This work is focused on the different supersymmetric extensions of the Landau model. We aim to fully solve each model and describe its energy levels, wavefunctions, Hilbert space and define a norm on it, as well as find symmetry generators and transformations with respect to them. Several possible generalizations were considered before. It was found for Landau model on the so called Superflag manifold as well as planar Superflag and Superplane Landau models that standard norm on the Hilbert space is not positive definite. Later for planar cases it was found that it is possible to fix this by introducing a new norm which will be invariant and positive definite. Surprisingly this procedure brings up "hidden" symmetries for the known super Landau models. In the dissertation we apply the same procedure for Landau model on superpshere and Superflag manifolds. It turns out that superpsherical Landau model is equivalent to the Superflag model with one of the parameters fixed. Because the model on superpshere can be recovered from the Superflag we will do calculations of corrected norm only for the Superflag. After this we develop a different generalization of the Superplane Landau model. Starting with Lagrangian in a superfield form we introduce two arbitrary functions of superfields K(Φ) and V(Φ) into the Lagrangian. We follow with the component form of Lagrangian. The quantization of the model is possible, and we will show that there is a reparametrization which turn equation of motion of the first scheme into the second set. Standard metric is again non-positive definite and we apply already known procedure to correct it. It will not be possible to solve Schrodinger equations in general with undefined K and V, so we consider one specific case which give us Landau model on a sphere with N = 2 supersymmetry, which put it apart from the superspherical Landau model, which have a superpshere for a target space but do not possess supersymmetry.

Relations between propagators and Green functions of Hamiltonians which are SUSY partners have been obtained. New exact propagators for the family of multi-well, time-dependent and non-hermitian potentials have been calculated. Non-conservative SUSY transformation has been studied in the case of multichannel Schrodinger equation with different thresholds. Spectrum (bound/virtual states and resonances) of the non-conservative SUSY partner of zero potential has been founded. Exactly solvable model of the magnetic induced Feshbach resonance has been constructed. This model was tested in the case of Rb$^{85}$. Conservative SUSY transformations of the first and the second order have been studied in the case of multichannel Schrodinger equation with equal thresholds. Transformations which introduce non-trivial coupling between scattering channels have been founded. The first order SUSY transformation which preserves $S$-matrix eigen-phase shifts and modifies mixing parameter has been founded in the case of two channel scattering with partial waves of different parities. In the case of coinciding parities we have found the second order SUSY transformation which preserves $S$-matrix eigen-phase shifts and modifies mixing parameter. Phenomenological two channel $^3S_1-^3D_1$ neutron-proton potential has been obtained by using single channel, phase equivalent and coupling SUSY transformations applied to zero potential.

The Atiyah-Singer index theorem, the Euler number, and the Hirzebruch signature are derived via the supersymmetric path integral. Concisely, the supersymmetric path integral is a combination of a bosonic and a femionic path integral. The action in the supersymmetric path integral includes here bosonic, fermionic- and isospin fields (backgroundfields), where the cross terms in the Lagrangian are nicely eliminated due to scaling of the fields and using techniques from spontaneous breaking of supersymmetry (that give rise to a mechanism, analogous to the Higgs-mechanism, but here regarding the so called superparticles instead).  Thus, the supersymmetric path integral is a product of three pathintegrals over the three given fields, respectively, that can be evaluated exactly by means of Gaussian integrals. The closely related Witten index is a measure of the failure of spontaneous breaking of supersymmetry. In addition, the basic concepts of supersymmetry breaking are reviewed.

