Pseudodifferential operators on Hilbert space riggings with associated psi *-algebras and generalized Hörmander classes

The present thesis is concerned with certain aspects of differential and pseudodifferential operators on infinite dimensional spaces. We aim to generalize classical operator theoretical concepts of pseudodifferential operators on finite dimensional spaces to the infinite dimensional case. At first we summarize some facts about the canonical Gaussian measures on infinite dimensional Hilbert space riggings. Considering the naturally unitary group actions in $L^2(H_-,gamma)$ given by weighted shifts and multiplication with $e^{iSkp{t}{cdot}_0}$ we obtain an unitary equivalence $F$ between them. In this sense $F$ can be considered as an abstract Fourier transform. We show that $F$ coincides with the Fourier-Wiener transform. Using the Fourier-Wiener transform we define pseudodifferential operators in Weyl- and Kohn-Nirenberg form on our Hilbert space rigging. In the case of this Gaussian measure $gamma$ we discuss several possible Laplacians, at first the Ornstein-Uhlenbeck operator and then pseudo-differential operators with negative definite symbol. In the second case, these operators are generators of $L^2_gamma$-sub-Markovian semi-groups and $L^2_gamma$-Dirichlet-forms. In 1992 Gramsch, Ueberberg and Wagner described a construction of generalized Hörmander classes by commutator methods. Following this concept and the classical finite dimensional description of $Psi_{ro,delta}^0$ ($0leqdeltaleqroleq 1$, $delta< 1$) in the $C^*$-algebra $L(L^2)$ by Beals and Cordes we construct in both cases generalized Hörmander classes, which are $Psi^*$-algebras. These classes act on a scale of Sobolev spaces, generated by our Laplacian. In the case of the Ornstein-Uhlenbeck operator, we prove that a large class of continuous pseudodifferential operators considered by Albeverio and Dalecky in 1998 is contained in our generalized Hörmander class. Furthermore, in the case of a Laplacian with negative definite symbol, we develop a symbolic calculus for our operators. We show some Fredholm-criteria for them and prove that these Fredholm-operators are hypoelliptic. Moreover, in the finite dimensional case, using the Gaussian-measure instead of the Lebesgue-measure the index of these Fredholm operators is still given by Fedosov's formula. Considering an infinite dimensional Heisenberg group rigging we discuss the connection of some representations of the Heisenberg group to pseudo-differential operators on infinite dimensional spaces. We use this connections to calculate the spectrum of pseudodifferential operators and to construct generalized Hörmander classes given by smooth elements which are spectrally invariant in $L^2(H_-,gamma)$. Finally, given a topological space $X$ with Borel measure $mu$, a locally compact group $G$ and a representation $B$ of $G$ in the group of all homeomorphisms of $X$, we construct a Borel measure $mu_s$ on $X$ which is invariant under $B(G)$.

Messung der Zerfallsasymmetrien der radiativen Hyperonzerfälle Xi 0 Lambda gamma und Xi 0 Sigma 0 gamma mit dem NA48/1-Experiment

