Infinite Number Of Eigenvalues Research Articles

Infinite number of eigenvalues of $$2{\times}2$$ operator matrices: Asymptotic discrete spectrum

We study an unbounded $$2{\times}2$$ operator matrix $$ \mathcal{A} $$ in the direct product of two Hilbert spaces. We obtain asymptotic formulas for the number of eigenvalues of $$ \mathcal{A} $$ . We consider a $$2{\times}2$$ operator matrix $$ \mathcal{A} _\mu$$ , where $$\mu>0$$ is the coupling constant, associated with the Hamiltonian of a system with at most three particles on the lattice $$ \mathbb{Z} ^3$$ . We find the critical value $$\mu_0$$ of the coupling constant $$\mu$$ for which $$ \mathcal{A} _{\mu_0}$$ has an infinite number of eigenvalues. These eigenvalues accumulate at the lower and upper bounds of the essential spectrum. We obtain an asymptotic formula for the number of such eigenvalues in both the left and right parts of the essential spectrum.

Spectral Asymptotics of the Dirichlet Laplacian on a Generalized Parabolic Layer

We perform quantitative spectral analysis of the self-adjoint Dirichlet Laplacian $${\mathsf {H}}$$ on an unbounded, radially symmetric (generalized) parabolic layer $${\mathcal {P}}\subset {\mathbb {R}}^3$$. It was known before that $${\mathsf {H}}$$ has an infinite number of eigenvalues below the threshold of its essential spectrum. In the present paper, we find the discrete spectrum asymptotics for $${\mathsf {H}}$$ by means of a consecutive reduction to the analogous asymptotic problem for an effective one-dimensional Schrodinger operator on the half-line with the potential the behaviour of which far away from the origin is determined by the geometry of the layer $${\mathcal {P}}$$ at infinity.

Eigenfunctions for quasi-Laplacian

To study the regularity of heat flow, Lin and Wang (1999) introduced the quasi-harmonic sphere, which is a harmonic map from M=(Rm,e−|x|22(m−2)ds02) to N with finite energy. Here ds02 is Euclidean metric in Rm. Ding and Yongqiang (2006) showed that if the target is a sphere, any equivariant quasi-harmonic spheres is discontinuous at infinity. The metric g=e−|x|22(m−2)ds02 is quite singular at infinity and it is not complete. In this paper, we mainly study the eigenfunction of Quasi-Laplacian Δg=e|x|22(m−2)(Δg0−∇g0h⋅∇g0)=e|x|22(m−2)Δh for h=|x|24. In particular, we show that there are infinite number of eigenvalues (of the quasi-Laplacian Δg ) of which the corresponding eigenfunctions are discontinuous at ∞ and any non-constant eigenfunction of drifted Laplacian Δh=Δg0−∇g0h⋅∇g0 is discontinuous at infinity.

Bound states of a two-boson system on a two-dimensional lattice

We consider a Hamiltonian of a two-boson system on a two-dimensional lattice Z2. The Schrodinger operator H(k1, k2) of the system for k1 = k2 = π, where k = (k1, k2) is the total quasimomentum, has an infinite number of eigenvalues. In the case of a special potential, one eigenvalue is simple, another one is double, and the other eigenvalues have multiplicity three. We prove that the double eigenvalue of H(π,π) splits into two nondegenerate eigenvalues of H(π, π − 2β) for small β > 0 and the eigenvalues of multiplicity three similarly split into three different nondegenerate eigenvalues. We obtain asymptotic formulas with the accuracy of β2 and also an explicit form of the eigenfunctions of H(π, π −2β) for these eigenvalues.

Chain of point-like potentials in ℛ3 and infiniteness of the number of bound states

Infinite chain of point-like potentials having the Hamiltonian with infinite number of eigenvalues below the continuous spectrum is constructed. The background of the model is the theory of self-adjoint extensions of symmetric operators in the Hilbert space. The analogous example of the Hamiltonian is obtained for the system of three-dimensional waveguides coupled through point-like windows.

Robust Control and Time-Domain Specifications for Systems of Delay Differential Equations via Eigenvalue Assignment

An approach to eigenvalue assignment for systems of linear time-invariant (LTI) delay differential equations (DDEs), based upon the solution in terms of the matrix Lambert W function, is applied to the problem of robust control design for perturbed LTI systems of DDEs, and to the problem of time-domain response specifications. Robust stability of the closed-loop system can be achieved through eigenvalue assignment combined with the real stability radius concept. For a LTI system of DDEs with a single delay, which has an infinite number of eigenvalues, the recently developed Lambert W function-based approach is used to assign a dominant subset of them, which has not been previously feasible. Also, an approach to time-domain specifications for the transient response of systems of DDEs is developed in a way similar to systems of ordinary differential equations using the Lambert W function-based approach.

