arXiv subjects
Publications and source records attributed to Jens Wirth.
In this paper we consider a singular wave equation with distributional and more singular non-distributional coefficients and develop tools and techniques for the phase-space analysis of such problems. In particular we provide a detailed analysis for the interaction of singularities of solutions with strong singularities of the coefficient in a model problem of recent interest.
In this paper, we study the group Fourier transform and the Kohn-Nirenberg quantization for homogeneous Lie groups as mappings between certain Gelfand triples. For this, we restrict our considerations to the case, where the homogeneous Lie group $G$ admits irreducible unitary representations, that are square integrable modulo the center $Z(G)$ of $G$, and where $\dim Z(G)=1$. Replacing the Schwartz space by a certain subspace $\mathcal S_*(G) \hookrightarrow \mathcal S(G)$, we characterise the range of the group Fourier transform on $\mathcal S_*(G)$ and construct distributions and Gelfand triples around $L^2(G,μ)$ and its Fourier image $L^2(\hat G,\hat μ)$, such that the Fourier transform becomes a Gelfand triple isomorphism. We give results on the multiplication of distributions with a large class of vector valued smooth functions and use this to establish the Kohn-Nirenberg quantization as an isomorphism for our Gelfand triples and provide an explicit formula for the Kohn-Nirenberg symbol of an operator.
In this paper we consider a Klein-Gordon model with time-dependent periodic coefficients. The aim is to investigate how the presence of the mass term influences energy estimates with respect to the case of vanishing mass, already treated in [18]. The approach is based on a diagonalisation argument for high frequencies and a contradiction argument for bounded frequencies.
This paper considers Hamiltonians with localised potentials and gives a variational characterisation of resonant coupling parameters, which allow to provide estimates for the first resonant parameter and in turn also to provide bounds for resonant free regions. As application we provide a constructive approach to calculate the first resonant parameter for Yukawa type potentials in $\mathbb R^3$.
We consider hyperbolic equations with time-dependent coefficients and develop an abstract framework to derive the asymptotic behaviour of the representation of solutions for large times. We are dealing with generic situations where the large-time asymptotics is of hyperbolic type. Our approach is based on diagonalisation procedures combined with asymptotic integration arguments.
Scatterings of electrons at quasiparticles or photons are very important for many topics in solid state physics, e.g., spintronics, magnonics or photonics, and therefore a correct numerical treatment of these scatterings is very important. For a quantum-mechanical description of these scatterings Fermi's golden rule is used in order to calculate the transition rate from an initial state to a final state in a first-order time-dependent perturbation theory. One can calculate the total transition rate from all initial states to all final states with Boltzmann rate equations involving Brillouin zone integrations. The numerical treatment of these integrations on a finite grid is often done via a replacement of the Dirac delta distribution by a Gaussian. The Dirac delta distribution appears in Fermi's golden rule where it describes the energy conservation among the interacting particles. Since the Dirac delta distribution is a not a function it is not clear from a mathematical point of view that this procedure is justified. We show with physical and mathematical arguments that this numerical procedure is in general correct, and we comment on critical points.
In this paper we consider a wave model with non-effective mass and dissipation terms and provide asymptotic descriptions of its representation of solutions. In particular we conclude sharp estimates for a corresponding energy and estimates of dispersive type.
In this paper we prove Lp multiplier theorems for invariant and non-invariant operators on compact Lie groups in the spirit of the well-known Hormander-Mikhlin theorem on Rn and its variants on tori Tn. We also give applications to a-priori estimates for non-hypoelliptic operators. Already in the case of tori we get an interesting refinement of the classical multiplier theorem.
In this paper we develop the functional calculus for elliptic operators on compact Lie groups without the assumption that the operator is a classical pseudo-differential operator. Consequently, we provide a symbolic descriptions of complex powers of such operators. As an application, we give a constructive symbolic proof of the Gårding inequality for operators in $(ρ,δ)$-classes in the setting of compact Lie groups.
In this note we provide some precise estimates explaining the diffusive structure of partially dissipative systems with time-dependent coefficients satisfying a uniform Kalman rank condition. Precisely, we show that under certain (natural) conditions solutions to a partially dissipative hyperbolic system are asymptotically equivalent to solutions of a corresponding parabolic equation. The approach is based on an elliptic WKB analysis for small frequencies in combination with exponential stability for large frequencies due to results of Beauchard and Zuazua and arguments of perturbation theory.
In this note we announce Lp multiplier theorems for invariant and non-invariant operators on compact Lie groups in the spirit of the well-known Hormander-Mikhlin theorem on Rn and its variants on tori Tn. Applications are given to the mapping properties of pseudo-differential operators on Lp-spaces and to a-priori estimates for non-hypoelliptic operators.
These notes provide an introduction to and a survey on recent results about the long-time behaviour of solutions to hyperbolic partial differential equations with time-dependent coefficients. Particular emphasis is given also to questions about the sharpness of estimates. The selection of materials is based on the mini-courses taught by the first author at the CRM, Barcelona, and by the second author at Aalto University, Helsinki, both in 2011.
In this note we develop tools to study the Cauchy problem for the system of thermo-elasticity in higher dimensions. The theory is developed for general homogeneous anisotropic media under non-degeneracy conditions. For degenerate cases a method of treatment is sketched and for the cases of cubic media and hexagonal media detailed studies are provided.
This paper is devoted to the study of time-dependent hyperbolic systems and the derivation of dispersive estimates for their solutions. It is based on a diagonalisation of the full symbol within adapted symbol classes in order to extract the essential information about representations of solutions. This is combined with a multi-dimensional van der Corput lemma to derive dispersive estimates.
In this note we provide dispersive estimates for Fourier integrals with parameter-dependent phase functions in terms of geometric quantities of associated families of Fresnel surfaces. The results are based on a multi-dimensional van der Corput lemma due to the first author. Applications to dispersive estimates for hyperbolic systems and scalar higher order hyperbolic equations are also discussed.
In this paper we give several global characterisations of the Hormander class of pseudo-differential operators on compact Lie groups. The result is applied to give criteria for the ellipticity and the global hypoellipticity of pseudo-differential operators in terms of their matrix-valued full symbols. Several examples of the first and second order globally hypoelliptic differential operators are given. Where the global hypoelliptiticy fails, one can construct explicit examples based on the analysis of the global symbols.
The aim of this exposition is to explain basic ideas behind the concept of diffusive wavelets on spheres in the language of representation theory of Lie groups and within the framework of the group Fourier transform given by Peter-Weyl decomposition of $L^2(G)$ for a compact Lie group $G$. After developing a general concept for compact groups and their homogeneous spaces we give concrete examples for tori -which reflect the situation on $R^n$- and for spheres $S^2$ and $S^3$.
We consider wave models with lower order terms and recollect some recent results on energy and dispersive estimates for their solution based on symbolic type estimates for coefficients and partly stabilisation conditions. The exposition is complemented by a collection of open problems.