Der radiative Zerfall eines Hyperons in ein leichteres Hyperon und ein Photon erlaubt eine Untersuchung der Struktur der elektroschwachen Wechselwirkung von Hadronen. Dazu wird die Zerfallsasymmetrie $alpha$ betrachtet. Sie beschreibt die Verteilung des Tochterhyperons bezüglich der Polarisation $vec{P}$ des Mutterhyperons mit $dN / d cos(Theta) propto 1 + alpha |vec{P}| cos(Theta)$, wobei $Theta$ der Winkel zwischen $vec{P}$ und dem Impuls des Tochterhyperons ist. Von besonderem Interesse ist der radiative Zerfall $Xi^0 to Lambda gamma$, für den alle Rechnungen auf Quarkniveau eine positive Asymmetrie vorhersagen, wohingegen bisher eine negative Asymmetrie von $alpha_{Lambda gamma} = -0,73 +- 0,17$ gemessen wurde. Ziel dieser Arbeit war es, die bisherigen Messungen zu überprüfen und die Asymmetrie mit einer deutlich höheren Präzision zu bestimmen. Ferner wurden die Zerfallsasymmetrie des radiativen Zerfalls $Xi^0 to Sigma^0 gamma$ ermittelt und zum Test der angewandten Analysemethode der gut bekannte Zerfall $Xi^0 to Lambda pi^0$ herangezogen. Während der Datennahme im Jahr 2002 zeichnete das NA48/1-Experiment am CERN gezielt seltene $K_S$- und Hyperonzerfälle auf. Damit konnte der weltweit größte Datensatz an $Xi^0$-Zerfällen gewonnen werden, aus dem etwa 52.000 $Xi^0 to Lambda gamma$-Zerfälle, 15.000 $Xi^0 to Sigma^0 gamma$-Zerfälle und 4 Mill. $Xi^0 to Lambda pi^0$-Zerfälle mit nur geringem Untergrund extrahiert wurden. Ebenso wurden die entsprechenden $antiXi$-Zerfälle mit etwa einem Zehntel der obigen Ereigniszahlen registriert. Die Bestimmung der Zerfallsasymmetrien erfolgte durch den Vergleich der gemessene Daten mit einer detaillierten Detektorsimulation und führte zu den folgenden Resultaten dieser Arbeit: $alpha_{Lambda gamma} = -0,701 +- 0,019_{stat} +- 0,064_{sys}$, $alpha_{Sigma^0 gamma} = -0,683 +- 0,032_{stat} +- 0,077_{sys}$, $alpha_{Lambda pi^0} = -0,439 +- 0,002_{stat} +- 0,056_{sys}$, $alpha_{antiLambda gamma} = 0,772 +- 0,064_{stat} +- 0,066_{sys}$, $alpha_{antiSigma^0 gamma} = 0,811 +- 0,103_{stat} +- 0,135_{sys}$, $alpha_{antiLambda pi^0} = 0,451 +- 0,005_{stat} +- 0,057_{sys}$. Somit konnte die Unsicherheit der $Xi^0 to Lambda gamma$-Zerfallsasymmetrie auf etwa ein Drittel reduziert werden. Ihr negatives Vorzeichen und damit der Widerspruch zu den Vorhersagen der Quarkmodellrechnungen ist so zweifelsfrei bestätigt. Mit den zum ersten Mal gemessenen $antiXi$-Asymmetrien konnten zusätzlich Grenzen auf eine mögliche CP-Verletzung in den $Xi^0$-Zerfällen, die $alpha_{Xi^0} neq -alpha_{antiXi}$ zur Folge hätte, bestimmt werden.

Sorption and direct speciation of neptunium(V) on aluminium oxide and montmorillonite

This study comprised batch experiments, direct speciation studies via EXAFS, and modelling with the 2SPNE SC/CE model to elucidate the mechanisms of Np(V) sorption on montmorillonite and, for reference, on γ-Al2O3. The sorption of pM 239Np(V) and µM 237Np(V) on montmorillonite (STx-1, 4 g/L) and γ-Al2O3 (0.5 g/L) was studied at room temperature in the presence and absence of ambient CO2 covering a pH-range from 2.5 (STx-1) or 5 (γ-Al2O3) to 10.5 with 0.01 or 0.1M NaClO4 as background electrolyte. The Np(V) uptake was determined by γ spectroscopy of the supernatants and calculated as percentage as well as distribution coefficient Kd. Sorption starts from pH ~6 and, under exclusion of CO2, increases continuously, while, in the presence of ambient air, it reaches a maximum at pH ~8.5 (γ-Al2O3: log Kd max ≈ 4 mL/g; STx-1: log Kd max ≈ 2.7 mL/g). Beyond that it decreases again due to the formation of queous neptunium carbonate complexes. Furthermore, neptunium sorption on montmorillonite is influenced by ionic strength at pH <6 through ion exchange processes pointing towards the formation of outer-sphere surface complexes there. Isotherms measured at the sorption maximum showed the precipitation of resumably neptunium carbonate complexes above 3∙10^-5 M under ambient air conditions. Additionally, they indicated progressive saturation of the sorption sites of γ-Al2O3. At selected pH (STx-1: 5.0, 7.0, 8.0, 8.5, 9.0, 9.5; γ-Al2O3: 8.5, 9.5) EXAFS samples were prepared as wet pastes with µM 237Np and measured at room temperature in fluorescence mode at ANKA and ESRF. Several spectra were averaged and analysed with EXAFSPAK and FEFF 8.20 employing models of NaNpO2(CO3) or soddyite, (UO2)2SiO4∙2(H2O). The shorter atomic distances of the neptunyl ion at pH 5 compared to the others hinted at the retention of the hydration shell and, thus, at outer-sphere sorption. On average the bond lengths for Np(V) sorbed on STx-1 at high pH were Oax ≈ 1.84 Å and Oeq ≈ 2.53 Å. At high pH, ternary neptunyl carbonate surface complexes could be identified for montmorillonite (C ≈ 3.00 Å), but not for γ-Al2O3, where an interaction of neptunium with the aluminium surface atoms according to the soddyite model gave better agreement with the experimental data. However, neither structure as suggested by the two models could be excluded for both systems rendering a combination most likely. Modelling of the sorption data provided further evidence for the existence of ternary neptunium carbonate surface complexes in both cases. The results of this study can aid environmental risk assessment for clay-based nuclear waste repositories by providing valuable input data for simulations of radionuclide migration from a final disposal site.