Browsing by Author "Turgut, Osman Teoman"

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Citation - WoS: 5
Citation - Scopus: 6
Existence of Hamiltonians for Some Singular Interactions on Manifolds
(American Institute of Physics, 2012-04) Doğan, Çağlar; Erman, Fatih; Erman, Fatih; Turgut, Osman Teoman; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
The existence of the Hamiltonians of the renormalized point interactions in two and three dimensional Riemannian manifolds and that of a relativistic extension of this model in two dimensions are proven. Although it is much more difficult, the proof of existence of the Hamiltonian for the renormalized resolvent for the non-relativistic Lee model can still be given. To accomplish these results directly from the resolvent formula, we employ some basic tools from the semigroup theory.
Citation - WoS: 2
Non-Relativistic Lee Model in Two-Dimensional Riemannian Manifolds
(American Institute of Physics, 2012) Erman, Fatih; Erman, Fatih; Turgut, Osman Teoman; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
This 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]
Citation - WoS: 1
Citation - Scopus: 2
Nondegeneracy of the Ground State for Nonrelativistic Lee Model
(American Institute of Physics, 2014-08) Erman, Fatih; Erman, Fatih; Malkoç, Berkin; Turgut, Osman Teoman; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
In the present work, we first briefly sketch the construction of the nonrelativistic Lee model on Riemannian manifolds, introduced in our previous works. In this approach, the renormalized resolvent of the system is expressed in terms of a well-defined operator, called the principal operator, so as to obtain a finite formulation. Then, we show that the ground state of the nonrelativistic Lee model on compact Riemannian manifolds is nondegenerate using the explicit expression of the principal operator that we obtained. This is achieved by combining heat kernel methods with positivity improving semi-group approach and then applying these tools directly to the principal operator, rather than the Hamiltonian, without using cut-offs.
Citation - WoS: 2
Citation - Scopus: 2
A Perturbative Approach To the Tunneling Phenomena
(Frontiers Media S.A., 2019-05) Erman, Fatih; Erman, Fatih; Turgut, Osman Teoman; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of Technology
The double-well potential is a good example, where we can compute the splitting in the bound state energy of the system due to the tunneling effect with various methods, namely path-integral, WKB, and instanton calculations. All these methods are non-perturbative and there is a common belief that it is dif fi cult to fi nd the splitting in the energy due to the barrier penetration from a perturbative analysis. However, we will illustrate by explicit examples including singular potentials (e.g., Dirac delta potentials supported by points and curves and their relativistic extensions) it is possible to fi nd the splitting in the bound state energies by developing some kind of perturbation method.