Eigenvalues Λn Research Articles

The smallest eigenvalue of the Hankel matrices associated with a perturbed Jacobi weight

In this paper, we study the large N behavior of the smallest eigenvalue λN of the (N+1)×(N+1) Hankel matrix, HN=(μj+k)0≤j,k≤N, generated by the γ dependent Jacobi weight w(z,γ)=e−γzzα(1−z)β,z∈[0,1],γ∈R,α>−1,β>−1. Applying the arguments of Szegö, Widom and Wilf, we obtain the asymptotic representation of the orthonormal polynomials PN(z), z∈C﹨[0,1], with the weight w(z,γ)=e−γzzα(1−z)β. Using the polynomials PN(z), we obtain the theoretical expression of λN, for large N. We also display the smallest eigenvalue λN for sufficiently large N, computed numerically.

Smoothness and approximative properties of solutions of the singular nonlinear Sturm-Liouville equation

It is known that the eigenvalues λn(n = 1, 2, ...) numbered in decreasing order and taking the multiplicity of the self-adjoint Sturm-Liouville operator with a completely continuous inverse operator L^{−1} have the following property (*) λn → 0, when n → ∞, moreover, than the faster convergence to zero so the operator L^{−1} is best approximated by finite rank operators. The following question: - Is it possible for a given nonlinear operator to indicate a decreasing numerical sequence characterized by the property (*)? naturally arises for nonlinear operators. In this paper, we study the above question for the nonlinear Sturm-Liouville operator. To solve the above problem the theorem on the maximum regularity of the solutions of the nonlinear Sturm-Liouville equation with greatly growing and rapidly oscillating potential in the space L2(R) (R = (−∞, ∞)) is proved. Twosided estimates of the Kolmogorov widths of the sets associated with solutions of the nonlinear SturmLiouville equation are also obtained. As is known, the obtained estimates of Kolmogorov widths give the opportunity to choose approximation apparatus that guarantees the minimum possible error.

Sharp polynomial bounds for certain C0-groups generated by operators with non-basis family of eigenvectors

Sharp polynomial bounds for norms of C0-groups generated by operators with purely imaginary eigenvalues λn=iln⁡n, n∈N, and complete minimal non-basis family of eigenvectors, constructed recently by G. Sklyar and V. Marchenko in [17], are obtained. Besides, it is shown that these C0-groups do not have a maximal asymptotics. For the more general case of behaviour of the spectrum of operators we present the proof of one lemma concerning the behaviour of j-th differences of sequences of complex exponentials exp⁡(itf(n)), n∈N, that is used in [17] to obtain bounds from above for norms of corresponding C0-groups. Also the growth properties of the resolvent for generators of constructed C0-groups are discussed.

Bounds for the largest and the smallest Aα eigenvalues of a graph in terms of vertex degrees

Let G be a graph with adjacency matrix A(G) and with D(G) the diagonal matrix of its vertex degrees. Nikiforov defined the matrix Aα(G), with α∈[0,1], asAα(G)=αD(G)+(1−α)A(G). The largest and the smallest eigenvalues of Aα(G) are respectively denoted by ρ(G) and λn(G). In this paper, we present a tight upper bound for ρ(G) in terms of the vertex degrees of G for α≠1. For any graph G,ρ(G)≤maxu∼w⁡{α(du+dw)+α2(du+dw)2+4(1−2α)dudw2}. If G is connected, for α∈[0,1), the equality holds if and only if G is regular or bipartite semi-regular. As an application, we solve the problem raised by Nikiforov in [15]. For the smallest eigenvalue λn(G) of a bipartite graph G of order n with no isolated vertices, for α∈[0,1), thenλn(G)≥minu∼w⁡{α(du+dw)−α2(du+dw)2+4(1−2α)dudw2}. If G is connected and α≠1/2, the equality holds if and only if G is regular or semi-regular.

The convective eigenvalues of the one–dimensional p–Laplacian as p → 1

This paper studies the limit behavior as p→1 of the eigenvalue problem{−(|ux|p−2ux)x−c|ux|p−2ux=λ|u|p−2u,0<x<1,u(0)=u(1)=0. We point out that explicit expressions for both the eigenvalues λn and associated eigenfunctions are not available (see [16]). In spite of this hindrance, we obtain the precise values of the limits limp→1+⁡λn. In addition, a complete description of the limit profiles of the eigenfunctions is accomplished. Moreover, the formal limit problem as p→1 is also addressed. The results extend known features for the special case c=0 ([6], [28]).

