Foi calculado microscopicamente a seção de choque total de reação para os sistemas ANTPOT. 12 C + ANTPOT. 12 C, ANTPOT. 1-2 C+ ANTPOT. 40 Ca, ANTPOT. 12 C + ANTPOT. 90 Zr, ANTPOT. 12 C + ANTPOT. 208 Pb, ANTPOT. 40 Ca + ANTPOT. 40 Ca, ANTPOT. 40 Ca + ANTPOT. 208 Pb, ANTPOT. 90 Zr + ANTPOT. 90 Zr, ANTPOT. 90 Zr + ANTPOT. 208 Pb e ANTPOT. 208 Pb + ANTPOT. 208 Pb numa ampla faixa de energia. Foi usada a expressão WKB para a defasagem imaginária, na representação do parâmetro do impacto. A parte imaginária do potencial óptico foi construída, usando o primeiro termo da teoria de espalhamento múltiplo, incorporando na sua expressão o principio de Pauli. A inclusão da interação nuclear e coulombiana mostrou ser importante. Os resultados teóricos não se apresentam em bom acordo com os poucos dados experimentais existentes, a baixas energias. Este resultado foi atribuído a fraca absorção contida no potencial imaginário, que contém o processo de knock-out quase-livre como mecanismo dominante de reação. / The total reaction cross section of the systems 12C +12C, 12C + 40Ca, 12C + 90Zr, 12C + 208 Pb, 40Ca + 40Ca, 40Ca + 208Pb, 90Zr + 90Zr, 90 Zr + 208 Pb and 20B Pb + 208 Pb for a wide range of energies has been calculated microscopically. A WKB expression for the imaginary part of the optical potential has been constructed by using the first term of multiple scattering theory with the effect of Pauli blocking incorporated into it. The inclusion of the nuclear and Coulomb interactions is shown to be important. The theoretical results do not show very good agreement with the experimental data at lower energies. This is attributed to the weak absorption contained in the imaginary potential of the tpApB interaction, wich contained only quasi-free knock-out as the dominant reaction mechanism.

La technique de la microscopie à grille de balayage (SGM) consiste à mesurer la conductance d'un gaz bidimensionnel d'électrons (2DEG) sous l'influence d'une pointe balayant la surface de l'échantillon. Dans ce travail, une approche analytique complétée par des simulations numériques est développée pour étudier la relation entre les mesures SGM et les propriétés électroniques locales dans des systèmes mésoscopiques. La correspondance entre la réponse SGM et la densité locale partielle (PLDOS) est étudiée pour un contact quantique entouré d’un 2DEG en présence ou en absence de désordre, pour une pointe perturbative ou non perturbative. Une correspondance SGM-PLDOS parfaite est trouvée pour des transmissions entières et des pointes locales. La dégradation de la correspondance en dehors de cette situation est étudiée. D’autre part, la liaison entre la réponse SGM et la transformée de Hilbert de la densité locale est discutée. Pour étudier le rôle de la force de la pointe sur la conductance SGM, une formule analytique donnant la conductance totale est obtenue. Dans le cas d'une pointe à taille finie nous proposons une méthode basée sur les fonctions de Green permettant de calculer la conductance en connaissant les propriétés non-perturbées. En plus, nous avons étudié la dépendance des branches de la PLDOS en fonction de l’énergie de Fermi. / The scanning gate microscopy (SGM) technique consists in measuring the conductance of a two dimensional electron gas (2DEG) under the influence of a scanning tip. In this work, an analytical approach complemented by numerical simulations is developed to study the connection between SGM measurements and local electronic properties in mesoscopic devices. The connection between the SGM response and the partial local density of states (PLDOS) is studied for the case of a quantum point contact surrounded by clean or disordered 2DEG for perturbative or non-perturbative, local or extended tips. An SGM-PLDOS correspondence is found for integer transmissions and local tips. The degradation of this correspondence out of these conditions is studied. Moreover, a presumed link between the SGM response and the Hilbert transform of the LDOS is discussed. To study the role of the tip strength, an analytical formula giving the full conductance in the case of local tips is obtained. Furthermore, a Green function method enabling to calculate the quantum conductance in the presence of a finite size tip in terms of the unperturbed properties is proposed. Finally the dependence of the PLDOS branches on the Fermi energy is studied.

