Eigenvalues Of Finite Multiplicity Research Articles

Some new results on the property (R) under compact perturbations

The property (R) for a bounded linear operator T on a Hilbert space means that the points λ in the approximate point spectrum of T for which T − λI is upper semi-Browder are exactly the isolated points of the spectrum of T which are eigenvalues of finite multiplicity. For an operator T, if T + K satisfies the property (R) for all compact operators K, then T is said to have the stability of the property (R). In this paper, we give new criteria for the fulfillment of the stability of the property (R) for an operator. Moreover, the property (R) for complex symmetric operators is discussed.

Spectral and Scattering Properties of Quantum Walks on Homogenous Trees of Odd Degree

For unitary operators $U_0,U$ in Hilbert spaces ${\mathcal H}_0,{\mathcal H}$ and identification operator $J:{\mathcal H}_0\to{\mathcal H}$, we present results on the derivation of a Mourre estimate for $U$ starting from a Mourre estimate for $U_0$ and on the existence and completeness of the wave operators for the triple $(U,U_0,J)$. As an application, we determine spectral and scattering properties of a class of anisotropic quantum walks on homogenous trees of odd degree with evolution operator $U$. In particular, we establish a Mourre estimate for $U$, obtain a class of locally $U$-smooth operators, and prove that the spectrum of $U$ covers the whole unit circle and is purely absolutely continuous, outside possibly a finite set where $U$ may have eigenvalues of finite multiplicity. We also show that (at least) three different choices of free evolution operators $U_0$ are possible for the proof of the existence and completeness of the wave operators.

M2-branes on a constant flux background

We describe a compactified Supermembrane, or M2-brane, with 2-form fluxes generated by constant three-forms that are turned on a 2-torus of the target space M9×T2. We compare this theory with the one describing a 11D M2-brane formulated on M9×T2 target space subject to an irreducible wrapping condition. We show that the flux generated by the bosonic 3-form under consideration is in a one to one correspondence to the irreducible wrapping condition. After a canonical transformation both Hamiltonians are exactly the same up to a constant shift in one particular case. Consequently both of them, share the same spectral properties. We conclude that the Hamiltonian of the M2-brane with 2-form target space fluxes on a torus has a purely discrete spectrum with eigenvalues of finite multiplicity and it can be considered to describe a new sector of the microscopic degrees of freedom of M-theory. We also show that the total membrane momentum in the direction associated to the flux condition acquires a quantized contribution in correspondence to the flux units that have been turned on.

On the Spectrum of Differential Operators Under Riemannian Coverings

For a Riemannian covering p :M_{2} rightarrow M_{1}, we compare the spectrum of an essentially self-adjoint differential operator D_{1} on a bundle E_{1} rightarrow M_{1} with the spectrum of its lift D_{2} on p^{*}E_{1} rightarrow M_{2}. We prove that if the covering is infinite sheeted and amenable, then the spectrum of D_{1} is contained in the essential spectrum of any self-adjoint extension of D_{2}. We show that if the deck transformations group of the covering is infinite and D_{2} is essentially self-adjoint (or symmetric and bounded from below), then D_{2} (or the Friedrichs extension of D_{2}) does not have eigenvalues of finite multiplicity and in particular, its spectrum is essential. Moreover, we prove that if M_{1} is closed, then p is amenable if and only if it preserves the bottom of the spectrum of some/any Schrödinger operator, extending a result due to Brooks.

Property (m) under perturbations

A Banach space operator is said to be obeys property (m) if the isolated points of the spectrum σ(T) of T which are eigenvalues of finite multiplicity are exactly those points λ of the spectrum for which T - λ is an upper semi-Browder. In this article, we study the stability of property (m), for a bounded operator acting on a Banach space, under perturbation by finite rank operators, by nilpotent operators, quasi-nilpotent operators, Riesz operator or algebraic operators commuting with T.

