Spectral Operators Research Articles

On multilinear spectral cluster estimates for manifolds with boundary

We prove bilinear and trilinear estimates for the spectral cluster operator on two and three-dimensional compact manifolds with boundary. These are the natural analogs of earlier estimates for the boundaryless case of Burq, G\'erard, and Tzvetkov~\cite{bgtbilin}, \cite{bgtmultilin}. Our theorem reduces to establishing inequalities over small cubes whose size depends on frequency. After rescaling, these inequalities follow from mixed $L^p$ norm estimates on squarefunctions associated to the wave equation.

On the characterization of scalar type spectral operators

The paper is concerned with conditions guaranteeing that a bounded operator in a reflexive Banach space is a scalar type spectral operator. The cases where the spectrum of the operator lies on the real axis and on the unit circle are studied separately.

R-bounded Representations of L1 (G)

We investigate R-bounded representations $$\Psi: L^{1}\left( G\right) \rightarrow {\mathcal{L}}\left( X\right) $$ , where X is a Banach space and G is a lca group. Observing that Ψ induces a (strongly continuous) group homomorphism $$U:G\rightarrow {\mathcal{L}}\left( X\right) $$ , we are then able to analyze certain classical homomorphisms U (e.g. translations in L p (G)) from the viewpoint of R-boundedness and the theory of scalar-type spectral operators.

Note on Z2 symmetries of the Knizhnik-Zamolodchikov equation

We continue the study of hidden Z2 symmetries of the four-point sl(2)̂k Knizhnik-Zamolodchikov equation initiated by Giribet [Phys. Lett. B 628, 148 (2005)]. Here, we focus our attention on the four-point correlation function in those cases where one spectral flowed state of the sector ω=1 is involved. We give a formula that shows how this observable can be expressed in terms of the four-point function of non spectral flowed states. This means that the formula holding for the winding violating four-string scattering processes in AdS3 has a simple expression in terms of the one for the conservative case, generalizing what is known for the case of three-point functions, where the violating and the nonviolating structure constants turn out to be connected one to each other in a similar way. What makes this connection particularly simple is the fact that, unlike what one would naively expect, it is not necessary to explicitly solve the five-point function containing a single spectral flow operator to this end. Instead, nondiagonal functional relations between different solutions of the Knizhnik-Zamolodchikov equation turn out to be the key point for this short path to exist. Considering such functional relation is necessary but it is not sufficient; besides, the formula also follows from the relation existing between correlators in both Wess-Zumino-Novikov-Witten (WZNW) and Liouville conformal theories.

On spectral radius algebras and normal operators

We associate to each normal operator N an algebra BN that contains the commutant ofN. For a subclass of normal operators we demonstrate thatBN has a nontrivial invariant subspace. Further, we show that BN properly contains the commutant of N so that the invariant subspace result is stronger then the existence of a hyperin- variant subspace.

Littlewood-Paley theorem for Schrödinger operators

Let H be a Schrodinger operator on ℝ n. Under a polynomial decay condition for the kernel of its spectral operator, we show that the Besov spaces and Triebel-Lizorkin spaces associated with H are well defined. We further give a Littlewood-Paley characterization of Lp spaces in terms of dyadic functions of H. This generalizes and strengthens the previous result when the heat kernel of H satisfies certain upper Gaussian bound.

On The Eigenvalues of the Spectral Second Order Differentiation Operator and Application to the Boundary Observability of the Wave Equation

The behaviour of the eigenvalues of the spectral second-order differentiation operator is studied and the results are used to investigate the boundary observability of the one dimensional wave equation approximated with a spectral Galerkin method. New explicit estimates of the discrete eigenvalues are given. These estimates improve the previous results on the subject especially for the portion of eigenvalues converging exponentially to those of the continuous problem. Although the boundary observability property of the discretized wave equation is not uniform with respect to the discretization parameter, we show that a uniform observability estimate can be obtained by filtering out the highest eigenmodes.

Remarks on the application of the Keldysh theorem to the completeness of root subspace of non-self-adjoint operators and comments on “Spectral operators generated by Timoshenko beam model”

In this note, we show, by means of an example, that the assumption of compactness in the Keldysh theorem in proving the completeness of root subspaces of non-self-adjoint operators cannot be ignored. This indicates a gap of the above mentioned paper and many other related papers in the proof of completeness for the root subspaces of beam-related problems.