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<p>This thesis consists of two papers on the spectral geometry of locally symmetric spaces of Riemannian and Lorentzian signature. Both works are concerned with the idea of relating analysis on such spaces to structures on their boundaries.</p><p>The first paper is motivated by a conjecture of Patterson on the Selberg zeta function of Kleinian groups. We consider geometrically finite hyperbolic cylinders with non-compact Riemann surfaces of finite area as cross sections. For these cylinders, we present a detailed investigation of the Bunke-Olbrich extension operator under the assumption that the cross section of the cylinder has one cusp. We establish the meromorphic continuation of the extension of Eisenstein series and incomplete theta series through the limit set. Furthermore, we derive explicit formulas for the residues of the extension operator in terms of boundary values of automorphic eigenfunctions.</p><p>The motivation for the second paper comes from conformal geometry in Lorentzian signature. We prove the existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces.</p>

Le sujet de cette thèse est l’étude de certaines équations de physique mathématique. Dans un premier temps, on étudie les résonances et la fonction de décalage spectral pour les opérateurs de Dirac semi-classique et de Schrödinger magnétique en dimension 3. On dé?nit les résonances comme des valeurs propres d’un opérateur non-autoadjoint obtenu par distortion complexe. Pour l’opérateur de Dirac, on majore le nombre de résonances par O(h-3) où h ? 0 est le paramètre semi-classique. Dans le cas de Schrödinger magnétique, l’opérateur de référence génère des valeurs propres de multipli- cité in?nie plongées dans le spectre continu. Dans une couronne centrée en une de ces valeurs propres et de rayons (r, 2r), on établit une borne supérieure, quand r ? 0, du nombre de résonances. Une approximation de type Breit-Wigner de la dérivée de la fonction de décalage spectral en fonction des résonances et une formule de trace locale sont obtenues pour ces deux opérateurs. De plus, on prouve une formule asymptotique de Weyl pour la fonction de décalage spectral pour l’opérateur de Dirac avec un potentiel électro-magnétique. Dans un deuxième temps, on s’intéresse à l’opérateur de Dirac semi-classique en dimension 1 avec un potentiel ayant des limites constantes mais pas nécessairement les mêmes à ±8. En utilisant la méthode BKW complexe, on construit des solutions analytiques de l’opérateur de Dirac. On étudie la théorie de la di?usion en fonction des solutions entrantes et sortantes. On obtient une asymptotique semi-classique de la matrice de di?usion dans di?érents cas, notamment dans le cas où le paradoxe de Klein apparaît. Le calcul des valeurs propres et des résonances est aussi traité pour l’opérateur de Dirac semi-classique unidimensionnel. / In this thesis, we consider equations of mathematical physics. First, we study the reso- nances and the spectral shift function for the semi-classical Dirac operator and the magnetic Schrö- dinger operator in three dimensions. We de?ne the resonances as the eigenvalues of a non-selfadjoint operator obtained by complex distortion. For the Dirac operator, we establish an upper bound O(h-3), as the semi-classical parameter h tends to 0, for the number of resonances. In the Schrödinger magne- tic case, the reference operator has in?nitely many eigenvalues of in?nite multiplicity embedded in its continuous spectrum. In a ring centered at one of this eigenvalues with radiuses (r, 2r), we establish an upper bound, as r tends to 0, of the number of the resonances. A Breit-Wigner approximation formula for the derivative of the spectral shift function related to the resonances and a local trace formula are obtained for the considered operators. Moreover, we prove a Weyl-type asymptotic of the SSF for the Dirac operator with an electro-magnetic potential. Secondly, we consider the semi-classical Dirac ope- rator on R with potential having constant limits, not necessarily the same at ±8. Using the complex WKB method, we construct analytic solutions for the Dirac operator. We study the scattering theory in terms of incoming and outgoing solutions. We obtain an asymptotic expansion, with respect to the semi-classical parameter h, of the scattering matrix in di?erent cases, in particular, in the case when the Klein paradox occurs. Quantization conditions for the resonances and for the eigenvalues of the one-dimensional Dirac operator are also obtained.