Weyl’s theorem for upper triangular operator matrix and perturbations

Let and be infinite dimensional separable complex Hilbert spaces and T a bounded linear operator on . ‘Weyl’s theorem’ holds for operator T when the complement in the spectrum of the Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity. Let . In this paper, we characterize the invariant subset and the diagonal operator matrix satisfying upper triangular operator matrix , where .

Property $(aBw)$ and perturbations

A bounded linear operator $T$ acting on a Banach space $X$ satisfies property $(aBw)$, a strong version of a-Weyl's theorem, if the complement in the approximate point spectrum $\sigma_{a}(T)$ of the upper semi-B-Weyl spectrum $\sigma_{USBW}(T)$ is the set of all isolated points of approximate point spectrum which are eigenvalues of finite multiplicity. In this paper we investigate the property $(aBw)$ in connection with Weyl type theorems. In particular, we show that $T$ satisfies property $(aBw)$ if and only if $T$ satisfies a-Weyl's theorem and $\sigma_{USBW}(T)=\sigma_{USW}(T)$, where $\sigma_{USW}(T)$ is the upper semi-Weyl spectrum of $T$. The preservation of property $(aBw)$ is also studied under commuting nilpotent, quasi-nilpotent, power finite rank or Riesz perturbations. The theoretical results are illustrated by some concrete examples.

Property $(aw)$ and perturbations

A bounded linear operator $T\in\mathbf{L}(\mathbb{X})$ acting on a Banach space satisfies property $(aw)$, a variant of Weyl's theorem, if the complement in the spectrum $\sigma(T)$ of the Weyl spectrum $\sigma_w(T)$ is the set of all isolated points of the approximate-point spectrum which are eigenvalues of finite multiplicity. In this article we consider the preservation of property $(aw)$ under a finite rank perturbation commuting with $T$, whenever $T$ is polaroid, or $T$ has analytical core $K(T-\lambda_0 I)=\{0\}$ for some $\lambda_0\in \mathbb{C}$. The preservation of property $(aw)$ is also studied under commuting nilpotent or under injective quasi-nilpotent perturbations or under Riesz perturbations. The theory is exemplified in the case of some special classes of operators.

Sixth order differential operators with eigenvalue dependent boundary conditions

We consider eigenvalue problems for sixth-order ordinary differential equations. Such differential equations occur in mathematical models of vibrations of curved arches. With suitably chosen eigenvalue dependent boundary conditions, the problem is realized by a quadratic operator pencil. It is shown that the operators in this pencil are self-adjoint, and that the spectrum of the pencil consists of eigenvalues of finite multiplicity in the closed upper half-plane, except for finitely many eigenvalues on the negative imaginary axis.

An abstract semigroup approach to the third‐order Moore–Gibson–Thompson partial differential equation arising in high‐intensity ultrasound: structural decomposition, spectral analysis, exponential stability

This paper considers an abstract third‐order equation in a Hilbert space that is motivated by, and ultimately directed to, the “concrete” Moore–Gibson–Thompson Equation arising in high‐intensity ultrasound. In its simplest form, with certain specific values of the parameters, this third‐order abstract equation (with unbounded free dynamical operator) is not well‐posed. In general, however, in the present physical model, a suitable change of variable permits one to show that it has a special structural decomposition, with a precise, hyperbolic‐dominated driving part. From this, various attractive dynamical properties follow: s.c. group generation; a refined spectral analysis to include a specifically identified point in the continuous spectrum of the generator (so that it does not have compact resolvent) as an accumulation point of eigenvalues; and a consequent theoretically precise exponential decay with the same decay rate in various function spaces. In particular, the latter is explicit and sharp up to a finite number of (stable) eigenvalues of finite multiplicity. A computer‐based analysis confirms the theoretical spectral analysis findings. Moreover, it shows that the dynamic behavior of these unaccounted for finite‐dimensional eigenvalues are the ones that ultimately may dictate the rate of exponential decay, and which can be estimated with arbitrarily preassigned accuracy. Copyright © 2012 John Wiley & Sons, Ltd.

