarily close to minima, the Hessian has vanishingly small, possibly negative eigenvalues. Thus, tCG is applied to ill-conditioned, indefinite systems. Yet, the core theory underlying tCG is that of CG, which assumes a positive definite operator. Accordingly, we develop new tools to analyze the dynamics of CG in the presence of small eigenvalues of any sign, for the regime of interest to TR-tCG.

High energy scattering in the Unitary Toy Model

We continue exploring the Unitary Toy Model (UTM) as a playground for high energy collisions in QCD. Our new approach is based on the diagonalization of the evolution Hamiltonian. Part of the spectrum can be identified with intercepts of dressed Pomerons. Analogously to QCD, a multi-Pomeron expansion of the S-matrix is badly divergent asymptotic series. Yet we succeeded to establish resummation procedures resulting in a well behaved S-matrix. In addition the Hamiltonian possesses negative eigenvalues, which dominate the approach of the S-matrix to saturation. We are hopeful that important lessons about BFKL-based Pomeron calculus could be taken from the toy world to real QCD.

Improved Branch and Bound algorithm and an interpolation-based search algorithm for quadratic minimization with one negative eigenvalue

Improved Branch and Bound algorithm and an interpolation-based search algorithm for quadratic minimization with one negative eigenvalue

Two-sided Lieb–Thirring bounds

We prove upper and lower bounds for the number of eigenvalues of semi-bounded Schrödinger operators in all spatial dimensions. As a corollary, we obtain two-sided estimates for the sum of the negative eigenvalues of atomic Hamiltonians with Kato potentials. Instead of being in terms of the potential itself, as in the usual Lieb–Thirring result, the bounds are in terms of the landscape function, also known as the torsion function, which is a solution of (−Δ + V + M)uM = 1 in Rd; here M∈R is chosen so that the operator is positive. We further prove that the infimum of (uM−1−M) is a lower bound for the ground state energy E0 and derive a simple iteration scheme converging to E0.

Persistence and disappearance of negative eigenvalues in dimension two

We compute asymptotics of eigenvalues approaching the bottom of the continuous spectrum, and associated resonances, for Schrödinger operators in dimension two. We distinguish persistent eigenvalues, which have associated resonances, from disappearing ones, which do not. We illustrate the significance of this distinction by computing corresponding scattering phase asymptotics and numerical Breit–Wigner peaks. We prove all of our results for circular wells, and extend some of them to more general problems using recent resolvent techniques.

On spectral simplicity of the Hodge Laplacian and curl operator along paths of metrics

We prove that the curl operator on closed oriented 3 3 -manifolds, i.e., the square root of the Hodge Laplacian on its coexact spectrum, generically has 1 1 -dimensional eigenspaces, even along 1 1 -parameter families of C k \mathcal {C}^k Riemannian metrics, where k ≥ 2 k\geq 2 . We show further that the Hodge Laplacian in dimension 3 3 has two possible sources for nonsimple eigenspaces along generic 1 1 -parameter families of Riemannian metrics: either eigenvalues coming from positive and from negative eigenvalues of the curl operator cross, or an exact and a coexact eigenvalue cross. We provide examples for both of these phenomena. In order to prove our results, we generalize a method of Teytel [Comm. Pure Appl. Math. 52 (1999), pp. 917–934], allowing us to compute the meagre codimension of the set of Riemannian metrics for which the curl operator and the Hodge Laplacian have certain eigenvalue multiplicities. A consequence of our results is that while the simplicity of the spectrum of the Hodge Laplacian in dimension 3 3 is a meagre codimension 1 1 property with respect to the C k \mathcal {C}^k topology as proven by Enciso and Peralta-Salas in [Trans. Amer. Math. Soc. 364 (2012), pp. 4207–4224], it is not a meagre codimension 2 2 property.