京都大学 / 0048 / 新制・課程博士 / 博士(情報学) / 甲第22583号 / 情博第720号 / 新制||情||123(附属図書館) / 京都大学大学院情報学研究科数理工学専攻 / (主査)教授 中村 佳正, 教授 矢ヶ崎 一幸, 准教授 辻本 諭 / 学位規則第4条第1項該当 / Doctor of Informatics / Kyoto University / DFAM

Nonlinearities in quantum cascade lasers (QCL¡¯s) have wide applications in wavelength tunability and ultra-short pulse generation. In this thesis, optical nonlinearities in InGaAs/AlInAs-based mid-infrared (MIR) QCL¡¯s with quadruple resonant levels are investigated. Design optimization for the second-harmonic generation (SHG) of the device is presented. Performance characteristics associated with the third-order nonlinearities are also analyzed. The design optimization for SHG efficiency is obtained utilizing techniques from supersymmetric quantum mechanics (SUSYQM) with both material-dependent effective mass and band nonparabolicity. Current flow and power output of the structure are analyzed by self-consistently solving rate equations for the carriers and photons. Nonunity pumping efficiency from one period of the QCL to the next is taken into account by including all relevant electron-electron (e-e) and longitudinal (LO) phonon scattering mechanisms between the injector/collector and active regions. Two-photon absorption processes are analyzed for the resonant cascading triple levels designed for enhancing SHG. Both sequential and simultaneous two-photon absorption processes are included in the rate-equation model. The current output characteristics for both the original and optimized structures are analyzed and compared. Stronger resonant tunneling in the optimized structure is manifested by enhanced negative differential resistance. Current-dependent linear optical output power is derived based on the steady-state photon populations in the active region. The second-harmonic (SH) power is derived from the Maxwell equations with the phase mismatch included. Due to stronger coupling between lasing levels, the optimized structure has both higher linear and nonlinear output powers. Phase mismatch effects are significant for both structures leading to a substantial reduction of the linear-to-nonlinear conversion efficiency. The optimized structure can be fabricated through digitally grading the submonolayer alloys by molecular beam epitaxy (MBE). In addition to the second-order nonlinearity, performance characteristics brought by the third-order nonlinearities are also discussed, which include third-harmonic generation (THG) and intensity dependent (Kerr) refractive index. Linear to third-harmonic (TH) conversion efficiency is evaluated based on the phase-mismatched condition. The enhanced self-mode-locking (SML) effect over a typical three-level laser is predicted, which will stimulate further investigations of pulse duration shortening by structures with multiple harmonic levels.

Exact solution of the Schr&ouml / dinger equation with some potentials is obtained. The normal and supersymmetric cases are considered. Deformed ring-shaped potential is solved in the parabolic and spherical coordinates. By taking appropriate values for the parameter q, similar results are obtained for Hulth&eacute / n and exponential type screened potentials. Similarly, Morse, P&ouml / schl-Teller and Hulth&eacute / n potentials are solved for the supersymmetric case. Supersymmetric solution of PT-/non-PT-symmetric and non-Hermitian Morse potential is also studied. The Nikiforov-Uvarov and Hamiltonian Hierarchy methods are used in the calculations. Eigenfunctions and corresponding energy eigenvalues are calculated analytically. Results are in good agreement with ones obtained before.

