arXiv subjects

Hellmut Baumgärtel

Publications and source records attributed to Hellmut Baumgärtel.

9 recordsLinked to original sources

arXiv2018

Remarks to the Resonance-Decay Problem in Quantum Mechanics from a mathematical point of view

The description of bumps in scattering cross-sections by Breit-Wigner amplitudes led in the framework of the mathematical Physics to its formulation as the so-called Resonance-Decay Problem. It consists of a spectraltheoretical component and the connection of this component with the construction of decaying states. First the note quotes a solution for scattering systems, where the absolutely continuous parts of the Hamiltonians are semi-bounded and the scattering matrix is holomorphic in the upper half plane. This result uses the approach developed by Lax and Phillips, where the energy scale is extended to the whole real axis. The relationship of the spectraltheoretical part of its solution and corresponding solutions obtained by other approaches is explained in the case of the Friedrichs model. A No-Go theorem shows the impossibility of the total solution within the specific framework of non-relativistic quantum mechanics. This points to the importance of the Lax-Phillips approach. At last, a solution is presented, where the scattering matrix is meromorphic in the upper half plane.

math-ph↗

arXiv2017

On the border lines between the regions of distinct solution type for solutions of the Friedmann equation satisfying the Hubble condition

It is well-known that there are four distinct basic types (two Big Bang types, Lemaitre and Big Crunch type) for solutions of the general Friedmann equation with positive cosmological constant, where radiation and matter do not couple (see e.g. [2, p.7]. In that paper the system of case distinction parameters contains a "critical radiation parameter" $σ_{cr}$. The present note contains the constructive description of the so-called {\em border lines} between Big Bang/Big Crunch type and Big Bang/Lemaitre type for so-called Hubble solutions of the Friedmann equation by two smooth function branches, expressing the cosmological constant as unique functions of the matter and radiation density (which is considered as a parameter). These functions satisfy simple asymptotic relations w.r.t. the matter density. They are constructed as the solutions of the equation $σ=σ_{cr}.

gr-qc↗

arXiv2015

On the frontier lines between the regions of invariant solution type for solutions of the Friedmann equation satisfying the Hubble condition

It is well-known that there are four distinct basic types (two BigBang types, Lemaitre and BigCrunch type) for solutions of the general Friedmann equation with positive cosmological constant, where radiation and matter do not couple. In the note it is shown that the "frontier lines" between BigBang/BigCrunch and BigBang/Lemaitre are given by two smooth function branches, expressing the cosmological constant as unique functions of the matter and radiation density, which satisfy a simple asymptotic relation w.r.t. the matter density. The proof is based on the solution of the equation $σ=σ_{cr}$, where $σ$ is the radiation invariant of the Friedmann equation and $σ_{cr}$ the "critical radiation parameter" (see [3]).

gr-qc↗

arXiv2011

The general Friedmann equation: a mathematical point of view

The note presents a classification of the relevant distinct types of solutions of the general Friedmann equation without assuming a priori restrictions for the parameters occurring in this equation. The emphasis is on the case of a non-vanishing cosmological constant. The classification uses algebraic criteria. The result is: There are four distinct basic types of models. Explicit formulas for decisive terms are given. Characteristic mutual relations of cosmological constant, mass and radiation density to distinguish between the models are calculated.

astro-ph.CO↗

arXiv2010

Decay Semigroups for the Resonances of Quantum Mechanical Scattering Systems

For selected classes of quantum mechanical Hamiltonians a canonical association of a decay semigroup is presented. The spectrum of the generator of this semigroup is a pure eigenvalue spectrum and it coincides with the set of all resonances. The essential condition for the results is the meromorphic continuability of the scattering matrix onto $\Bbb{C}\setminus(-\infty,0]$ and the rims $\Bbb{R}_{-}\pm i0$. Further finite multiplicity is assumed. The approach is based on an adaption of the Lax-Phillips scattering theory to semi-bounded Hamiltonians. It is applied to trace class perturbations with analyticity conditions. A further example is the potential scattering for central-symmetric potentials with compact support and angular momentum 0.

math-ph↗

arXiv2006

The eigenvalue problem for the resonances of the infinite-dimensional Friedrichs model on the positive half line with Hilbert-Schmidt perturbations