Stability of nonlinear hybrid dynamical systems with time delay via sum of squares decomposition

Considering an infinite number of eigenvalues for time delay systems, it is difficult to determine their stability. We have developed a new approach for the stability test of time delay nonlinear hybrid systems. Construction of Lyapunov functions for hybrid systems is generally a difficult task, but once these functions are found, stability’s analysis of the system is straight-forward. In this paper both delay-independent and delay-dependent stability tests are proposed, based on the construction of appropriate Lyapunov-Krasovskii functionals. The methodology is based on the sum of squares decomposition of multivariate polynomials and the algorithmic construction is achieved through the use of semidefinite programming. The reduction techniques provide numerical solution of large-scale instances; otherwise they will be computationally infeasible to solve. The introduced method can be used for hybrid systems with linear or nonlinear vector fields. Finally simulation results show the correctness and validity of the designed method.

Effective Prüfer angles and relative oscillation criteria

We present a streamlined approach to relative oscillation criteria based on effective Prüfer angles adapted to the use at the edges of the essential spectrum. Based on this we provided a new scale of oscillation criteria for general Sturm–Liouville operators which answer the question whether a perturbation inserts a finite or an infinite number of eigenvalues into an essential spectral gap. As a special case we recover and generalize the Gesztesy–Ünal criterion (which works below the spectrum and contains classical criteria by Kneser, Hartman, Hille, and Weber) and the well-known results by Rofe-Beketov including the extensions by Schmidt.

The continuous spectrum for the M / M / c queue

Transient solutions for M/M/c queues are important for many purposes, in particular for staffing facilities such as call centers. In this article, we show how to use spectral analysis to find such solutions. The difficulty is that unless the number in line is bounded, one has to deal with matrices of infinite size, and hence with a countable infinite number of eigenvalues. This problem can be overcome by noting that the spectrum is dense with few exceptions. We also show how many discrete eigenvalues remain. Our theory may also work to obtain spectra for other infinite-dimensional matrices. Numerical properties of our approach are explored.

Robust Regulation of Infinite-Dimensional Systems with Infinite-Dimensional Exosystems

In this paper robust regulation problem for infinite-dimensional systems with infinite-dimensional exosystems is discussed. The exosystems considered in this paper have infinite number of eigenvalues on the imaginary axis and thus include periodic signals. It is shown that there exists a feedback controller which robustly regulates the class of signals generated by the exosystem and strongly stabilizes the closed-loop system. As far as the authors know, the result is new even for finite-dimensional systems.

Effective Masses for Zigzag Nanotubes in Magnetic Fields

We consider the Schr\"odinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes in magnetic field. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We obtain identities and a priori estimates in terms of effective masses and gap lengths.

Schrödinger Operators on Zigzag Nanotubes

We consider the Schrodinger operator with a periodic potential on quasi-1D models of zigzag single-wall carbon nanotubes. The spectrum of this operator consists of an absolutely continuous part (intervals separated by gaps) plus an infinite number of eigenvalues with infinite multiplicity. We describe all compactly supported eigenfunctions with the same eigenvalue. We define a Lyapunov function, which is analytic on some Riemann surface. On each sheet, the Lyapunov function has the same properties as in the scalar case, but it has branch points, which we call resonances. We prove that all resonances are real. We determine the asymptotics of the periodic and antiperiodic spectrum and of the resonances at high energy. We show that there exist two types of gaps: i) stable gaps, where the endpoints are periodic and anti-periodic eigenvalues, ii) unstable (resonance) gaps, where the endpoints are resonances (i.e., real branch points of the Lyapunov function). We describe all finite gap potentials. We show that the mapping: potential \(\rightarrow\) all eigenvalues is a real analytic isomorphism for some class of potentials.

Expression for the number of eigenvalues of a Friedrichs model

We consider the self-adjoint operator of a generalized Friedrichs model whose essential spectrum may contain lacunas. We obtain a formula for the number of eigenvalues lying on an arbitrary interval outside the essential spectrum of this operator. We find a sufficient condition for the discrete spectrum to be finite. Applying the formula for the number of eigenvalues, we show that there exist an infinite number of eigenvalues on the lacuna for a particular Friedrichs model and obtain the asymptotics for the number of eigenvalues.