In dieser Arbeit stellen wir verschiedene Tests der (M)atrixtheorie-Vermutung vor. Die (M)atrixtheorievermutung besagt, dass die Dynamik von M-Theorie, der Urtheorie, die alle bekannten Stringtheorien und auch elfdimensionale Supergravitation als bestimmte Grenzfaelle enthalten soll, durch ein quantenmechanisches Matrixmodell gegeben ist. Insbesondere untersuchen wir Streuprozesse sowohl aus Sicht des Matrixmodells, als auch aus Sicht der Supergravitation, und vergleichen die resultierenden S-Matrixelemente. Wir finden beeindruckende Uebereinstimmung zwischen den beiden Theorien, solange wir uns auf klassische Supergravitation beschraenken. Sobald wir auch Quanteneffekte auf der Supergravitationsseite einbeziehen, hat diese Uebereinstimmung keinen Bestand. Des weiteren untersuchen wir die Frage, ob Loesungen der klassischen Matrixmodell-Bewegungsgleichungen mit Impulsuebertrag existieren, und finden eine negative Antwort. / In this thesis, we present several tests of the M(atrix)-Model conjecture that asserts that the dynamics of M-Theory, the eleven-dimensional Ur-theory containing all known string theories and also eleven-dimensional supergravity in specific limits, is given by a quantum mechanical matrix model. In particular, scattering processes are analyzed both from the M(atrix)-Model and from the supergravity perspective and the corresponding S-matrix elements are compared. We find impressive agreement between these two theories as long as only classical supergravity is considered. If one includes also quantum effects on the supergravity side, the agreement does not persist. In addition to these calculations, the question of the existence of classical solutions to the M(atrix)-Model equations of motion with momentum transfer is addressed and answered negatively.

This thesis consists of two papers on the spectral geometry of locally symmetric spaces of Riemannian and Lorentzian signature. Both works are concerned with the idea of relating analysis on such spaces to structures on their boundaries. The first paper is motivated by a conjecture of Patterson on the Selberg zeta function of Kleinian groups. We consider geometrically finite hyperbolic cylinders with non-compact Riemann surfaces of finite area as cross sections. For these cylinders, we present a detailed investigation of the Bunke-Olbrich extension operator under the assumption that the cross section of the cylinder has one cusp. We establish the meromorphic continuation of the extension of Eisenstein series and incomplete theta series through the limit set. Furthermore, we derive explicit formulas for the residues of the extension operator in terms of boundary values of automorphic eigenfunctions. The motivation for the second paper comes from conformal geometry in Lorentzian signature. We prove the existence and uniqueness of a sequence of differential intertwining operators for spherical principal series representations, which are realized on boundaries of anti de Sitter spaces. Algebraically, these operators correspond to homomorphisms of generalized Verma modules. We relate these families to the asymptotics of eigenfunctions on anti de Sitter spaces.