Made available in DSpace on 2016-08-18T18:19:30Z (GMT). No. of bitstreams: 1 Dissertacao Genilson.pdf: 3934400 bytes, checksum: f88bea8a3bd3a04623dda92abb3f6064 (MD5) Previous issue date: 2012-08-17 / FUNDAÇÃO DE AMPARO À PESQUISA E AO DESENVOLVIMENTO CIENTIFICO E TECNOLÓGICO DO MARANHÃO / We studied the localization of fermionic fields in a ring-like topological defect constructed with a real scalar field ϕ and immersed in (3, 1)−dimensional spacetime. The process of localization of fermions in the ring is analyzed by studying a Yukawa-like coupling ¯ ΨF(ϕ)Ψ, where F(ϕ) is a function of the real scalar field. Using the usual γ−matrices in (1, 1)−dimensions, we express the Dirac spinor in terms of its chiral left-handed and right-handed components. The amplitudes of fermions support Schr¨odinger-like equations allowing a probabilistic interpretation of them. In our case, to analyze the existence of resonances, we consider the coupling F(ϕ) = (1 − ϕ2)2. The eigenvalue equations are solved using numerical procedures. The eigenvalues that characterize the resonances are obtained for the two chiralities that show exactly the same, confirming the qualitative analysis of the potential are supersymmetric partners. / Estudamos a localização de campos fermiônicos num defeito topológico tipo anel construído com um campo escalar real ϕ e imerso no espaço-tempo de (3, 1)−dimensões. O processo de localização de férmions no anel é analisado estudando um acoplamento tipo Yukawa ¯ΨF(ϕ)Ψ, onde F(ϕ) é uma função do campo escalar real. Utilizando as matrizes−Γ usuais em (1, 1) dimensões, dividimos o espinor de Dirac em termos das duas componentes de quiralidade de mão-esquerda e mão-direita. As amplitudes dos f´ermions suportam equações tipo-Schrodinger possibilitando uma interpretação probabilística das mesmas. Em nosso caso específico, para analisarmos a existência de ressonâncias consideramos o acoplamento F(ϕ) = (1 − ϕ2)2. As equações de autovalores são solucionadas usando procedimentos numéricos. Os autovalores que caracterizam as ressonâncias são obtidas para as duas quiralidades que se mostram exatamente iguais confirmando a análise qualitativa dos potenciais serem parceiros supersimétricos.

Relations between propagators and Green<p>functions of Hamiltonians which are SUSY partners have been obtained. New exact propagators for the family of multi-well, time-dependent and non-hermitian potentials have been calculated.<p><p>Non-conservative SUSY transformation has been studied in<p>the case of multichannel Schrodinger equation with different thresholds. Spectrum (bound/virtual states and resonances) of the<p>non-conservative SUSY partner of zero potential has been founded. <p><p>Exactly solvable model of the magnetic induced Feshbach resonance<p>has been constructed. This model was tested in the case of Rb$^{85}$.<p><p>Conservative SUSY transformations of the first and the second order have been studied in the case of multichannel Schrodinger equation with equal thresholds. Transformations which introduce non-trivial coupling between scattering channels have been founded. <p><p>The first order SUSY transformation which preserves $S$-matrix eigen-phase shifts and<p>modifies mixing parameter has been founded in the case of two channel scattering with partial waves of different parities. In the case of coinciding parities we have found the second order SUSY transformation which preserves $S$-matrix eigen-phase shifts and modifies mixing parameter. <p><p>Phenomenological two channel $^3S_1-^3D_1$<p>neutron-proton potential has been obtained by using single channel, phase equivalent and coupling SUSY transformations applied to zero potential. / Doctorat en Sciences de l'ingénieur / info:eu-repo/semantics/nonPublished

The supersymmetric solutions of PT-/non-PT symmetric and Hermitian/non-Hermitian forms of quantum systems are obtained by solving the Schr&Auml / odinger equation with the deformed Morse, Hulth&para / en, P&Auml / oschl-Teller, Hyperbolic Kratzer-like, Screened Coulomb, and Exponential-Cosine Screened Coulomb (ECSC) potentials. The Hamiltonian hi- erarchy method is used to get the real energy eigenvalues and corresponding wave functions.

The work of this Ph.D. thesis in mathematics concerns the problem of determining existence, uniqueness, and structure of zero-energy states in supersymmetric matrix models, which arise from a quantum mechanical description of the physics of relativistic membranes, reduced Yang-Mills gauge theory, and of nonperturbative features of string theory, respectively M-theory. Several new approaches to this problem are introduced and considered in the course of seven scientific papers, including: construction by recursive methods (Papers A and D), deformations and alternative models (Papers B and C), averaging with respect to symmetries (Paper E), and weighted supersymmetry and index theory (Papers F and G). The mathematical tools used and developed for these approaches include Clifford algebras and associated representation theory, structure of supersymmetric quantum mechanics, as well as spectral theory of (matrix-) Schrödinger operators. / QC20100629