Property (R) under Perturbations

Property (R) holds for a bounded linear operator $${T \in L(X)}$$ , defined on a complex infinite dimensional Banach space X, if the isolated points of the spectrum of T which are eigenvalues of finite multiplicity are exactly those points λ of the approximate point spectrum for which λI − T is upper semi-Browder. In this paper we consider the permanence of this property under quasi nilpotent, Riesz, or algebraic perturbations commuting with T.

On the Removal of Finite Discrete Spectrum by Coefficient Stripping

We prove for a large class of operators, J , including block Jacobi matrices, if \sigma(J) \setminus [\alpha,\beta] is a finite set, each point of which is an eigenvalue of finite multiplicity, then a finite coefficient stripped, J_N , has \sigma(J_N)\subset [\alpha,\beta] . We use an abstract Dirichlet decoupling.

On the spectral theory and dispersive estimates for a discrete Schrödinger equation in one dimension

Based on the recent work [Komech et al., “Dispersive estimates for 1D discrete Schrödinger and Klein-Gordon equations,” Appl. Anal. 85, 1487 (2006)] for compact potentials, we develop the spectral theory for the one-dimensional discrete Schrödinger operator, Hϕ=(−Δ+V)ϕ=−(ϕn+1+ϕn−1−2ϕn)+Vnϕn. We show that under appropriate decay conditions on the general potential (and a nonresonance condition at the spectral edges), the spectrum of H consists of finitely many eigenvalues of finite multiplicities and the essential (absolutely continuous) spectrum, while the resolvent satisfies the limiting absorption principle and the Puiseux expansions near the edges. These properties imply the dispersive estimates ‖eitHPa.c.(H)‖lσ2→l−σ2≲t−3/2 for any fixed σ>52 and any t>0, where Pa.c.(H) denotes the spectral projection to the absolutely continuous spectrum of H. In addition, based on the scattering theory for the discrete Jost solutions and the previous results by Stefanov and Kevrekidis [“Asymptotic behaviour of small solutions for the discrete nonlinear Schrödinger and Klein-Gordon equations,” Nonlinearity 18, 1841 (2005)], we find new dispersive estimates ‖eitHPa.c.(H)‖l1→l∞≲t−1/3, which are sharp for the discrete Schrödinger operators even for V=0.

Property (w) for perturbations of polaroid operators

A bounded linear operator T∈L(X) acting on a Banach space satisfies property (w), a variant of Weyl’s theorem, if the complement in the approximate point spectrum σa(T) of the Weyl essential approximate-point spectrum σwa(T) is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this note, we study the stability of property (w) for a polaroid operator T acting on a Banach space, under perturbations by finite rank operators, by nilpotent operators and, more generally, by algebraic operators commuting with T.

Weyl's theorem for perturbations of paranormal operators

A bounded linear operator T ∈ L(X) on a Banach space X is said to satisfy if the complement in the spectrum of the Weyl spectrum is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this paper we show that if T is a paranormal operator on a Hilbert space, then T + K satisfies Weyl's theorem for every algebraic operator K which commutes with T.

Property [formula omitted] and perturbations

A bounded linear operator T ∈ L ( X ) defined on a Banach space X satisfies property ( w ) , a variant of Weyl's theorem, if the complement in the approximate point spectrum σ a ( T ) of the Weyl essential approximate spectrum σ wa ( T ) coincides with the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this note, we study the stability of property ( w ) , for a bounded operator T acting on a Banach space, under perturbations by finite rank operators, by nilpotent operator and quasi-nilpotent operators commuting with T.

