Erman, Fatih
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Erman, F
Erman, F.
Erman, F.
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fatiherman@iyte.edu.tr
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04.02. Department of Mathematics
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Current Staff
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Documents
25
Citations
157
h-index
7
Documents
25
Citations
148
Scholarly Output
26
Articles
21
Views / Downloads
51878/9384
Supervised MSc Theses
4
Supervised PhD Theses
1
WoS Citation Count
95
Scopus Citation Count
99
Patents
0
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0
WoS Citations per Publication
3.65
Scopus Citations per Publication
3.81
Open Access Source
23
Supervised Theses
5
| Journal | Count |
|---|---|
| Journal of Mathematical Physics | 4 |
| Frontiers in Physics | 3 |
| European Physical Journal Plus | 2 |
| Communications in Theoretical Physics | 1 |
| European Journal of Physics | 1 |
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26 results
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Article Citation - WoS: 3Citation - Scopus: 3A Perturbative Approach To the Tunneling Phenomena(Frontiers Media S.A., 2019) Erman, Fatih; Turgut, Osman TeomanThe 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.Article Citation - WoS: 16Citation - Scopus: 16A Singular One-Dimensional Bound State Problem and Its Degeneracies(Springer Verlag, 2017) Erman, Fatih; Gadella, Manuel; Tunalı, Seçil; Uncu, HaydarWe give a brief exposition of the formulation of the bound state problem for the one-dimensional system of N attractive Dirac delta potentials, as an N× N matrix eigenvalue problem (ΦA= ωA). The main aim of this paper is to illustrate that the non-degeneracy theorem in one dimension breaks down for the equidistantly distributed Dirac delta potential, where the matrix Φ becomes a special form of the circulant matrix. We then give elementary proof that the ground state is always non-degenerate and the associated wave function may be chosen to be positive by using the Perron-Frobenius theorem. We also prove that removing a single center from the system of N delta centers shifts all the bound state energy levels upward as a simple consequence of the Cauchy interlacing theorem.Article Citation - WoS: 1Citation - Scopus: 1The Harmonic Oscillator Potential Perturbed by a Combination of Linear and Non-Linear Dirac Delta Interactions With Application To Bose–einstein Condensation(Elsevier B.V., 2024) Akyüz,C.; Erman,F.; Uncu,H.In this paper, we study the bound state analysis of a one dimensional nonlinear version of the Schrödinger equation for the harmonic oscillator potential perturbed by a δ potential, where the nonlinear term is taken to be proportional to δ(x)|ψ(x)|2ψ(x). The bound state wave functions are explicitly found and the bound state energy of the system is algebraically determined by the solution of an implicit equation. Then, we apply this model to the Bose–Einstein condensation of a Bose gas in a harmonic trap with a dimple potential. We propose that the many-body interactions of the Bose gas can be effectively described by the nonlinear term in the Schrödinger equation. Then, we investigate the critical temperature, the condensate fraction, and the density profile of this system numerically. © 2024 Elsevier B.V.Article Citation - WoS: 5Citation - Scopus: 5A Many-Body Problem With Point Interactions on Two-Dimensional Manifolds(IOP Publishing Ltd., 2013) Erman, Fatih; Turgut, O. TeomanA non-perturbative renormalization of a many-body problem, where non-relativistic bosons living on a two-dimensional Riemannian manifold interact with each other via the two-body Dirac delta potential, is given by the help of the heat kernel defined on the manifold. After this renormalization procedure, the resolvent becomes a well-defined operator expressed in terms of an operator (called principal operator) which includes all the information about the spectrum. Then, the ground state energy is found in the mean-field approximation and we prove that it grows exponentially with the number of bosons. The renormalization group equation (or Callan-Symanzik equation) for the principal operator of the model is derived and the beta function is exactly calculated for the general case, which includes all particle numbers.Article Citation - WoS: 4Citation - Scopus: 4On Schrödinger Operators Modified by Δ Interactions(Academic Press, 2023) Akbaş, Kaya Güven; Erman, Fatih; Turgut, O. TeomanWe study the spectral properties of a Schrödinger operator H0 modified by δ interactions and show explicitly how the poles of the new Green's function are rearranged relative to the poles of original Green's function of H0. We prove that the new bound state energies are interlaced between the old ones, and the ground state energy is always lowered if the δ interaction is attractive. We also derive an alternative perturbative method of finding the bound state energies and wave functions under the assumption of a small coupling constant in a somewhat heuristic manner. We further show that these results can be extended to cases in which a renormalization process is required. We consider the possible extensions of our results to the multi center case, to δ interaction supported on curves, and to the case, where the particle is moving in a compact two-dimensional manifold under the influence of δ interaction. Finally, the semi-relativistic extension of the last problem has been studied explicitly. © 2023 Elsevier Inc.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]Article Citation - WoS: 1Citation - Scopus: 2Nondegeneracy of the Ground State for Nonrelativistic Lee Model(American Institute of Physics, 2014) Erman, Fatih; Malkoç, Berkin; Turgut, Osman TeomanIn 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: 16Citation - Scopus: 15On Scattering From the One-Dimensional Multiple Dirac Delta Potentials(Institute of Physics Publishing, 2018) Erman, Fatih; Gadella, Manuel; Uncu, HaydarIn this paper, we propose a pedagogical presentation of the Lippmann-Schwinger equation as a powerful tool, so as to obtain important scattering information. In particular, we consider a one-dimensional system with a Schrödinger-type free Hamiltonian decorated with a sequence of N attractive Dirac delta interactions. We first write the Lippmann-Schwinger equation for the system and then solve it explicitly in terms of an N × N matrix. Then, we discuss the reflection and the transmission coefficients for an arbitrary number of centres and study the threshold anomaly for the N = 2 and N = 4 cases. We also study further features like the quantum metastable states and resonances, including their corresponding Gamow functions and virtual or antibound states. The use of the Lippmann-Schwinger equation simplifies our analysis enormously and gives exact results for an arbitrary number of Dirac delta potentials.Master Thesis Two-Particle Schrodinger Operators With Point Interactions(01. Izmir Institute of Technology, 2020) Kızılkaya, Melih; Erman, FatihIn this thesis, a singular quantum mechanical problem, where two particles interact with each other through Dirac delta potentials in the plane, has been considered. The proof for the existence of a self-adjoint Hamiltonian operator for the model is given by using some operator theory techniques and renormalization idea in quantum field theory. Moreover, some necessary background for unbounded operators is reviewed in order to make the thesis as self-contained as possible.
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