Nth Eigenvalue Research Articles

Discontinuity of the [formula omitted]th eigenvalue for a vibrating beam equation

The present paper is concerned with the eigenvalues of a problem describing a vibrating beam. The continuity and differentiability of the eigenvalues with respect to the parameters are well understood. Here the discontinuity of the nth eigenvalue of the problem is completely characterized. The asymptotic behavior of the nth eigenvalue at the discontinuous points and the monotonicity of the nth eigenvalue are investigated.

First-order nonlinear eigenvalue problems involving functions of a general oscillatory behavior

Eigenvalue problems arise in many areas of physics, from solving a classical electromagnetic problem to calculating the quantum bound states of the hydrogen atom. In textbooks, eigenvalue problems are defined for linear problems, particularly linear differential equations such as time-independent Schrödinger equations. Eigenfunctions of such problems exhibit several standard features independent of the form of the underlying equations. As discussed in Bender et al (2014 J. Phys. A: Math. Theor. 47 235204), separatrices of nonlinear differential equations share some of these features. In this sense, they can be considered eigenfunctions of nonlinear differential equations, and the quantized initial conditions that give rise to the separatrices can be interpreted as eigenvalues. We introduce a first-order nonlinear eigenvalue problem involving a general class of functions and obtain the large-eigenvalue limit by reducing it to a random walk problem on a half-line. The introduced general class of functions covers many special functions such as the Bessel and Airy functions, which are themselves solutions of second-order differential equations. For instance, in a special case involving the Bessel functions of the first kind, i.e. for y′(x) = J ν (xy), we show that the eigenvalues asymptotically grow as 241/42 n 1/4. We also introduce and discuss nonlinear eigenvalue problems involving the reciprocal gamma and the Riemann zeta functions, which are not solutions to simple differential equations. With the reciprocal gamma function, i.e. for y′(x) = 1/Γ(−xy), we show that the nth eigenvalue grows factorially fast as , where r k is the kth root of the digamma function.

Strong coupling asymptotics for δ-interactions supported by curves with cusps

Let Γ⊂R2 be a simple closed curve which is smooth except at the origin, at which it has a power cusp and coincides with the curve |x2|=x1p for some p>1. We study the eigenvalues of the Schrödinger operator Hα with the attractive δ-potential of strength α>0 supported by Γ, which is defined by its quadratic formH1(R2)∋u↦∬R2|∇u|2dx−α∫Γu2ds, where ds stands for the one-dimensional Hausdorff measure on Γ. It is shown that if n∈N is fixed and α is large, then the well-defined nth eigenvalue En(Hα) of Hα behaves asEn(Hα)=−α2+22p+2Enα6p+2+O(α6p+2−η), where the constants En>0 are the eigenvalues of an explicitly given one-dimensional Schrödinger operator determined by the cusp, and η>0. Both main and secondary terms in this asymptotic expansion are different from what was observed previously for the cases when Γ is smooth or piecewise smooth with non-zero angles.

Dependence of eigenvalues of Sturm–Liouville problems with eigenparameter dependent boundary conditions

This paper is concerned with regular Sturm–Liouville problems with eigenparameter dependent boundary conditions. We obtain that the eigenvalues are not only continuously but also smoothly dependent on the parameters of the problem, in particular, eigenparameter-dependent boundary condition matrix. Moreover the differential expression of the eigenvalues with respect to the data is given. The dependence of the nth eigenvalue as a function of the parameters of the problem is also investigated by Pru¨fer transformation and the inequalities of eigenvalues.

On comparing Hecke eigenvalues of cusp forms

Let f(z) be a primitive holomorphic cusp form of even integral weight k for the full modular group. Denote its nth Hecke eigenvalue or normalized Fourier coefficient by \(\lambda_{f}(n)\). Let g(z) be another distinct primitive holomorphic cusp form of even integral weight \(\ell\) for the full modular group. In this paper, we establish that the set $$\{p| \lambda_f(p^j)<\lambda_g(p^j)\} \quad \mbox{with } 1\leq j \leq 8$$ has analytic density at least \(\frac{1}{16[\frac{j+1}{2}]^{2}}\), and the set $$\{p| \lambda_f(p^j)^2<\lambda_g(p^j)^2\} \quad \mbox{with } 1\leq j \leq 4$$ has analytic density at least \(\frac{1}{4j(j+1)^{2}}\).