On the simplicity of eigenvalues of two nonhomogeneous Euler-Bernoulli beams connected by a point mass

In this paper we consider a linear system modeling the vibrations of two nonhomogeneous Euler-Bernoulli beams connected by a point mass. This system is generated by the following equationsρ(x)ytt(t,x)+(σ(x)yxx(t,x))xx−(q(x)yx(t,x))x=0,t>0,x∈(−1,0)∪(0,1),Mytt(t,0)=(Ty(t,x))|x=0−−(Ty(t,x))|x=0+,t>0, with hinged boundary conditions at both ends, where M>0, ρ(x)>0,σ(x)>0, q(x)≥0 and Ty=(σ(x)yxx)x−q(x)yx for x∈(−1,0)∪(0,1). We prove that all the associated eigenvalues (λn)n≥1 are algebraically simple, furthermore the corresponding eigenfunctions (ϕn)n≥1 satisfy ϕn′(−1)Tϕn(−1)>0 and ϕn′(1)Tϕn(1)<0 for all n≥1. These results are essential for the proof of the exact boundary controllability problem related to this system by means of the non-harmonic Fourier series method.

A Pseudospectral solution of a bistable Fokker–Planck equation that models protein folding

We consider the one-dimensional bistable Fokker–Planck equation proposed by Polotto et al. (2018), with specific drift and diffusion coefficients so as to model protein folding. In this paper, a pseudospectral method is used to solve the Fokker–Planck equation in terms of the eigenvalues (λn) and eigenfunctions (ψn) of the Fokker–Planck operator. Nonclassical polynomials, constructed orthogonal with respect to the equilibrium distribution of the Fokker–Planck equation, are used as basis functions. The eigenvalues determined with the pseudospectral method are compared with the Wentzel–Kramers–Brillouin (WKB) and the SUperSYmmetric (SUSY) Wentzel–Kramers–Brillouin (SWKB) approximations. The eigenvalues calculated differ significantly from those reported by Polotto et al. A detailed study of the role of the lowest non-zero eigenvalue, λ1, to model the rate coefficient for the transition between the bistable states is provided.

Energy dependent potential problems for the one dimensional [formula omitted]-Laplacian operator

In this work we analyze a nonlinear eigenvalue problem for the p-Laplacian operator with zero Dirichlet boundary conditions. We assume that the problem has a potential which depends on the eigenvalue parameter, and we show that, for n big enough, there exists a real eigenvalue λn, and their corresponding eigenfunctions have exactly n nodal domains.We characterize the asymptotic behavior of these eigenvalues, obtaining two terms in the asymptotic expansion of λn in powers of n.Finally, we study the inverse nodal problem in the case of energy dependent potentials, showing that some subset of the zeros of the corresponding eigenfunctions is enough to determine the main term of the potential.

Spectral properties of a sequence of matrices connected to each other via Schur complement and arising in a compartmental model

Abstract We consider a sequence of real matrices An which is characterized by the rule that An−1 is the Schur complement in An of the (1,1) entry of An, namely −en, where en is a positive real number. This sequence is closely related to linear compartmental ordinary differential equations. We study the spectrum of An. In particular,we show that An has a unique positive eigenvalue λn and {λn} is a decreasing convergent sequence. We also study the stability of An for small n using the Routh-Hurwitz criterion.

Characterization of the potential smoothness of one-dimensional Dirac operator subject to general boundary conditions and its Riesz basis property

The one-dimensional Dirac operator with periodic potential V=(0P(x)Q(x)0), where P,Q∈L2([0,π]) subject to periodic, antiperiodic or a general strictly regular boundary condition (bc), has discrete spectrums. It is known that, for large enough |n| in the disk centered at n of radius 1/2, the operator has exactly two (periodic if n is even or antiperiodic if n is odd) eigenvalues λn+ and λn− (counted according to multiplicity) and one eigenvalue μnbc corresponding to the boundary condition (bc). We prove that the smoothness of the potential could be characterized by the decay rate of the sequence |δnbc|+|γn|, where δnbc=μnbc−λn+ and γn=λn+−λn−. Furthermore, it is shown that the Dirac operator with periodic or antiperiodic boundary condition has the Riesz basis property if and only if supγn≠0⁡|δnbc||γn| is finite.