Self-similar blow-up solutions in the generalised Korteweg-de Vries equation: spectral analysis, normal form and asymptotics

In the present work we revisit the problem of the generalised Korteweg–de Vries equation parametrically, as a function of the relevant nonlinearity exponent, to examine the emergence of blow-up solutions, as traveling waveforms lose their stability past a critical point of the relevant parameter p, here at p = 5. We provide a normal form of the associated collapse dynamics, and illustrate how this captures the collapsing branch bifurcating from the unstable traveling branch. We also systematically characterise the linearisation spectrum of not only the traveling states, but importantly of the emergent collapsing waveforms in the so-called co-exploding frame where these waveforms are identified as stationary states. This spectrum, in addition to two positive real eigenvalues which are shown to be associated with the symmetries of translation and scaling invariance of the original (non-exploding) frame features complex patterns of negative eigenvalues that we also fully characterise. We show that the phenomenology of the latter is significantly affected by the boundary conditions and is far more complicated than in the corresponding symmetric Laplacian case of the nonlinear Schrödinger problem that has recently been explored. In addition, we explore the dynamics of the unstable solitary waves for p > 5 in the co-exploding frame.

Completely degenerate equilibria of the Kuramoto model on networks

Kuramoto Networks contain non-hyperbolic equilibria whose stability is sometimes difficult to determine. We consider the extreme case in which all Jacobian eigenvalues are zero. In this case linearizing the system at the equilibrium leads to a Jacobian matrix which is zero in every entry. We call these equilibria completely degenerate. We prove that they exist for certain intrinsic frequencies if and only if the underlying graph is bipartite, and that they do not exist for generic intrinsic frequencies. In the case of zero intrinsic frequencies, we prove that they exist if and only if the graph has an Euler circuit such that the number of steps between any two visits at the same vertex is a multiple of 4. The simplest example is the cycle graph with 4 vertices. We prove that graphs with this property exist for every number of vertices N⩾6 and that they become asymptotically rare for N large. Regarding stability, we prove that for any choice of intrinsic frequencies, any coupling strength and any graph with at least one edge, completely degenerate equilibria are not Lyapunov stable. As a corollary, we obtain that stable equilibria in Kuramoto Networks must have at least one strictly negative eigenvalue.

Spectral structure of the Neumann–Poincaré operator on axially symmetric functions

We consider the Neumann–Poincaré operator on a three-dimensional axially symmetric domain which is generated by rotating a planar domain around an axis which does not intersect the planar domain. We investigate its spectral structure when it is restricted to axially symmetric functions. If the boundary of the domain is smooth, we show that there are infinitely many axially symmetric eigenfunctions and derive Weyl-type asymptotics of the corresponding eigenvalues. We also derive the leading order terms of the asymptotic limits of positive and negative eigenvalues. The coefficients of the leading order terms are related to the convexity and concavity of the domain. If the boundary of the domain is less regular, we derive decay estimates of the eigenvalues. The decay rate depends on the regularity of the boundary. We also consider the domains with corners and prove that the essential spectrum of the Neumann–Poincaré operator on the axially symmetric three-dimensional domain is non-trivial and contains that of the planar domain.

Modelling the role of tourism in the spread of HIV: A case study from Malaysia

This study aimed to assess the role of tourism in the spread of human immunodeficiency virus (HIV) using Malaysian epidemiological data on HIV and acquired immunodeficiency syndrome (AIDS) incidence from 1986 to 2004. A population-level mathematical model was developed with the following compartments: the population susceptible to HIV infection, the clinically confirmed HIV-positive population, the population diagnosed with AIDS, and the tourist population. Additionally, newborns infected with HIV were considered. Sensitivity analyses and variations in fixed parameter values were used to explore the effect of changes to various parameter values on HIV incidence in the model. It was determined that variations in the rate of HIV-positive inbound tourist entries and the rate of foreign tourist exits (i.e., the duration of time tourists spent in Malaysia) significantly impacted the predicted incidence of HIV and AIDS in Malaysia. The model's fit to observed HIV and AIDS incidence was evaluated, resulting in adjusted R2 values of 53.3% and 53.2% for HIV and AIDS, respectively. Furthermore, the reproduction number (R0) was also calculated to quantify the stability of the HIV endemicity in Malaysia. The findings suggest that a steady-state level of HIV in Malaysia is achievable based on the low value of R0 = 0.0136, and the disease-free equilibrium was stable from the negative eigenvalues obtained, which is encouraging from a public health perspective. The Partial Rank Correlation Coefficient (PRCC) values between the proportion of newborns born HIV-positive, the rate of Malaysian tourist entries returning home after contracting HIV, and the rate of foreign tourist exits have a significant impact on the R0. The methods provide a framework for epidemiological modelling of HIV spread through transient population groups. The model results suggest that the role of tourism should not be overlooked within the set of available measures to mitigate the spread of HIV.

