WoS İndeksli Yayınlar Koleksiyonu / WoS Indexed Publications Collection
Permanent URI for this collectionhttps://hdl.handle.net/11147/7150
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Article Citation - WoS: 1Citation - Scopus: 1Completeness of Energy Eigenfunctions for the Reflectionless Potential in Quantum Mechanics(Aip Publishing, 2024) Erman, Fatih; Turgut, O. TeomanThere are a few exactly solvable potentials in quantum mechanics for which the completeness relation of the energy eigenstates can be explicitly verified. In this article, we give an elementary proof that the set of bound (discrete) states together with the scattering (continuum) states of the reflectionless potential form a complete set. We also review a direct and elegant derivation of the energy eigenstates with proper normalization by introducing an analog of the creation and annihilation operators of the harmonic oscillator problem. We further show that, in the case of a single bound state, the corresponding wave function can be found from the knowledge of continuum eigenstates of the system. Finally, completeness is shown by using the even/odd parity eigenstates of the Hamiltonian, which provides another explicit demonstration of a fundamental property of quantum mechanical Hamiltonians.Article Explicit Derivation of the Propagator for a Point Interaction in Three Dimensional Hyperbolic Space(Springer/plenum Publishers, 2024) Erman, FatihThe explicit expression for the propagator of the Dirac delta potential in three dimensional hyperbolic spaces is derived using the integral transform of the Krein's type of the resolvent formula, obtained after the renormalization procedure.Article Citation - WoS: 2Non-Relativistic Lee Model in Two-Dimensional Riemannian Manifolds(American Institute of Physics, 2012) Erman, Fatih; Turgut, Osman TeomanThis work is a continuation of our previous work [F. Erman and O. T. Turgut, J. Math. Phys. 48, 122103 ( 2007)], where we constructed the non-relativistic Lee model in three-dimensional Riemannian manifolds. Here we renormalize the two-dimensional version by using the same methods and the results are shortly given since the calculations are basically the same as in the three-dimensional model. We also show that the ground state energy is bounded from below due to the upper bound of the heat kernel for compact and Cartan-Hadamard manifolds. In contrast to the construction of the model and the proof of the lower bound of the ground state energy, the mean field approximation to the two-dimensional model is not similar to the one in three dimensions and it requires a deeper analysis, which is the main result of this paper. (C) 2012 American Institute of Physics. [http://dx.doi.org/10.1063/1.4705355]