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.     

On the Discrete Poincaré–Friedrichs Inequalities for Nonconforming Approximations of the Sobolev Space H 1

We present a direct proof of the discrete Poincaré–Friedrichs inequalities for a class of nonconforming approximations of the Sobolev space H ¹(Ω), indicate optimal values of the constants in these inequalities, and extend the discrete Friedrichs inequality onto domains only bounded in one direction. We consider a polygonal domain Ω in two or three space dimensions and its shape-regular simplicial triangulation. The nonconforming approximations of H ¹(Ω) consist of functions from H ¹ on each element such that the mean values of their traces on interelement boundaries coincide. The key idea is to extend the proof of the discrete Poincaré–Friedrichs inequalities for piecewise constant functions used in the finite volume method. The results have applications in the analysis of nonconforming numerical methods, such as nonconforming finite element or discontinuous Galerkin methods.     

Stability of Hodge decompositions in finite element spaces of differential forms in arbitrary dimension

We elaborate on the interpretation of some mixed finite element spaces in terms of differential forms. In particular we define regularization operators which, combined with the standard interpolators, enable us to prove discrete Poincaré–Friedrichs inequalities and discrete Rellich compactness for finite element spaces of differential forms of arbitrary degree on compact manifolds of arbitrary dimension.     

Spectral properties of the generalized Friedrichs model

We consider the self-adjoint generalized Friedrichs model with small values of the “coupling parameter.” In this case, we completely investigate the spectrum of the model and the structure of its eigenvectors (both ordinary and generalized). The constructions we use are based on an analysis of the resolvent of the Friedrichs operator and on the corresponding scattering theory.     

On the Discrete Friedrichs Inequality for Nonconforming Finite Elements

In this paper we establish the validity of the discrete Friedrichs inequality for piecewise linear Crouzeix-Raviart nonconforming finite elements in polygonal domains. It represents an extension of an important result proven for a polygonal convex domain to a general polygonal nonconvex domain. This result has applications in the analysis of exterior approximations of partial differential equations as, e.g., the Navier-Stokes equations and convection-diffusion problems.     

Embedded eigenvalues and resonances of a generalized Friedrichs model

The existence of resonances and embedded eigenvalues of a multidimensional generalized Friedrichs model is studied. The existence of a Friedrichs model with a given number of eigenvalues located within the continuous spectrum is proved. The existence of resonances is shown, and the widths of these resonances are calculated.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 103, No. 1, pp. 54–62, April, 1995.     

The connection between the rigged Hilbert space and the complex scaling approaches for resonances. The Friedrichs model

The rigged Hilbert space approach and the complex scaling method are compared for the Friedrichs model. It is shown that they define the same resonance pole. The choice of the test space for the rigged Hilbert space approach is discussed.     

Matrix analysis and the friedrichs operator of a quadrature domain

The Friedrichs operator of a quadrature domain is a finite rank operator which appears in the joint diagonalization of the Bergman space scalar product and its real part, regarded as a real symmetric form. This note relates some general aspects of the theory of pairs of hermitian or symmetric forms to the specific case of the Friedrichs operator.     

On the number of eigenvalues of a matrix operator

We consider a matrix operator H in the Fock space. We prove the finiteness of the number of negative eigenvalues of H if the corresponding generalized Friedrichs model has the zero eigenvalue (0 = min σ _ess(H)). We also prove that H has infinitely many negative eigenvalues accumulating near zero (the Efimov effect) if the generalized Friedrichs model has zero energy resonance. We obtain asymptotics for the number of negative eigenvalues of H below z as z → −0.     

Hyperbolic systems equivalent to the wave equation

Some relation along an additional characteristic is constructed for a class of Friedrichs hyperbolic systems to which the wave equation is reduced. The statement is proven about preservation of a vortex. The conditions are stated for the solutions to Friedrichs hyperbolic systems of this class to remain solutions to the initial wave equation.     

New Formulae for the Wave Operators for a Rank One Interaction

We prove new formulae for the wave operators for a Friedrichs scattering system with a rank one perturbation, and we derive a topological version of Levinson’s theorem for this model.     

Non-factorizable extensions of the Liouville operator: the Friedrichs model

We construct for the Friedrichs model extensions of the Liouville operator which do not reduce to any extension of the corresponding Hamilton operator. These extensions originate in the Brussels–Austin approach to irreversibility and acquire meaning in suitable rigged Hilbert spaces (RHS) associated with the Hilbert–Schmidt space, known as rigged Liouville space (RLS).     

Crouzeix–Raviart boundary elements

This paper establishes a foundation of non-conforming boundary elements. We present a discrete weak formulation of hypersingular integral operator equations that uses Crouzeix–Raviart elements for the approximation. The cases of closed and open polyhedral surfaces are dealt with. We prove that, for shape regular elements, this non-conforming boundary element method converges and that the usual convergence rates of conforming elements are achieved. Key ingredient of the analysis is a discrete Poincaré–Friedrichs inequality in fractional order Sobolev spaces. A numerical experiment confirms the predicted convergence of Crouzeix–Raviart boundary elements. Norbert Heuer is supported by Fondecyt-Chile under grant no. 1080044. F.-J. Sayas is partially supported by MEC-FEDER Project MTM2007-63204 and Gobierno de Aragón (Grupo Consolidado PDIE).     

Some spectral properties of the generalized Friedrichs model

The spectrum and the resonance of the generalized Friedrichs model are studied. The existence of the wave operators and of the scattering operator is established. The structure of the resonances, the connection between resonances and eigenvalues, and the singularities of the scattering matrix are investigated.Translated from Trudy Seminara imeni I. G. Petrovskogo, No. 11, pp. 210–238, 1986.     

Friedrichs systems for the three-dimensional wave equation

The three-dimensional wave equation is reduced to a Friedrichs symmetric hyperbolic system. We describe all these reductions and find those of them that preserve the velocity of propagation of perturbations. We also exhibit transformations of a Friedrichs system under the Lorentz transformation of coordinates. The construction of the reduction of the wave equation and justification of the properties of this reduction are based on the use of quaternions.     

Examples of Gamow vectors

We present three selected physical examples of Gamow vectors to illustrate their physical significance, namely, central potential scattering, one-dimensional lattice with impurities in electric field (Stark) and a simple model for unstable elementary particles (Friedrichs).     

Asymptotic Expansions and Numerical Methods for Compressible and Low Mach Number Fluid Flow

The results of a formal asymptotic low Mach number analysis [5, 6] of the Euler equations of gas dynamics are used to extend the validity of a numerical method from the simulation of compressible inviscid flow fields to the low Mach number regime. Although, different strategies are applicable [7, 8, 5, 9] in this context we focus our view to a preconditioning technique recently proposed by Guillard and Viozat [16]. We present a finite volume approximation of the governing equations using a Lax‐Friedrichs scheme whereby a preconditioning of the incorporated numerical dissipation is employed. A discrete asymptotic analysis proves the validity of the scheme in the low Mach number regime.