Sequence Of Eigenvalues Research Articles

Rate optimal estimation of quadratic functionals in inverse problems with partially unknown operator and application to testing problems

We consider the estimation of quadratic functionals in a Gaussian sequence model where the eigenvalues are supposed to be unknown and accessible through noisy observations only. Imposing smoothness assumptions both on the signal and the sequence of eigenvalues, we develop a minimax theory for this problem. We propose a truncated series estimator and show that it attains the optimal rate of convergence if the truncation parameter is chosen appropriately. Consequences for testing problems in inverse problems are equally discussed: in particular, the minimax rates of testing for signal detection and goodness-of-fit testing are derived.

Uniform local Lipschitz continuity of eigenvalues with respect to the potential in L^1[a,b

The present paper shows that the eigenvalue sequence $\{\lambda_n(q)\}_{n\geqslant 1}$ of regular Sturm-Liouville eigenvalue problem with certain monotonic weights is uniformly Lipschitz continuous with respect to the potential $q$ on any bounded subset of $L^1([a,b],\mathbb{R})$.

The Use of the Inverse Problem of Spectral Analysis to Forecast Time Series

The paper proposes a new method to forecast time series. We assume that a time series is a sequence of eigenvalues of a discrete self-adjoint operator acting in a Hilbert space. In order to construct such an operator, we use the theory of solving inverse problems of spectral analysis. The paper gives a theoretical justification for the proposed method. An algorithm for solving the inverse problem is given. Also, we give an example of constructing a differential operator whose eigenvalues practically coincide with a given time series.

Applications of a Class of Herglotz Operator Pencils

We identify a class of operator pencils, arising in a number of applications, which have only real eigenvalues. In the one-dimensional case we prove a novel version of the Sturm oscillation theorem: if the dependence on the eigenvalue parameter is of degree $k$, then the real axis can be partitioned into a union of $k$ disjoint intervals, each of which enjoys a Sturm oscillation theorem: on each interval there is an increasing sequence of eigenvalues that are indexed by the number of roots of the associated eigenfunction. One consequence of this is that it guarantees that the spectra of these operator pencils have finite accumulation points, implying that the operators do not have compact resolvents. As an application we apply this theory to an epidemic model and several species dispersal models arising in biology.

Morse index and symmetry for elliptic problems with nonlinear mixed boundary conditions

AbstractWe consider an elliptic problem of the type $$\left\{ {\matrix{ {-\Delta u = f(x,u)\quad } \hfill & {{\rm in}\,\Omega } \hfill \cr {u = 0} \hfill & {{\rm on}\,\Gamma _1} \hfill \cr {\displaystyle{{\partial u} \over {\partial \nu }} = g(x,u)} \hfill & {{\rm on}\,\Gamma _2} \hfill \cr } } \right.$$ where Ω is a bounded Lipschitz domain in ℝN with a cylindrical symmetry, ν stands for the outer normal and $\partial \Omega = \overline {\Gamma _1} \cup \overline {\Gamma _2} $.Under a Morse index condition, we prove cylindrical symmetry results for solutions of the above problem.As an intermediate step, we relate the Morse index of a solution of the nonlinear problem to the eigenvalues of the following linear eigenvalue problem $$\left\{ {\matrix{ {-\Delta w_j + c(x)w_j = \lambda _jw_j} \hfill & {{\rm in }\Omega } \hfill \cr {w_j = 0} \hfill & {{\rm on }\Gamma _1} \hfill \cr {\displaystyle{{\partial w_j} \over {\partial \nu }} + d(x)w_j = \lambda _jw_j} \hfill & {{\rm on }\Gamma _2} \hfill \cr } } \right.$$ For this one, we construct sequences of eigenvalues and provide variational characterization of them, following the usual approach for the Dirichlet case, but working in the product Hilbert space L2 (Ω) × L2(Γ2).

