arXiv subjects
Publications and source records attributed to Özlem Yeşiltaş.
We have generalized the solutions of the radial Dirac equation with a tensor potential under spin and pseudospin symmetry limits to exceptional orthogonal Hermite polynomials family. We have obtained new general rational potential models which are the generalization of the nonlinear isotonic potential families and also energy dependent.
The Dirac equation on the toroidal surface is studied. The non-constant Fermi velocity functions are used in the Dirac Hamiltonian to get trigonometric Scarf and Rosen-Morse potentials using the method given in [19], then, the exact solutions are written. On the other hand, consecutive mappings are used to get a trigonometric Scarf I-like potential model and the solutions are obtained in the second part of the paper.
The exact solutions of the (2+1) dimensional Dirac equation on the torus and the new extension and generalization of the trigonometric Poschl-Teller potential families in terms of the torus parameters are obtained. Supersymmetric quantum mechanical techniques are used to get the extended potentials when the inner and outer radii of the torus are both equal and inequal. In addition, using the aspects of the Lie algebraic approaches, the iso(2; 1) algebra is also applied to the system where we have arrived at the spectrum solutions of the extended potentials using the Casimir operator that matches with the results of the exact solutions.
Two dimensional massless Dirac Hamiltonian under the influence of hyperbolic magnetic fields is mentioned in curved space. Using a spherical surface parametrization, the Dirac operator on the sphere is presented and the system is given as two supersymmetric partner Hamiltonians which coincides with the position dependent mass Hamiltonians. We introduce two ansatzes for the component of the vector potential to acquire effective solvable models, which are Rosen Morse II potential and the model given in \cite{mid} whose bound states are Jacobi $X_1$ type polynomials, and we adapt our work to these special models under some parameter restrictions. The energy spectrum and the eigenvectors are found for Rosen Morse II potential. On the other hand, complete solutions are given for the second system. The vector and the effective potentials with their eigenvalues are sketched for each system.
Two dimensional effective Hamiltonian for a massless Dirac electron interacting with a hyperbolic magnetic field is discussed within PT symmetry. Factorization method and polynomial procedures are used to solve Dirac equation for the constant Fermi velocity and the effective potential which is complex Scarf II potential. The more general effective Scarf II potential models are also obtained within pseudo-supersymmetry. Finally, an extension of Panella and Roy's work [12] to the both PT symmetric and real Scarf II partner potentials are given using the position dependent Fermi velocity.
Searching for non-Hermitian (parity-time)$\mathcal{PT}$-symmetric Hamiltonians \cite{bender} with real spectra has been acquiring much interest for fourteen years. In this article, we have introduced a $\mathcal{PT}$ symmetric non-Hermitian Hamiltonian model which is given as $\hat{\mathcal{H}}=ω(\hat{b}^\dagger\hat{b}+\frac{1}{2})+ α(\hat{b}^{2}-(\hat{b}^\dagger)^{2})$ where $ω$ and $α$ are real constants, $\hat{b}$ and $\hat{b^\dagger}$ are first order differential operators. Moreover, Pseudo-Hermiticity that is a generalization of $\mathcal{PT}$ symmetry has been attracting a growing interest \cite{mos}. Because the Hamiltonian $\mathcal{H}$ is pseudo-Hermitian, we have obtained the Hermitian equivalent of $\mathcal{H}$ which is in Sturm- Liouville form leads to exactly solvable potential models which are effective screened potential and hyperbolic Rosen-Morse II potential. $\mathcal{H}$ is called pseudo-Hermitian, if there exists a Hermitian and invertible operator $η$ satisfying $\mathcal{H^\dagger}=η\mathcal{H} η^{-1}$. For the Hermitian Hamiltonian $h$, one can write $h=ρ\mathcal{H} ρ^{-1}$ where $ρ=\sqrtη$ is unitary. Using this $ρ$ we have obtained a physical Hamiltonian $h$ for each case. Then, the Schrödinger equation is solved exactly using Shape Invariance method of Supersymmetric Quantum Mechanics \cite{susy1}. Mapping function $ρ$ is obtained for each potential case.
We have obtained the metric operator $Θ=\exp T$ for the non-Hermitian Hamiltonian model $H=ω(a^†a+1/2)+α(a^{2}-a^{†^{2}})$. We have also found the intertwining operator which connects the Hamiltonian to the adjoint of its pseudo-supersymmetric partner Hamiltonian for the model of hyperbolic Rosen-Morse II potential.
Within the ideas of pseudo-supersymmetry, we have studied a non-Hermitian Hamiltonian $H_{-}=ω(ξ^† ξ+\1/2)+αξ^{2}+βξ^{†2}$, where $α\neq β$ and $ξ$ is a first order differential operator, to obtain the partner potentials $V_{+}(x)$ and $V_{-}(x)$ which are new isotonic and isotonic nonlinear oscillators, respectively, as the Hermitian equivalents of the non-Hermitian partner Hamiltonians $H_{\pm}$. We have provided an algebraic way to obtain the spectrum and wavefunctions of a nonlinear isotonic oscillator. The solutions of $V_{-}(x)$ which are Hermitian counterparts of Swanson Hamiltonian are obtained under some parameter restrictions that are found. Also, we have checked that if the intertwining operator satisfies $η_{1} H_{-}=H_{+} η_{1}$, where $η_{1}=ρ^{-1} \mathcal{A} ρ$ and $\mathcal{A}$ is the first order differential operator, which factorizes Hermitian equivalents of $H_{\pm}$.