Spectral Properties of Sturm–Liouville Problems with Strongly Singular Potentials

This paper studies a class of singular differential equations $$\begin{aligned} -\left( \frac{\mathrm {d}}{{\mathrm {d}}x}-\frac{k}{x^{l}}-v\right) \left( \frac{{\mathrm {d}}}{{\mathrm {d}}x}+\frac{k}{x^{l}}+v\right) y=\lambda y \text { on }J=(0,1) , \end{aligned}$$ where $$k\ge \frac{1}{2},1\le l<2$$ and $$v\in L^{1}(J, {\mathbb {R}} )$$ which is bounded below. Using the prufer transformation, we get the oscillation property of the eigenfunctions. In particular, the location of eigenvalues are also described. Furthermore, we establish the continuous dependence of nth eigenvalue on the boundary condition.

Eigenvalues of Robin Laplacians in infinite sectors

AbstractFor , let denote the infinite planar sector of opening 2α, urn:x-wiley:0025584X:media:mana201600314:mana201600314-math-0003and be the Laplacian in , , with the Robin boundary condition , where stands for the outer normal derivative and . The essential spectrum of does not depend on the angle α and equals , and the discrete spectrum is non‐empty if and only if . In this case we show that the discrete spectrum is always finite and that each individual eigenvalue is a continous strictly increasing function of the angle α. In particular, there is just one discrete eigenvalue for . As α approaches 0, the number of discrete eigenvalues becomes arbitrary large and is minorated by with a suitable , and the nth eigenvalue of behaves as urn:x-wiley:0025584X:media:mana201600314:mana201600314-math-0018and admits a full asymptotic expansion in powers of α2. The eigenfunctions are exponentially localized near the origin. The results are also applied to δ‐interactions on star graphs.

Geometric aspects of self-adjoint Sturm–Liouville problems

In this paper we use U(2), the group of 2 × 2 unitary matrices, to parametrize the space of all self-adjoint boundary conditions for a fixed Sturm–Liouville equation on the interval [0, 1]. The adjoint action of U(2) on itself naturally leads to a refined classification of self-adjoint boundary conditions – each adjoint orbit is a subclass of these boundary conditions. We give explicit parametrizations of those adjoint orbits of principal type, i.e. orbits diffeomorphic to the 2-sphere S2, and investigate the behaviour of the nth eigenvalue λnas a function on such orbits.

Extremal norms of the potentials recovered from inverse Dirichlet problems

Consider the Sturm–Liouville eigenvalue problem , where , and its spectrum is denoted by . For a real number λ, define and . We will set up a formula for explicitly in terms of λ and specify where the infimum can be attained. As an application, we will give the extremal values of the nth eigenvalue of the Dirichlet problem for potentials on a sphere , . The proofs are based on a new Lyapunov-type inequality for Sturm–Liouville equations with potentials.

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.

Component-based reduced basis for parametrized symmetric eigenproblems

A component-based approach is introduced for fast and flexible solution of parameter-dependent symmetric eigenproblems. Considering a generalized eigenproblem with symmetric stiffness and mass operators, we start by introducing a “ σ-shifted” eigenproblem where the left hand side operator corresponds to an equilibrium between the stiffness operator and a weighted mass operator, with weight-parameter σ>0. Assuming that σ=λ n >0, the nth real positive eigenvalue of the original eigenproblem, then the shifted eigenproblem reduces to the solution of a homogeneous linear problem. In this context, we can apply the static condensation reduced basis element (SCRBE) method, a domain synthesis approach with reduced basis (RB) approximation at the intradomain level to populate a Schur complement at the interdomain level. In the Offline stage, for a library of archetype subdomains we train RB spaces for a family of linear problems; these linear problems correspond to various equilibriums between the stiffness operator and the weighted mass operator. In the Online stage we assemble instantiated subdomains and perform static condensation to obtain the “ σ-shifted” eigenproblem for the full system. We then perform a direct search to find the values of σ that yield singular systems, corresponding to the eigenvalues of the original eigenproblem. We provide eigenvalue a posteriori error estimators and we present various numerical results to demonstrate the accuracy, flexibility and computational efficiency of our approach. We are able to obtain large speed and memory improvements compared to a classical Finite Element Method (FEM), making our method very suitable for large models commonly considered in an engineering context.