Hill's spectral curves and the invariant measure of the periodic KdV equation

This paper analyses the periodic spectrum of Schrödinger's equation −f″+qf=λf when the potential is real, periodic, random and subject to the invariant measure νNβ of the periodic KdV equation. This νNβ is the modified canonical ensemble, as given by Bourgain (1994) [7], and νNβ satisfies a logarithmic Sobolev inequality. Associated concentration inequalities control the fluctuations of the periodic eigenvalues (λn). For β,N>0 small, there exists a set of positive νNβ measure such that (±2(λ2n+λ2n−1))n=0∞ gives a sampling sequence for Paley–Wiener space PW(π) and the reproducing kernels give a Riesz basis. Let (μj)j=1∞ be the tied spectrum; then (2μj−j) belongs to a Hilbert cube in ℓ2 and is distributed according to a measure that satisfies Gaussian concentration for Lipschitz functions. The sampling sequence (μj)j=1∞ arises from a divisor on the spectral curve, which is hyperelliptic of infinite genus. The linear statistics ∑jg(λ2j) with test function g∈PW(π) satisfy Gaussian concentration inequalities.

Spectrally accurate Stokes eigen-modes on isosceles triangles

We numerically study the Stokes eigen-modes in two dimensions on isosceles triangles with apex angle θ=π/3,π/2, and 2π/3 by using two spectral solvers, i.e., a Lagrangian collocation method with a weak formulation for the primitive variables and a Legendre–Galerkin method for the stream-function. We compute the first 6,400 Stokes eigen-modes. With 72 collocation points in each spatial dimension, the eigen-values λn for n ≤ 400 can be obtained with spectral accuracy and at least ten significant digits. We show the symmetry of the Stokes eigen-modes dictated by the geometry of the bounded flow domain. From the spectrally accurate data of the Stokes eigen-modes, the following features are observed. First, the n-dependence of the spectrum λn obeys the Weyl asymptotic formula in two dimensional space: λn=C1n+C2n+o(n). Second, for an isosceles triangle with legs of unit length, the θ-dependence of the spectrum λn can be accurately approximated by λn(θ)/λn(π/2) ≈ 1/(sin θ), as a consequence the volume-dependence of the coefficient C1 in the Weyl asymptotic formula. And third, a linear stream function-vorticity correlation is observed in the interior of the flow domain.

Signed Graphs with extremal least Laplacian eigenvalue

A signed graph is a pair Γ=(G,σ), where G=(V(G),E(G)) is a graph and σ:E(G)→{+,−} is the corresponding sign function. For a signed graph we consider the Laplacian matrix defined as L(Γ)=D(G)−A(Γ), where D(G) is the matrix of vertex degrees of G and A(Γ) is the (signed) adjacency matrix. It is well-known that Γ is balanced, that is, each cycle contains an even number of negative edges, if and only if the least Laplacian eigenvalue λn=0. Therefore, if Γ is not balanced, then λn>0. We show here that among unbalanced connected signed graphs of given order the least eigenvalue is minimal for an unbalanced triangle with a hanging path, while the least eigenvalue is maximal for the complete graph with the all-negative sign function.

The eigenvalue ratio for a class of densities

We investigate the nature of the eigenvalues for vibrating strings with the density functionρ=ρ(x,t)={−xif −1≤x≤0txif 0≤x≤1 where t>0. The nth eigenvalue λn(t) has a monotonicity property when t is changed. By means of Bessel functions, we obtain the limits of λn(t) as t→0 and as t→∞. We also prove that the minimum of the ratio λ2(t)/λ1(t) for t>0 occurs at t=1.

Spectral decay of time and frequency limiting operator

For fixed c, the Prolate Spheroidal Wave Functions (PSWFs) ψn,c form a basis with remarkable properties for the space of band-limited functions with bandwidth c. They have been largely studied and used after the seminal work of D. Slepian, H. Landau and H. Pollack. Many of the PSWFs applications rely heavily on the behavior and the decay rate of the eigenvalues (λn(c))n≥0 of the time and frequency limiting operator, which we denote by Qc. Hence, the issue of the accurate estimation of the spectrum of this operator has attracted a considerable interest, both in numerical and theoretical studies. In this work, we give an explicit integral approximation formula for these eigenvalues. This approximation holds true starting from the plunge region where the spectrum of Qc starts to have a fast decay. As a consequence of our explicit approximation formula, we give a precise description of the super-exponential decay rate of the λn(c). Also, we mention that the described approximation scheme provides us with fairly accurate approximations of the λn(c) with low computational load, even for very large values of the parameters c and n. Finally, we provide the reader with some numerical examples that illustrate the different results of this work.