Error Investigation of a Finite Element Approximation for a Nonlinear Sturm–Liouville Problem

A positive definite differential eigenvalue problem with coefficients depending nonlinearly on the spectral parameter is studied. The problem is formulated as a variational eigenvalue problem in a Hilbert space with bilinear forms depending nonlinearly on the spectral parameter. The variational problem has an increasing sequence of positive simple eigenvalues that correspond to a normalized system of eigenfunctions. The variational problem is approximated by a finite element mesh scheme on a uniform grid with Lagrangian finite elements of arbitrary order. Error estimates for approximate eigenvalues and eigenfunctions are proved depending on the mesh size and the eigenvalue size. The results obtained are generalizations of well-known results for differential eigenvalue problems with linear dependence on the spectral parameter.

Spectral Analysis of a Viscoelasticity Problem

An eigenvalue problem associated with small movements of a viscoelastic body fixed on the boundary of a bounded domain is studied. The spectrum of the problem is proved to lie in a vertical strip bounded away from the imaginary axis and to be symmetric about the real axis. The essential spectrum of the problem consists of a finite number of points on the real axis. There are two sequences of complex conjugate eigenvalues condensing toward infinity. Under certain additional conditions, the spectrum that does not lie on the real axis is bounded away from it.

A nonlinear elliptic eigenvalue–transmission problem with Neumann boundary condition

Let $$\varOmega \subset \mathbb {R}^N$$ , $$N\ge 2$$ , be a bounded domain which is divided into two sub-domains $$\varOmega _1$$ and $$\varOmega _2$$ . Consider in $$\varOmega $$ an eigenvalue–transmission problem associated with the p-Laplacian acting in $$\varOmega _1$$ and the q-Laplacian acting in $$\varOmega _2$$ , $$1<p<q$$ , with Dirichlet–Neumann conditions on the interface separating the two sub-domains $$\varOmega _1$$ and $$\varOmega _2$$ [see (1.1)]. The main result Theorem 2.1 states the existence of a sequence of eigenvalues for this eigenvalue problem. The proof is based on the Lusternik–Schnirelmann principle. Using the method of Lagrange multipliers for constrained minimization problems, we show (see Theorem 2.2) that if $$2\le p<q$$ then there exists an eigenfunction in any set of the form $$\begin{aligned} \left\{ u\in W^{1,p}(\varOmega );\, u|_{\varOmega _2}\in W^{1,q}(\varOmega _2), \, \frac{1}{p}\int _{\varOmega _1}|u|^p + \frac{1}{q}\int _{\varOmega _2}|u|^q = \alpha \right\} , \quad \alpha >0. \end{aligned}$$ The case of Robin conditions on $$\partial \varOmega $$ and the Riemannian setting are also addressed.

Perturbations of globally hypoelliptic operators on closed manifolds

Analyzing the behavior at infinity of the sequence of eigenvalues given by matrix symbol of a invariant operator with respect to a fixed elliptic operator, we obtain necessary and sufficient conditions to ensure that perturbations of globally hypoelliptic operators continue to have this property. As an application, we recover classical results about perturbations of constant vector fields on the torus and extend them for more general classes of perturbations. Additionally, we construct examples of low order perturbations that destroy the global hypoellipticity, in the presence of diophantine phenomena

Estimation of the number of endmembers in hyperspectral data using a weight-sequence geometry method

The terrestrial reflection or emission spectrum obtained by the remote sensor is recorded in units of pixels. In most cases, a pixel usually contains many types of terrains. This pixel is a mixed pixel, and each of the terrains in the mixed pixels is called “endmember”. Estimating the number of endmembers is a significant step in many hyperspectral data mining techniques, such as target classification and endmember extraction. The paper proposes a separative detection method by the use of a weight-sequence geometry to estimate the number of endmembers. This method projects the spectral matrix into the orthogonal subspace by eigenvalue decomposition at first. Then, on the basis of the normalized eigenvalue sequence, the separative detection method innovatively uses a geometric criterion to find the separation point between the main factors and minor factors. Finally, the number of endmembers is determined by the sequence of the “separation point”. Validation through a series of simulated and real hyperspectral data, it indicates that the proposed method can accurately and rapidly detect the number of endmembers in the hyperspectral data without any prior information. In addition, the new method is also applicable to the ultra-high resolution remote spectral data in the future.

