Impasse of the Desire

Bu yazıda, öncelikle topolojik bir problem olan Königsberg Köprü Probleminin ne olduğu ve matematiksel olarak çıkmaza girmesinin nedenleri anlatılmış, bu problemin Freud’un rapor ettiği “Fare Adam” vakasının obsesif semptomları ile benzerliği ele alınmıştır. Daha sonra ise psikanalist Jacques Lacan’ın Fare Adam vakasını psiko-topolojik bakış açısı ile yorumlamasına (Cross-cap) ve belirtilerin sebeplerini R-Şeması ile açıklamasına yer verilmiştir. Lacan’ın Fare Adam vakasına ilişkin kavramsallaştırmasının, özellikle obsesif-kompulsif tarzda şikayetlerle başvuran kişilere yardımcı olabilmek için, alanda çalışan klinisyenlere ışık tutacağı değerlendirilmektedir.In this paper, first of all a topological problem, called as a Königsberg Bridges Problem and the impossibility of its solution with the acknowledged rules were explained. Next, the similarity in terms of the impossiblity of the Bridge Problem’s solutions and Freud’s Rat Man case’s symptoms were discussed. Finally, the interpretation of the Rat Man by Jacques Lacan using his psycho-topological explanations and R-Schema are explained in detail. The explanations made by Jacques Lacan about the Rat Man are thought to be helpful for those clinicians who work with the people, reporting obsessivecompulsive problems

Extended dynamical symmetries of Landau levels in higher dimensions

Continuum models for time-reversal (TR) invariant topological insulators (Tis) in d >= 3 dimensions are provided by harmonic oscillators coupled to certain SO(d) gauge fields. These models are equivalent to the presence of spin-orbit (SO) interaction in the oscillator Hamiltonians at a critical coupling strength (equivalent to the harmonic oscillator frequency) and leads to flat Landau Level (LL) spectra and therefore to infinite degeneracy of either the positive or the negative helicity states depending on the sign of the SO coupling. Generalizing the results of [1] to d >= 4, we construct vector operators commuting with these Hamiltonians and show that SO(d, 2) emerges as the non-compact extended dynamical symmetry. Focusing on the model in four dimensions, we demonstrate that the infinite degeneracy of the flat spectra can be fully explained in terms of the discrete unitary representations of SO(4,2), i.e. the doubletons. The degeneracy in the opposite helicity branch is finite, but can still be explained exploiting the complex conjugate doubleton representations. Subsequently, the analysis is generalized to d-dimensions, distinguishing the cases of odd and even d. We also determine the spectrum generating algebra in these models and briefly comment on the algebraic organization of the LL states w.r.t. an underlying "deformed" AdS geometry as well as on the organization of the surface states under open boundary conditions in view of our results

Integrable and superintegrable systems with spin

A system of two particles with spin s=0 and s=1/2 respectively, moving in a plane is considered. It is shown that such a system with a nontrivial spin-orbit interaction can allow an 8 dimensional Lie algebra of first-order integrals of motion. The Pauli equation is solved in this superintegrable case and reduced to a system of ordinary differential equations when only one first-order integral exists.Comment: 12 page

Integrable and superintegrable systems with spin in three-dimensional Euclidean space

A systematic search for superintegrable quantum Hamiltonians describing the interaction between two particles with spin 0 and 1/2, is performed. We restrict to integrals of motion that are first-order (matrix) polynomials in the components of linear momentum. Several such systems are found and for one non-trivial example we show how superintegrability leads to exact solvability: we obtain exact (nonperturbative) bound state energy formulas and exact expressions for the wave functions in terms of products of Laguerre and Jacobi polynomials.Comment: 23 page

The Integrability of Pauli System in Lorentz Violating Background

We systematically analyze the integrability of a Pauli system in Lorentz violating background at the non-relativistic level both in two- and three-dimensions. We consider the non-relativistic limit of the Dirac equation from the QED sector of the so-called Standard Model Extension by keeping only two types of background couplings, the vector a_mu and the axial vector b_mu. We show that the spin-orbit interaction comes as a higher order correction in the non-relativistic limit of the Dirac equation. Such an interaction allows the inclusion of spin degree non-trivially, and if Lorentz violating terms are allowed, they might be comparable under special circumstances. By including all possible first-order derivative terms and considering the cases a\ne 0, b\ne 0, and b_0\ne 0 one at a time, we determine the possible forms of constants of motion operator, and discuss the existence or continuity of integrability due to Lorentz violating background.Comment: 19 page

Superintegrable systems with spin and second-order integrals of motion

We investigate a quantum nonrelativistic system describing the interaction of two particles with spin 1/2 and spin 0, respectively. We assume that the Hamiltonian is rotationally invariant and parity conserving and identify all such systems which allow additional integrals of motion that are second order matrix polynomials in the momenta. These integrals are assumed to be scalars, pseudoscalars, vectors or axial vectors. Among the superintegrable systems obtained, we mention a generalization of the Coulomb potential with scalar potential $V_0=\frac{\alpha}{r}+\frac{3\hbar^2}{8r^2}$ and spin orbital one $V_1=\frac{\hbar}{2r^2}$.Comment: 32 page

Doğrusal olmayan evrim denklemlerinin uzatma yapıları, backlund dönüşümleri ve painleve analizi

The Wahlquist-Estabrook prolongation technique and the Painleve analysis, used for testing the integrability of nonlinear evolution equations, are considered and applied both to the Drinfel'd-Sokolov system of equations, indeed known to be one of the coupled Korteweg-de Vries (KdV) systems, and Kersten-Krasil'shchik coupled KdV-mKdV equations. Some new Backlund transformations for the Drinfel'd-Sokolov system of equations are also found.Ph.D. - Doctoral Progra