Asymptotic Behavior of Eigenvalues of Hankel Operators

We consider compact Hankel operators realized in `(Z+) as infinite matrices Γ with matrix elements h(j + k). Roughly speaking, we show that if h(j) ∼ (b1 + (−1)b−1)j(log j)−α as j → ∞ for some α > 0, then the eigenvalues of Γ satisfy λn (Γ) ∼ c±n−α as n→∞. The asymptotic coefficients c± are explicitly expressed in terms of the asymptotic coefficients b1 and b−1. Similar results are obtained for Hankel operators Γ realized in L(R+) as integral operators with kernels h(t+s). In this case the asymptotics of eigenvalues λn (Γ) are determined by the behaviour of h(t) as t→ 0 and as t→∞.

Ramsey numbers, graph eigenvalues, and a conjecture of Cao and Yuan

Let N be a positive integer and R(N,N) denote the Ramsey number (see [15] or [11]) such that any graph with at least R(N,N) vertices contains a clique with N vertices or an independent set with N vertices. We show that any graph G with order at least R(N,N) must have its Nth largest eigenvalue λN(G)≥−1, and that any graph G with order at least R(N+1,N+1) must have its Nth smallest eigenvalue λN′(G)≤0, where the bounds −1 and 0 are best possible. This reveals some connection between graph spectra and Ramsey numbers, and enables us to give bounds of some eigenvalues for graphs with order greater than certain numbers. Moreover, it leads to a disproof of a conjecture on limit points of graph eigenvalues posted by Cao and Yuan [3] in 1995.Finally we post some problems for further research.

Constructing polynomials of minimal growth

In [2] a cyclic diagonal operator on the space of functions analytic on the unit disk with eigenvalues (λn) is shown to admit spectral synthesis if and only if for each j there is a sequence of polynomials (pn) such that limn→∞pn(λk) = δj,k and lim supn→∞ supk>j |pn(λk)|1/k ≤ 1. The author also shows, through contradiction, that certain classes of cyclic diagonal operators are synthetic. It is the intent of this paper to use the aforementioned equivalence to constructively produce examples of synthetic diagonal operators. In particular, this paper gives two different constructions for sequences of polynomials that satisfy the required properties for certain sequences to be the eigenvalues of a synthetic operator. Along the way we compare this to other results in the literature connecting polynomial behavior ([4] and [9]) and analytic continuation of Dirichlet series ([1]) to the spectral synthesis of diagonal operators.

ONE-DIMENSIONAL QUASI-RELATIVISTIC PARTICLE IN THE BOX

The eigenvalues and eigenfunctions of the one-dimensional quasi-relativistic Hamiltonian (-ℏ2c2d2/dx2 + m2c4)1/2 + V well (x) (the Klein–Gordon square-root operator with electrostatic potential) with the infinite square well potential V well (x) are studied. Eigenvalues represent energies of a "massive particle in the box" quasi-relativistic model. Approximations to eigenvalues λn are given, uniformly in n, ℏ, m, c and a, with error less than C1ℏca-1 exp (-C2ℏ-1mca)n-1. Here 2a is the width of the potential well. As a consequence, the spectrum is simple and the nth eigenvalue is equal to (nπ/2 - π/8)ℏc/a + O(1/n) as n → ∞. Non-relativistic, zero mass and semi-classical asymptotic expansions are included as special cases. In the final part, some L2 and L∞ properties of eigenfunctions are studied.

Trace formulae for differential pencils with spectral parameter dependent boundary conditions

For the eigenvalues λn of a differential operator, the series , generally speaking, diverges; however, it can be regularized by subtracting from λn the first terms of the asymptotic expansion, which interfere with the convergence of the series. The sum of such a regularized series is called the trace of Gelfand–Levitan type. A second‐order differential pencil on a finite interval with spectral parameter dependent boundary conditions is considered. We derive the regularized trace formulae of Gelfand–Levitan type for this operator. Copyright © 2013 John Wiley &amp; Sons, Ltd.