Integrable nonlinear ladder system with background-controlled intersite resonant coupling

A new spectral problem on one-dimensional lattices is found allowing consistently to support the zero-curvature representation for a wide class of integrable nonlinear ladder systems. The modified recurrence technique for obtaining an infinite set of conservation laws is developed and some basic conserved quantities are explicitly derived. The eigenvalue problems associated with the limiting spectral operator for the special case of rapidly vanishing boundary conditions on Schrodinger-type fields and finite background condition on a concomitant field are solved and the domains of analyticity of Jost functions are presented both analytically and graphically. This particular example shows that the original auxiliary spectral problem is basically of fourth order and must generate a set of four distinct Jost functions that have to be involved in the procedure of inverse scattering transform. Moreover, there exists a critical background value of accompanying field which separates two principally different possibilities in the organization of analyticity domains of Jost functions. This crossover should inevitably lead to qualitative rearrangements in the structure of model solutions. Thus already in the limit of low-amplitude excitations we strictly observe the loss of stability regarding the linear spectrum of Schrodinger subsystem just above the critical background value of practically unexcited concomitant field, whereas in the stability region the structure of linear spectrum is essentially controlled by the magnitude of background level via effective modification of both intersite resonant coupling and self-site coupling.

The free surface of a coupled ocean–atmosphere model due to forcing effects

In this work, we consider external effects on a coupled shallow water ocean–atmosphere model. The forced shallow water equation due to the wind action is studied through a decomposition technique. Induced free perturbations of the permanent response are identified by using a spectral operator basis that is generated by a dynamical Green function. This later was determined through a spectral technique. By using the semi-lagrangian method, we compute the non-linear response of the full model due to a shear stress that comes from the action of the wind at the ocean surface.

When is a non-self-adjoint Hill operator a spectral operator of scalar type?

We derive necessary and sufficient conditions for a one-dimensional periodic Schrödinger (i.e., Hill) operator H = − d 2 / d x 2 + V in L 2 ( R ) to be a spectral operator of scalar type. The conditions demonstrate the remarkable fact that the property of a Hill operator being a spectral operator is independent of smoothness (or even analyticity) properties of the potential V. To cite this article: F. Gesztesy, V. Tkachenko, C. R. Acad. Sci. Paris, Ser. I 343 (2006).

Winding strings in AdS3

Correlation functions of one unit spectral flowed states in string theory on AdS_3 are considered. We present the modified Knizhnik-Zamolodchikov and null vector equations to be satisfied by amplitudes containing states in winding sector one and study their solution corresponding to the four point function including one w=1 field. We compute the three point function involving two one unit spectral flowed operators and find expressions for amplitudes of three w=1 states satisfying certain particular relations among the spins of the fields. Several consistency checks are performed.

Three-point function in perturbed Liouville gravity

Abstract Three-point correlation function in perturbed conformal field theory coupled to two-dimensional quantum gravity (perturbed Liouville gravity) is explicitly computed by using the free field approach. The representation considered here is the one recently proposed in [G. Giribet, Nucl. Phys. B 737 (2006) 209] to describe the string theory in AdS 3 space. Consequently, this computation extends previous results which presented free field calculations of particular cases of string amplitudes, and confirms that the free field approach leads to the exact result. Remarkably, this representation allows to compute winding violating three-point functions without making use of the spectral flow operator.

Spectral synthesis and operator synthesis

Relations between spectral synthesis in the Fourier algebra $A(G)$ of a compact group $G$ and the concept of operator synthesis due to Arveson have been studied in the literature. For an $A(G)$-submodule $X$ of $\mathop {\rm VN}\nolimits (G)$, $X$-synthes

Phase recovery from a single fringe pattern using an orientational vector-field-regularized estimator

Recent studies have demonstrated that the phase recovery from a single fringe pattern with closed fringes can be properly performed if the modulo 2pi fringe orientation is estimated. For example, the fringe pattern in quadrature can be efficiently obtained in terms of the orientational phase spatial operator using fast Fourier transformations and a spiral phase spectral operator in the Fourier space. The computation of the modulo 2pi fringe orientation, however, is by far the most difficult task in the global process of phase recovery. For this reason we propose the demodulation of fringe patterns with closed fringes through the computation of the modulo 2pi fringe orientation using an orientational vector-field-regularized estimator. As we will show, the phase recovery from a single pattern can be performed in an efficient manner using this estimator, provided that it requires one to solve locally in the fringe pattern a simple linear system to optimize a regularized cost function. We present simulated and real experiments applying the proposed methodology.