On a Riesz basis of exponentials related to a family of analytic operators and application

In this paper, we are interested by the perturbed operator $$\begin{aligned} T(\varepsilon ):=T_0+\varepsilon T_1 +\varepsilon ^2T_2+\cdots +\varepsilon ^k T_k+\cdots \end{aligned}$$ where $$\varepsilon \in \mathbb {C}$$ , $$T_0$$ is a closed densely defined linear operator on a separable Hilbert space $$\mathcal{H}$$ with domain $$\mathcal{D}(T_0)$$ having isolated eigenvalues with multiplicity one whereas $$T_1, T_2,\ldots $$ are linear operators on $$\mathcal{H}$$ having the same domain $$\mathcal{D}\supset \mathcal{D}(T_0)$$ and satisfying a specific growing inequality. The basic idea here is to investigate under sufficient conditions the existence of Riesz bases of exponentials, where the exponents corresponding as a sequence of eigenvalues of $$T(\varepsilon )$$ , can be developed as entire series of $$\varepsilon $$ . An application to a nonself-adjoint problem describing the radiation of a vibrating structure in a light fluid is presented.

Finite Element Modeling of Eigenvibrations of a Loaded Bar

The nonlinear second-order differential eigenvalue problem describing eigenvibrations of a bar with elastically attached load is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. The sequence of eigenvalues corresponds to a system of normalized eigenfunctions. The initial nonlinear eigenvalue problem is approximated by the quadrature finite element method on a uniform grid. The existence and accuracy of approximate solutions are studied. Investigations of the present paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with elastically attached loads.

Asymptotic Behavior of Eigenvalues of a Boundary Value Problem for a Second-Order Elliptic Differential-Operator Equation with Spectral Parameter Quadratically Occurring in the Boundary Condition

The asymptotic behavior of eigenvalues of a boundary value problem for a secondorder differential-operator equation in a separable Hilbert space on a finite interval is studied for the case in which the same spectral parameter occurs linearly in the equation and quadratically in one of the boundary conditions. We prove that the problem has a sequence of eigenvalues converging to zero.

Bifurcation solutions of a free boundary problem modeling tumor growth with angiogenesis

In this paper we study bifurcation solutions from the unique radial solution of a free boundary problem modeling stationary state of tumors with angiogenesis. This model comprises two elliptic equations describing the distribution of the nutrient concentration σ=σ(x) and the inner pressure p=p(x). Unlike similar tumor models that have been intensively studied in the literature where Dirichlet boundary condition for σ is imposed, in this model the boundary condition for σ is a Robin boundary condition. Existence and uniqueness of a radial solution of this model have been successfully proved in a recently published paper [20]. In this paper we study existence of nonradial solutions by using the bifurcation method. Let {γk}k=2∞ be the sequence of eigenvalues of the linearized problem. We prove that there exists a positive integer k⁎⩾2 such that in the two dimension case for any k⩾k⁎, γk is a bifurcation point, and in the three dimension case for any even k⩾k⁎, γk is also a bifurcation point.

P⁢(x)p(x)-biharmonic operator involving the p⁢(x)p(x)-Hardy inequality

Abstract In this work, we investigate the spectrum denoted by Λ for the p ⁢ ( x ) {p(x)} -biharmonic operator involving the Hardy term. We prove the existence of at least one non-decreasing sequence of positive eigenvalues of this problem such that sup ⁡ Λ = + ∞ {\sup\Lambda=+\infty} . Moreover, we prove that inf ⁡ Λ &gt; 0 {\inf\Lambda&gt;0} if and only if the domain Ω satisfies the p ⁢ ( x ) {p(x)} -Hardy inequality.

