Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection
Permanent URI for this collectionhttps://hdl.handle.net/11147/7148
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Article Citation - WoS: 4Citation - Scopus: 5On the Number of Bound States of Point Interactions on Hyperbolic Manifolds(World Scientific Publishing Co. Pte Ltd, 2017) Erman, Fatih; Erman, Fatih; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyWe study the bound state problem for N attractive point Dirac δ-interactions in two- and three-dimensional Riemannian manifolds. We give a sufficient condition for the Hamiltonian to have N bound states and give an explicit criterion for it in hyperbolic manifolds ℍ2 and ℍ3. Furthermore, we study the same spectral problem for a relativistic extension of the model on ℝ2 and ℍ2.Article Citation - WoS: 1Citation - Scopus: 1Recursion Formula for the Green's Function of a Hamiltonian for Several Types of Dirac Delta-Function Potentials in Curved Spaces(TUBITAK, 2016) Erman, Fatih; Erman, Fatih; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn this short article, we nonperturbatively derive a recursive formula for the Green's function associated with finitely many point Dirac delta potentials in one dimension. We extend this formula to the one for the Dirac delta potentials supported by regular curves embedded in two-dimensional manifolds and for the Dirac delta potentials supported by two-dimensional compact manifolds embedded in three-dimensional manifolds. Finally, this formulation allows us to find the recursive formula of the Green's function for the point Dirac delta potentials in two- and three-dimensional Riemannian manifolds, where the renormalization of coupling constant is required.Article Citation - WoS: 1Citation - Scopus: 2Nondegeneracy of the Ground State for Nonrelativistic Lee Model(American Institute of Physics, 2014) Erman, Fatih; Erman, Fatih; Turgut, Osman Teoman; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyIn 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.Article Citation - WoS: 6Citation - Scopus: 7Existence of Hamiltonians for Some Singular Interactions on Manifolds(American Institute of Physics, 2012) Doğan, Çağlar; Erman, Fatih; Turgut, Osman Teoman; 04.02. Department of Mathematics; 04. Faculty of Science; 01. Izmir Institute of TechnologyThe 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.