Nonlinear eigenvalue problems

This paper presents an asymptotic study of the differential equation y′(x) = cos [πxy(x)] subject to the initial condition y(0) = a. While this differential equation is nonlinear, the solutions to the initial-value problem bear a striking resemblance to solutions to the linear time-independent Schrödinger eigenvalue problem. As x increases from 0, y(x) oscillates and thus resembles a quantum wave function in a classically allowed region. At a critical value x = xcrit, where xcrit depends on a, the solution y(x) undergoes a transition; the oscillations abruptly cease and y(x) decays to 0 monotonically as x → ∞. This transition resembles the transition in a wave function at a turning point as one enters the classically forbidden region. Furthermore, the initial condition a falls into discrete classes; in the nth class of initial conditions an − 1 < a < an (n = 1, 2, 3, …), y(x) exhibits exactly n maxima in the oscillatory region. The boundaries an of these classes are the analogues of quantum-mechanical eigenvalues. An asymptotic calculation of an for large n is analogous to a high-energy semiclassical (WKB) calculation of eigenvalues in quantum mechanics. The principal result of this paper is that as n → ∞, , where A = 25/6. Numerical analysis reveals that the first Painlevé transcendent has an eigenvalue structure that is quite similar to that of the equation y′(x) = cos [πxy(x)] and that the nth eigenvalue grows with n like a constant times n3/5 as n → ∞. Finally, it is noted that the constant A is numerically very close to the lower bound on the power-series constant P in the theory of complex variables, which is associated with the asymptotic behavior of zeros of partial sums of Taylor series.

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.

Inequalities among eigenvalues of Sturm–Liouville problems with distribution potentials

This paper deals with the eigenvalue problems for the Sturm–Liouville operators generated by the differential expression L(y)=−(p(x)y′)′+q(x)y with singular coefficients q(x) in the sense of distributions. We obtain the inequalities among the eigenvalues corresponding to different self-adjoint boundary conditions. The continuity region, the differentiability and the monotonicity of the nth eigenvalue corresponding to the separated boundary conditions are given. Oscillation properties of the eigenfunctions of all the coupled Sturm–Liouville problems are characterized. The main results of this paper can also be applied to solve a class of transmission problems.

Dependence of eigenvalues of a class of fourth-order Sturm–Liouville problems on the boundary

In this paper, we consider the dependence of eigenvalues of a class of fourth-order Sturm–Liouville problems on the boundary. We show that the eigenvalues depend not only continuously but smoothly on boundary points, and that the derivative of the nth eigenvalue as a function of an endpoint satisfies a first order differential equation. In addition, we prove that as the length of the interval shrinks to zero all higher fourth-order Dirichlet eigenvalues march off to plus infinity, this is also true for the first (i.e., lowest) eigenvalue.

On the evaluation of prolate spheroidal wave functions and associated quadrature rules