Principal eigenvalue of mixed problem for the fractional Laplacian: Moving the boundary conditions

We analyze the behavior of the eigenvalues of the following nonlocal mixed problem{(−Δ)su=λ1(D)u in Ω,u=0 in D,Nsu=0 in N. Our goal is to construct different sequences of problems by modifying the configuration of the sets D and N, and to provide sufficient and necessary conditions on the size and the location of these sets in order to obtain sequences of eigenvalues that in the limit recover the eigenvalues of the Dirichlet or Neumann problem. We will see that the nonlocality plays a crucial role here, since the sets D and N can have infinite measure, a phenomenon that does not appear in the local case (see for example [6–8]).

A multidimensional Borg–Levinson theorem for magnetic Schrödinger operators with partial spectral data

We consider the multidimensional Borg–Levinson theorem of determining both the magnetic field dA and the electric potential V , appearing in the Dirichlet realization of the magnetic Schrödinger operator H=(-\rm i\nabla+A)^2+V on a bounded domain \Omega\subset\mathbb R^n , n\geq2 , from partial knowledge of the boundary spectral data of H . The full boundary spectral data are given by the set \{(\lambda_{k},{\partial_\nu \phi_{k}}_{|\partial\Omega})\colon k\geq1\} , where \{ \lambda_k\colon k \geq1 \} is the non-decreasing sequence of eigenvalues of H , \{ \phi_k\colon k\geq1 \} an associated Hilbertian basis of eigenfunctions and \nu is the unit outward normal vector to \partial\Omega . We prove that some asymptotic knowledge of (\lambda_{k},{\partial_\nu \phi_{k}}_{|\partial\Omega}) with respect to k\geq1 determines uniquely the magnetic field dA and the electric potential V .

Extreme Localization of Eigenfunctions to One-Dimensional High-Contrast Periodic Problems with a Defect

Following a number of recent studies of resolvent and spectral convergence of nonuniformly elliptic families of differential operators describing the behavior of periodic composite media with high contrast, we study the corresponding one-dimensional version that includes a “defect”: an inclusion of fixed size with a given set of material parameters. It is known that the spectrum of the purely periodic case without the defect and its limit, as the period $\varepsilon$ goes to zero, has a band-gap structure. We consider a sequence of eigenvalues $\lambda_\varepsilon$ that are induced by the defect and converge to a point $\lambda_0$ located in a gap of the limit spectrum for the periodic case. We show that the corresponding eigenfunctions are “extremely” localized to the defect, in the sense that the localization exponent behaves as $\exp(-\nu/\varepsilon),$ $\nu>0,$ which has not been observed in the existing literature. In two- and three-dimensional configurations, whose one-dimensional cross sections are described by the setting considered, this implies the existence of propagating waves that are localized to a vicinity of the defect. We also show that the unperturbed operators are norm-resolvent close to a degenerate operator on the real axis, which is described explicitly.

Investigation of eigenvibrations of a loaded bar

The differential eigenvalue problem describing eigenvibrations of a bar with fixed ends and attached load at an interior point is investigated. This problem has an increasing sequence of positive simple eigenvalues with limit point at infinity. To the sequence of eigenvalues, there corresponds a complete orthonormal system of eigenfunctions. We formulate limit differential eigenvalue problems and prove the convergence of the eigenvalues and eigenfunctions of the initial problem to the corresponding eigenvalues and eigenfunctions of the limit problems as load mass tending to infinity. The original differential eigenvalue problem is approximated by the finite difference method on a uniform grid. Error estimates for approximate eigenvalues and eigenfunctions are established. Theoretical results are illustrated by numerical experiments for a model problem. Investigations of this paper can be generalized for the cases of more complicated and important problems on eigenvibrations of beams, plates and shells with attached loads.

Eigenvibrations of a simply supported beam with elastically attached load

The nonlinear differential eigenvalue problem describing eigenvibrations of a simply supported beam with elastically attached load is investigated. The existence of an increasing sequence of positive simple eigenvalues with limit point at infinity is established. To the sequence of eigenvalues, there corresponds a system of normalized eigenfunctions. To illustrate the obtained theoretical results, the initial problem is approximated by the finite difference method on a uniform grid. The accuracy of approximate solutions is studied. Investigations of the present paper can be generalized for the cases of more complicated and important problems on eigenvibrations of plates and shells with elastically attached loads.