Bound states of atoms in a homogeneous magnetic field

AbstractWe consider Schrödinger operators of atoms in a homogeneous magnetic field. We prove that for each fixed value of the pseudomomentum, the corresponding operator has an infinite number of eigenvalues. The asymptotic behaviour of the counting function is studied for values of spectral parameter near the bottom of the essential spectrum. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

On the band gap structure of Hill's equation

We revisit the old problem of finding the stability and instability intervals of a second-order elliptic equation on the real line with periodic coefficients (Hill's equation). It is well known that the stability intervals correspond to the spectrum of the Bloch family of operators defined on a single period. Here we propose a characterization of the instability intervals. We introduce a new family of non-self-adjoint operators, formally equivalent to the Bloch ones but with an imaginary Bloch parameter, that we call exponential. We prove that they admit a countable infinite number of eigenvalues which, when they are real, completely characterize the intervals of instability of Hill's equation.

Eigenvalue Asymptotics for Weakly Perturbed Dirac and Schrödinger Operators with Constant Magnetic Fields of Full Rank

The even-dimensional Dirac and Schrödinger operators with a constant magnetic field of full rank have purely essential spectrum consisting of isolated eigenvalues, so-called Landau levels. For a sign-definite electric potential Vwhich tends to zero at infinity, not too fast, it is known for the Schrödinger operator that the number of eigenvalues near each Landau level is infinite and their leading (quasi-classical) asymptotics depends on the rate of decay for V. We show, both for Schrödinger and Dirac operators, that, for anysign-definite, bounded Vwhich tends to zero at infinity, there still are an infinite number of eigenvalues near each Landau level. For compactly supported V, we establish the non-classicalformula, not depending on V, describing how the eigenvalues converge to the Landau levels asymptotically.

Sturm–Liouville Problems with Finite Spectrum

For every positive integer n, we construct a class of regular self-adjoint and non-self-adjoint Sturm–Liouville problems with exactly n eigenvalues. These n eigenvalues can be located anywhere in the complex plane in the non-self-adjoint case and anywhere along the real line in the self-adjoint case. The latter complements the well-known general result for right-definite Sturm–Liouville problems with an infinite number of eigenvalues, which must go to infinity asymptotically like n2 does. With an appropriate and natural interpretation of a “zero,” the eigenfunctions have the usual zero properties.

Bifurcating fronts for the Taylor—Couette problem in infinite cylinders

We show the existence of bifurcating fronts for the weakly unstable Taylor—Couette problem in an infinite cylinder. These fronts connect a stationary bifurcating pattern, here the Taylor vortices, with the trivial ground state, here the Couette flow. In order to show the existence result we improve a method which was already used in establishing the existence of bifurcating fronts for the Swift—Hohenberg equation by Collet and Eckmann, 1986, and by Eckmann and Wayne, 1991. The existence proof is based on spatial dynamics and center manifold theory. One of the difficulties in applying center manifold theory comes from an infinite number of eigenvalues on the imaginary axis for vanishing bifurcation parameter. But nevertheless, a finite dimensional reduction is possible, since the eigenvalues leave the imaginary axis with different velocities, if the bifurcation parameter is increased. In contrast to previous work we have to use normalform methods and a non–standard cut–off function to obtain a center manifold which is large enough to contain the bifurcating fronts.

On Properties of Transfer Functions of a Flexible Structure

A flexible arm is the distributed parameter system with infinite number of eigenvalues. Its transfer function can be calculated directly from the partial differential equation model by using the Laplace transformation. On the other hand, two approaches of the modal analysis; the unconstrained mode method and the constrained mode method are employed to construct infinite dimensional lumped parameter dynamical system models by expanding the elastic deformation, which make two kinds of transfer functions. We obtain three different but equivalent transfer functions for the same system. Comparison of these transfer functions yields relationships on eigenvalues and parameters of these systems. To this end, the distributed parameter model transfer function is expressed as a ratio of two transcendental entire functions and these are expanded in infinite products. It is shown that the poles of unconstrained model transfer function and zeros of the constrained model transfer function coincide with those of the distributed parameter transfer function respectively.

Spectral properties of a linearized model of magnetohydrodynamics

One considers the frequency spectrum of the linear oscillations of a perfectly conducting nonviscous plasma with a nonzero pressure. It is shown that the essential spectrum for each of the harmonic modes is not empty and can consist of at most two finite intervals (or points). The discrete spectrum contains an infinite number of eigenvalues, diverging to +∞. One has found the first term of the asymptotics of this sequence. One shows the absence of the accumulation of the eigenvalues towards the endpoints of one of the intervals of the essential spectrum and also the absence of the eigenvalues to the left of the essential spectrum.