Cwikel–Lieb–Rozenblum type inequalities for Hardy–Schrödinger operator

We prove a Cwikel–Lieb–Rozenblum type inequality for the number of negative eigenvalues of the Hardy–Schrödinger operator −Δ−(d−2)2/(4|x|2)−W(x) on L2(Rd). The bound is given in terms of a weighted Ld/2-norm of W which is sharp in both large and small coupling regimes. We also obtain a similar bound for the fractional Laplacian.

Spurious negative eigenvalues of numerical variance-covariance matrices in many-body systems correlate with the existence of frozen degrees of freedom

The principal component analysis (PCA) is widely used to reduce the dimensionality of a dataset to its essential components. To perform PCA, the covariance matrix is constructed and its eigenvalues and eigenvectors are computed. In practical numerical applications, the tail of the sorted eigenvalues is sometimes found to contain negative eigenvalues, which are prohibited mathematically and are a pure consequence of finite-accuracy numerics. The present study suggests that in the case of a many-body dynamical system, the spurious negative eigenvalues of the covariance matrix may in fact be related to the frozen degrees of freedom in the system. Here, we outline the mathematical connection between the eigenvalues of the covariance matrix and the frozen degrees of freedom and validate the connection through two case studies: a model system of coupled harmonic oscillators and a molecular dynamics simulation of a small protein in solution.

Stability analysis of Reiner–Philippoff nanofluid flow due to sinusoidal waves past a nonuniform channel with Brownian and thermophoretic diffusions

The need for understanding peristaltic flow (PF) is growing frequently because of its applicability in several scientific and technical domains, particularly in biological and medical field. The primary objective of this analysis is to examine the PF in tapering network considering Reiner–Philippoff nanofluid (RPF-NF) under the influence of pseudoplastic fluid (PF) and dilatant fluid (DF) performance. In order to examine the heat and mass transportation inquiry, Buongiorno’s nanofluid model (BNFM) is taken into consideration. The suitable alterations are taken to transform the channel from static reference to stirring frame and, then used the dimensionless variables to transform the moving frame to dimensionless form. Since the Reynolds number (RN) is small and the wavelength is long, the underlying equations are simplified. The statistical consequences are gained through bvp4c technique. The motivation of the physical features on the liquefied crescendos is particularized using graphs as well as tables. For the immovability exploration, eigenvalues are figured. The lowest positive eigenvalues suggest the reliable resolutions; however, the negative eigenvalues denote the unreliable solution. It has been shown that altering the RPF parameter causes the velocity of fluid to switch from a dilatant liquid to a Newtonian fluid and from Newtonian to Pseudoplastic. The results show that the temperature curves ascend with increasing Brownian motion and thermophoretic factors and decrease with increasing Prandtl number. Additionally, a brief mathematical and graphical investigation of the effects of each key parameter on the flow characteristics is conducted.

Fundamental Limits in Measuring the Anisotropic Rotational Diffusion of Single Molecules.

Many biophysical techniques, such as single-molecule fluorescence correlation spectroscopy, Förster resonance energy transfer, and fluorescence anisotropy, measure the translation and rotation of biomolecules to quantify molecular processes at the nanoscale. These methods often simplify data analysis by assuming isotropic rotational diffusion, e.g., that molecules wobble within a circular cone. This simplification ignores the anisotropy present in many biological contexts that may cause molecules to exhibit different degrees of diffusion in different directions. Here, we loosen this assumption and establish a theoretical framework for describing and measuring anisotropic rotational diffusion using fluorescence imaging. We show that anisotropic wobble is directly quantified by the eigenvalues of a 3-by-3 positive-semidefinite Hermitian matrix M consisting of the second-order moments of a molecule's transition dipole μ. This formalism enables us to model the influence of unavoidable shot noise using a Hermitian perturbation matrix E; the eigenvalues of E directly bound errors in measurements of wobble via Weyl's inequality. Quantifying various perturbations E reveals that anisotropic wobble measurements are generally more sensitive to errors compared to quantifying isotropic wobble. Moreover, severe shot noise can induce negative eigenvalues in estimates of M, thereby causing the anisotropic wobble measurement to fail. Our analysis, using Fisher information, shows that techniques with worse orientation measurement sensitivity experience stronger perturbations E and require larger signal to background ratios to measure anisotropic rotational diffusion accurately. Our work provides deep insights for improving the state of the art in imaging the orientations and anisotropic rotational diffusion of single molecules.