As demonstrated by Slepian et al. in a sequence of classical papers (see Slepian (1983) [33], Slepian and Pollak (1961) [34], Landau and Pollak (1961) [18], Slepian and Pollak (1964) [35], Slepian (1965) [36]), prolate spheroidal wave functions (PSWFs) provide a natural and efficient tool for computing with bandlimited functions defined on an interval. Recently, PSWFs have been becoming increasingly popular in various areas in which such functions occur – this includes physics (e.g. wave phenomena, fluid dynamics), engineering (signal processing, filter design), etc.To use PSWFs as a computational tool, one needs fast and accurate numerical algorithms for the evaluation of PSWFs and related quantities, as well as for the construction of corresponding quadrature rules, interpolation formulas, etc. During the last 15 years, substantial progress has been made in the design of such algorithms – see, for example, Xiao et al. (2001) [40] (see also Bowkamp (1947) [6], Slepian and Pollak (1961) [34], Landau and Pollak (1961) [18], Slepian and Pollak (1964) [35] for some classical results).The complexity of many of the existing algorithms, however, is at least quadratic in the band limit c. For example, the evaluation of the nth eigenvalue of the prolate integral operator requires O(c2+n2) operations (see e.g. Xiao et al. (2001) [40]); the construction of accurate quadrature rules for the integration (and associated interpolation) of bandlimited functions with band limit c requires O(c3) operations (see e.g. Cheng et al. (1999) [8]). Therefore, while the existing algorithms are satisfactory for moderate values of c (e.g. c⩽103), they tend to be relatively slow when c is large (e.g. c⩾104).In this paper, we describe several numerical algorithms for the evaluation of PSWFs and related quantities, and design a class of PSWF-based quadratures for the integration of bandlimited functions. While the analysis is somewhat involved and will be published separately (currently, it can be found in Osipov and Rokhlin (2012) [27]), the resulting numerical algorithms are quite simple and efficient in practice. For example, the evaluation of the nth eigenvalue of the prolate integral operator requires O(n+c⋅logc) operations; the construction of accurate quadrature rules for the integration (and associated interpolation) of bandlimited functions with band limit c requires O(c) operations. All algorithms described in this paper produce results essentially to machine precision. Our results are illustrated via several numerical experiments.

Optimization of spectral functions of Dirichlet–Laplacian eigenvalues

We consider the shape optimization of spectral functions of Dirichlet–Laplacian eigenvalues over the set of star-shaped, symmetric, bounded planar regions with smooth boundary. The regions are represented using Fourier-cosine coefficients and the optimization problem is solved numerically using a quasi-Newton method. The method is applied to maximizing two particular nonsmooth spectral functions: the ratio of the nth to first eigenvalues and the ratio of the nth eigenvalue gap to first eigenvalue, both of which are generalizations of the Payne–Pólya–Weinberger ratio. The optimal values and attaining regions for n ⩽ 13 are presented and interpreted as a study of the range of the Dirichlet–Laplacian eigenvalues. For both spectral functions and each n, the optimal attaining region has multiplicity two nth eigenvalue.

On the Hersch-Payne-Schiffer inequalities for Steklov eigenvalues

We prove that the Hersch-Payne-Schiffer isoperimetric inequality for the nth nonzero Steklov eigenvalue of a bounded simply connected planar domain is sharp for all n ⩾ 1. The equality is attained in the limit by a sequence of simply connected domains degenerating into a disjoint union of n identical disks. Similar results are obtained for the product of two consecutive Steklov eigenvalues. We also give a new proof of the Hersch-Payne-Schiffer inequality for n = 2 and show that it is strict in this case.

Nodal solutions of second order nonlinear boundary value problems

We study the nonlinear boundary value problem (BVP) consisting of the equation −(p(t)y′)′ + q(t)y = w(t)f(y) on [a, b] and a general separated boundary condition (BC). By comparing it with a linear Sturm--Liouville problem (SLP) we obtain conditions for the existence and nonexistence of nodal solutions of this problem. More specifically, let λn, n = 0, 1, 2, . . ., be the nth eigenvalue of the corresponding linear SLP. Then the BVP has a pair of solutions with exactly n zeros in (a, b) if λn is in the interior of the range of f(y)/y; and does not have any solution with exactly n zeros in (a, b) if λn is outside this range. These conditions become necessary and sufficient when f(y)/y is monotone on (−∞, 0) and on (0, ∞). We also discuss the changes of the number of different types of nodal solutions as the equation or the BC changes. Our results are obtained without assuming the global existence and uniqueness of solutions of the corresponding initial value problems.

The index problem for eigenvalues for coupled boundary conditions and Fulton’s conjecture

This paper deals with regular self-adjoint Sturm–Liouville problems with coupled boundary conditions. The spectrum of the problems can be either bounded from below, or unbounded from both below and above. We consider the following question: given an eigenvalue, how to determine its index or indices efficiently? Using results on the level surfaces of the nth eigenvalue and certain inequalities among eigenvalues, the determination is converted into that of the index of the same eigenvalue for an appropriate separated boundary condition and hence can be achieved in terms of the Prufer angle. As an application of this complete solution of the index problem, we show that Fulton’s conjecture about such an index is true.