L q-solutions to the Cosserat spectrum in bounded and exterior domains

SummaryIfλ = 1 is an eigenvalue of infinite multiplicity and λ = 2 is an accumulation point of eigenvalues of finite multiplicity. For the

Spectral concentration phenomena for the Laplace operator with the Dirichlet boundary condition on a cavity

Let D1 ⊂ R3 be a non-empty simply connected open bounded set and Γ = ∂D1 be its boundary. We assume Γ to be a C2 surface and let (Γn)n∈N be a sequence of surfaces, such that Γn is closed, , and the surface measure of Γ\Γn is positive for n ∈ N. The set D1 is the cavity associated with (Γn)n∈N. Let HΓn, n ∈ N, HΓ be the self-adjoint operators on L2 associated with the minus Laplace operator with Dirichlet boundary condition on Γn, n ∈ N and on Γ, respectively. Note that, roughly speaking, HΓ is the operator associated with minus the Laplacian on R3\Γ with the Dirichlet boundary condition on Γ. A similar statement holds for HΓn, n ∈ N. We show that the spectral measure, dEΓn (λ), λ ∈ R, associated with the operator HΓn, n ∈ N, when n → +∞ converges to the spectral measure dEΓ (λ), λ ∈ R, associated with the operator HΓ. That is in a sense made precise later we prove that Σ (HΓn) → Σ (HΓ) when n → +∞, where Σ (HΓ) and Σ (HΓn), n ∈ N, are the spectra of HΓ and HΓn, n ∈ N, respectively. Moreover, we show that the point spectrum of HΓ is made of an infinite number of positive eigenvalues of finite multiplicity and that the point spectrum of HΓn, n ∈ N, is empty and that the essential spectrum of HΓn, n ∈ N, and of HΓ is continuous and is contained in [0, +∞). Under the extra hypothesis that Γ is a C4 surface and that the Gaussian curvature of Γ is positive at every point of Γ, we prove that the essential spectrum of HΓ is a continuous spectrum and is given by [0, +∞). So that the convergence of dEΓn (λ), λ ∈ R to dEΓ (λ), λ ∈ R, when n → +∞ can be interpreted as a spectral concentration phenomenon that consists in the fact that the limit of the spectrum of operators with purely continuous spectrum is the spectrum of an operator that is made of eigenvalues embedded in a continuous spectrum.

Gravity and Capillary-Gravity Periodic Travelling Waves for Two Superposed Fluid Layers, One Being of Infinite Depth

The mathematical study of 2D travelling waves in the potential flow of two superposed layers of perfect fluid, with free surface and interfaces (with or without surface tensions) and with the bottom layer of infinite depth, is set as an ill-posed reversible evolution problem, where the horizontal space variable plays the role of a “time”. We give the structure of the spectrum of the linearized operator near equilibrium. This spectrum contains a set of isolated eigenvalues of finite multiplicities, a small number of which lie near or on the imaginary axis, and the entire real axis constitutes the essential spectrum, where there is no eigenvalue, except 0 in some cases. We give a general constructive proof of bifurcating periodic waves, adapting the Lyapunov-Schmidt method to the present (reversible) case where 0 (which is “resonant”) belongs to the continuous spectrum. In particular we give the results for the generic case and for the 1 : 1 resonance case

Intermittency and regularized Fredholm determinants

We consider real-analytic maps of the interval I=[0,1] which are expanding everywhere except for a neutral fixed point at 0. We show that on a certain function space the associated Perron-Frobenius operator ℳ has a continuous and residual spectrum contained in the line-segment σ c =[0,1] and a point spectrum σ p which has no points of accumulation outside 0 and 1. Furthermore, points in σ p −{0,1} are eigenvalues of finite multiplicity. We construct a regularized Fredholm determinant d(λ) which has a holomorphic extension to λ∈ℂ−σ c and can be analytically continued from each side of σ c to an open neighborhood of σ c −{0,1} (on different Riemann sheets). In ℂ−σ c the zero-set of d(λ) is in one-to-one correspondence with the point spectrum of ℳ. Through the conformal transformation the function d∘λ(z) extends to a holomorphic function in a domain which contains the unit disc.