This thesis reports our work on transport related problems in mesoscopic physics using analytical as well as numerical techniques. Some of the problems we studied are: effect of interactions and static impurities on the conductance of a ballistic quantum wire[1], aspects of quantum charge pumping [2, 3, 4], DC and AC conductivity of a (dissipative) quantum Hall (edge) line junctions[5, 6], and junctions of three or more Luttinger liquid (LL)quantum wires[7]. This thesis begins with an introductory chapter which gives a brief glimpse of the underlying physical systems and the ideas and techniques used in our studies. In most of the problems we will look at the physical effects caused by e-e interactions and static scattering processes. In the second chapter we study the effects of a static impurity and interactions on the conductance of a 1D-quantum wire numerically. We use the non-equilibrium Green’s function (NEGF) formalism along with a self-consistent Hartree-Fock approximation to numerically study the effects of a single impurity and interactions between the electrons (with and without spin) on the conductance of a quantum wire [1]. We study the variation of the conductance with the wire length, temperature and the strength of the impurity and electron-electron interactions. We find our numerical results to be in agreement with the results obtained from the weak interaction RG analysis. We also discover that bound states produce large density deviations at short distances and have an appreciable effect on the conductance which is not captured by the renormalization group analysis. In the third chapter we use the equations of motion (EOM) for the density matrix and Floquet scattering theory to study different aspects of charge pumping of non-interacting electrons in a one-dimensional system. We study the effects of the pumping frequency, amplitude, band filling and finite bias on the charge pumped per cycle, and the spectra of the charge and energy currents in the leads[2]. The EOM method works for all values of parameters, and gives the complete time-dependences of the current and charge at any site of the system. In particular we study a system with oscillating impurities at several sites and our results agree with Floquet and adiabatic theory where these are applicable, and provides support for a mechanism proposed elsewhere for charge pumping by a traveling potential wave in such systems. For non-adiabatic and strong pumping, the charge and energy currents are found to have a marked asymmetry between the two leads, and pumping can work even against a substantial bias. We also study one-parameter charge pumping in a system where an oscillating potential is applied at one site while a static potential is applied in a different region [3]. Using Floquet scattering theory, we calculate the current up to second order in the oscillation amplitude and exactly in the oscillation frequency. For low frequency, the charge pumped per cycle is proportional to the frequency and therefore vanishes in the adiabatic limit. If the static potential has a bound state, we find that such a state has a significant effect on the pumped charge if the oscillating potential can excite the bound state into the continuum states or vice versa. In the fourth chapter we study the current produced in a Tomonaga-Luttinger liquid (TLL) by an applied bias and by weak, point-like impurity potentials which are oscillating in time[4]. We use bosonization to perturbatively calculate the current up to second order in the impurity potentials. In the regime of small bias and low pumping frequency, both the DC and AC components of the current have power law dependences on the bias and pumping frequencies with an exponent 2K−1 for spinless electrons, where Kis the interaction parameter. For K<1/2, the current grows large for special values of the bias. For non-interacting electrons with K= 1, our results agree with those obtained using Floquet scattering theory for Dirac fermions. We also discuss the cases of extended impurities and of spin-1/2 electrons. In chapter five, we present a microscopic model for a line junction formed by counter or co-propagating single mode quantum Halledges corresponding to different filling factors and calculate the DC [5] and AC[6] conductivity of the system in the diffusive transport regime. The ends of the line junction can be described by two possible current splitting matrices which are dictated by the conditions of both lack of dissipation and the existence of chiral commutation relations between the outgoing bosonic fields. Tunneling between the two edges of the line junction then leads to a microscopic understanding of a phenomenological description of line junctions introduced by Wen. The effect of density-density interactions between the two edges is considered exactly, and renormalization group (RG) ideas are used to study how the tunneling parameter changes with the length scale. The RG analysis leads to a power law variation of the conductance of the line junction with the temperature (or other energy scales) and the line junction may exhibit metallic or insulating phase depending on the strength of the interactions. Our results can be tested in bent quantum Hall systems fabricated recently. In chapter six, we study a junction of several Luttinger Liquid (LL) wires. We use bosonization with delayed evaluation of boundary conditions for our study. We first study the fixed points of the system and discuss RG flow of various fixed points under switching of different ‘tunneling’ operators at the junction. Then We study the DC conductivity, AC conductivity and noise due to tunneling operators at the junction (perturbative).We also study the tunneling density of states of a junction of three Tomonaga-Luttinger liquid quantum wires[7]. and find an anomalous enhancement in the TDOS for certain fixed points even with repulsive e-e interactions.

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