Bifurcations of a Laminated Circular Cylindrical Shell

In this paper, the stability and local bifurcations of a composite laminated circular cylindrical shell with radially pre-stretched membranes are explored by using analytical and numerical methods. On the basis of the four-dimensional averaged equation in the case of 1:1 internal resonance, three types of critical points are studied in detail. They are characterized as (1) one pair of purely imaginary eigenvalues and two negative eigenvalues; (2) a simple zero and one pair of purely imaginary eigenvalues; (3) two pairs of purely imaginary eigenvalues in nonresonant case. With the aid of normal form theory and Maple software, the steady-state solutions and the stability regions of the initial equilibrium solutions are obtained. The explicit expressions for the critical bifurcation curves leading to static bifurcation and Hopf bifurcation are also presented. The presence of Hopf bifurcation indicates that the circular cylindrical shell will flutter. The results contribute to the design of reasonable structure parameters to avoid flutter. Finally, numerical simulations are also presented to demonstrate the good agreement with the analytical predictions.

Exploring Tuberculosis: A Theoretical Framework for Infection Regulation and Eradication

In this paper we explored the study of tuberculosis, throughout human history, tuberculosis (TB) has been a persistent challenge due to its severe consequences for society. It is an infectious disease transmitted by Mycobacterium tuberculosis (MT),more than 150 million years ago, it was thought to have originated from the genus Mycobacterium. Our theoretical framework presents methods for regulating and eradicating tuberculosis infections. This has led to the compartmentalization of the model population and the analytical solution of the ensuing model equations. To validate the outcomes of the theoretical approach, a numerical simulation has been deployed. This model carried out six compartment stage, susceptible, latent for a short time period and latent for long time period, infected in hospital, random infected at public place and the recovered class. In this model we examined Disease free equilibrium points (DFEP), Basic reproduction number R_0. Positive boundaries solution is used to describe the infection at particular stage. If R_0<1, the DFE is locally asymptotically stable, using Jacobian matrix we explored the negative Eigen values. Lyapunov function is used to examined the global stability of endemic equilibrium. We used Carrying out a simulation using random data in a selected region,using random values in MATLAB, to simulate the result of the model.

Triangle-free signed graphs with small negative inertia index

Let Γ=(G,σ) be a signed graph with underlying graph G and sign function σ:E(G)→{+,−}. The number of negative eigenvalues (including multiplicities) of Γ, denoted by n(Γ), is called the negative inertia index of Γ. In this paper, we characterize the signed graphs Γ with n(Γ)=1, and the triangle-free signed graphs Γ with n(Γ)=2, respectively.

Study of some graph theoretical parameters for the structures of anticancer drugs

Eigenvalues have great importance in the field of mathematics, and their relevance extends beyond this area to include several other disciplines such as economics, chemistry, and numerous fields. According to our study, eigenvalues are utilized in chemistry to express a chemical compound’s numerous physical properties as well as its energy form. It is important to get a comprehensive understanding of the interrelationship underlying mathematics and chemistry. The anti-bonding phase is correlated with positive eigenvalues, whereas the bonding level is connected with negative eigenvalues. Additionally, the non-bonded level corresponds to eigenvalues of zero. This study focuses on the analysis of various structures of anticancer drugs, specifically examining their characteristic polynomials, eigenvalues of the adjacency matrix, matching number and nullity. Consequently, the selected structures of the aforementioned anticancer drugs exhibit stability since they are composed of closed-shell molecules, characterized by a nullity value of zero.

Semiclassical estimates for measure potentials on the real line

We prove an explicit weighted estimate for the semiclassical Schrödinger operator P = - h^{2} \partial^{2}_{x} + V(x;h) on L^{2}(\R) , with V(x;h) a finite signed measure, and where h >0 is the semiclassical parameter. The proof is a one-dimensional instance of the spherical energy method, which has been used to prove Carleman estimates in higher dimensions and in more complicated geometries. The novelty of our result is that the potential need not be absolutely continuous with respect to Lebesgue measure. Two consequences of the weighted estimate are the absence of positive eigenvalues for P , and a limiting absorption resolvent estimate with sharp h -dependence. The resolvent estimate implies exponential time-decay of the local energy for solutions to the corresponding wave equation with a compactly supported measure potential, provided there are no negative eigenvalues and no zero resonance, and provided the initial data have compact support.