Spectral element benchmark simulations of natural convection in two‐dimensional cavities

AbstractThis paper reports the results of spectral element simulations of natural convection in two‐dimensional cavities. In particular, a detailed comparison is performed with the reference data for the 8:1 cavity at Ra=3.4 × 105 recently described by Christon et al. [Int. J. Numer. Methods Fluids 2002; 40:953–980]. The Navier–Stokes equations augmented by the Boussinesq approximation to represent buoyancy effects are solved by a numerical method based on a spectral element discretization and operator splitting. The computed solutions agree closely with the reference data for both the square and the rectangular cavity configurations. Copyright © 2005 John Wiley & Sons, Ltd.

Atmospheric Stability in a Generalized Barotropic Model

Abstract An empirical modification of conventional barotropic dynamics is implemented to study the low-frequency variability (LFV) of the upper troposphere. Using the conservation of potential vorticity, generalized spectral barotropic operators that apply at single isentropic levels are constructed. In initial value calculations the empirical model shows improvement in skill compared to the conventional barotropic model, but it does not do significantly better than persistence. For short times, however, the empirically modified model shows a much closer resemblance to the observed streamfunction tendency. Overall, it is a significantly more accurate representation of the atmosphere than the conventional barotropic model. Normal, optimal, and singular modes of the modified model are calculated. The modes of the empirically modified model are more stable and more difficult to excite than those of the barotropic model. These results are consistent with previous studies that found barotropic dynamics deficient for the quantitative description of LFV. The singular modes of the modified operator have very similar patterns but explain less variance than those of the barotropic operator, which is consistent with the difficulty in detecting optimal patterns in observations. The modified barotropic operator is also more normal than the barotropic operator, and thus less variable.

Mini-Workshop: Nonlinear Spectral and Eigenvalue Theory

What is the state-of-the-art of abstract spectral and eigenvalue theory for nonlinear operators, and how may this theory be applied to nonlinear equations involving the p-Laplace operator? These two questions have provided the main focus of the Mini-Workshop. Accordingly, the main topics covered by the talks on this Mini-Workshop have been Of course, these three topics are not mutually independent, but there are various interconnections between them which are of particular interest. For example, sets of eigenvalues (point spectra) may be regarded, as in the linear case, as an important part of the spectrum; conversely, nonlinear eigenvalue theory is one of the historical roots of nonlinear spectral theory. Moreover, the p-Laplace operator is one of the most interesting homogeneous (though nonlinear) operators which may not only serve as a “model operator” in nonlinear eigenvalue problems, but also occurs quite frequently in various applications to physics, mechanics, and elasticity. The aim and scope of the Mini-Workshop was to bring together experts on nonlinear spectral analysis and operator theory, on the one hand, and more application-oriented specialists in eigenvalue problems for nonlinear partial differential equations (like the p-Laplace equation), on the other. As a result, 15 leading experts in the field from 10 different countries discussed recent progress and open problems in the theory, methods, and applications of spectra and eigenvalues of nonlinear operators.

Assessing the utility of spectral band operators to reduce the influence of total suspended solids on the relationship between chlorophyll concentration and the bidirectional reflectance factor in Amazon waters

Measurements of reflectance spectra, total suspended solids (TSS), chlorophyll a+phaeophytin (CLH) and dissolved organic carbon (DOC) plus pH, Secchi disk transparency and water depth were made in lakes associated with the Amazons and Negro rivers upstream from Manaus. Reflectance spectra were acquired between 400 and 900 nm using a spectroradiometer with a nominal resolution of 3 nm. Two spectral band operators designed to detect the CHL signal in waters with high TSS concentrations were examined with the aid of linear and nonlinear regression methods. Results showed that none of the proposed operators were successful in Amazon white water systems.

Functional calculus for the Laguerre operator

In this paper we study the boundedness of spectral multipliers associated to the multidimensional Laguerre operator . It is well known that, for special values of α, the analysis of the Laguerre operator can be interpreted as the analysis of the Ornstein-Uhlenbeck operator acting on “polyradial” functions. Exploiting this relation, we prove that if M is a bounded holomorphic function in the sector satisfying suitable Hormander type conditions on the boundary, then the spectral operator M( ) is bounded on L p with respect to the Laguerre measure. We also prove that holomorphy in the sector S is a necessary condition for multipliers whose norm is invariant under dilations.