A Gelfand triplet for the Hamiltonian H of the infinite-dimensional Friedrichs model on the positive half line with Hilbert-Schmidt perturbations is constructed such that exactly the resonances (poles of the inverse of the Livsic-matrix) are eigenvalues of the extension H^{\times} of H. The corresponding eigenantilinear forms are calculated explicitly. Using the wave matrices for the Abelian wave (Möller) operators the corresponding eigenantilinear forms for the unperturbed Hamiltonian $H_{0}$ turn out to be of pure Dirac type and can be characterized by their corresponding Gamov vector which is uniquely determined by restriction to the intersection of the Gelfand space for $H_{0}$ with $P_{+}H^{2}_{+}$, where $H^{2}_{+}$ is the Hardy space of the upper half plane. Simultaneously, this restriction yields a truncation of the unitary evolution $t\to e^{-itH_{0}}$ to the well-known decay semigroup for $t\geq 0$ of the Toeplitz type on $P_{+}H^{2}_{+}$. That is, exactly those eigenvectors $λ\to k(λ-ζ)^{-1}$, $k$ element of the multiplicity space K, of the decay semigroup have an extension to an eigenantilinear form for $H_{0}$ hence for H if $ζ$ is a resonance and k is from that subspace of K which is uniquely determined by its corresponding Dirac type antilinear form. Moreover, the scattering matrix which is meromorphic in the lower half plane has only simple poles there and the main part of its Laurent representation is a linear combination of Gamov vectors.

math-ph↗

arXiv2005

Generalized Eigenvectors for Resonances in the Friedrichs Model and Their Associated Gamov Vectors

A Gelfand triplet for the Hamiltonian H of the Friedrichs model on R with finite-dimensional multiplicity space K, is constructed such that exactly the resonances (poles of the inverse of the Livsic-matrix) are (generalized) eigenvalues of H. The corresponding eigen-antilinearforms are calculated explicitly. Using the wave matrices for the wave (Moller) operators the corresponding eigen-antilinearforms on the Schwartz space S for the unperturbed Hamiltonian are also calculated. It turns out that they are of pure Dirac type and can be characterized by their corresponding Gamov vector, which is uniquely determined by restriction of S to the intersection of S with the Hardy space of the upper half plane. Simultaneously this restriction yields a truncation of the generalized evolution to the well-known decay semigroup of the Toeplitz type for the positive half line on the Hardy space. That is: exactly those pre-Gamov vectors (eigenvectors of the decay semigroup) have an extension to a generalized eigenvector of H if the eigenvalue is a resonance and if the multiplicity parameter k is from that subspace of K which is uniquely determined by its corresponding Dirac type antilinearform.

math-ph↗

arXiv2004

Duality of compact groups and Hilbert C-systems for C-algebras with a nontrivial center

In the present paper we prove a duality theory for compact groups in the case when the C*-algebra A, the fixed point algebra of the corresponding Hilbert C*-system (F,G), has a nontrivial center Z and the relative commutant satisfies the minimality condition A.'\cap F = Z as well as a technical condition called regularity. The abstract characterization of the mentioned Hilbert C*-system is expressed by means of an inclusion of C*-categories T_\c < T, where T_cis a suitable DR-category and T a full subcategory of the category of endomorphisms of A. Both categories have the same objects and the arrows of T can be generated from the arrows of T_cand the center Z. A crucial new element that appears in the present analysis is an abelian group C(G), which we call the chain group of G, and that can be constructed from certain equivalence relation defined on G^, the dual object of G. The chain group, which is isomorphic to the character group of the center of G, determines the action of irreducible endomorphisms of A when restricted to Z. Moreover, C(G) encodes the possibility of defining a symmetry $ε$ also for the larger category T of the previous inclusion.

math.OA↗

arXiv2002

Twisted duality of the CAR-Algebra

We give a complete proof of the twisted duality property M(q)'= Z M(q^\perp) Z* of the (self-dual) CAR-Algebra in any Fock representation. The proof is based on the natural Halmos decomposition of the (reference) Hilbert space when two suitable closed subspaces have been distinguished. We use modular theory and techniques developed by Kato concerning pairs of projections in some essential steps of the proof. As a byproduct of the proof we obtain an explicit and simple formula for the graph of the modular operator. This formula can be also applied to fermionic free nets, hence giving a formula of the modular operator for any double cone.

math-ph↗