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On the Number of Bound States of Semirelativistic Hamiltonian With Dirac Delta Potentials in One Dimension
(National Research Council of Canada, 2018) Erman, Fatih
We study the bound state problem for semirelativistic N attractive Dirac delta-potentials in one dimension. We give a sufficient condition for the Hamiltonian to have N bound states and give an explicit criterion for it.
Citation - WoS: 3
Citation - Scopus: 3
A Perturbative Approach To the Tunneling Phenomena
(Frontiers Media S.A., 2019) Erman, Fatih; Turgut, Osman Teoman
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.
Citation - WoS: 4
Citation - Scopus: 4
Green's Function Formulation of Multiple Nonlinear Dirac Delta-Function Potential in One Dimension
(Elsevier, 2020) Erman, Fatih; Uncu, Haydar
In this work, we study the scattering problem of the general nonlinear finitely many Dirac delta potentials with complex coupling constants (or opacities in the context of optics) using the Green's function method and then find the bound state energies and the wave functions for the particular form of the nonlinearity in the case of positive real coupling constants. (C) 2020 Elsevier B.V. All rights reserved.
Citation - WoS: 16
Citation - Scopus: 15
On Scattering From the One-Dimensional Multiple Dirac Delta Potentials
(Institute of Physics Publishing, 2018) Erman, Fatih; Gadella, Manuel; Uncu, Haydar
In 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.
Citation - WoS: 2
Citation - Scopus: 3
Renormalization of Dirac Delta Potentials Through Minimal Extension of Heisenberg Algebra
(IOP Publishing Ltd., 2017) Erman, Fatih
We renormalize the model of multiple Dirac delta potentials in two and three dimensions by regularizing it through the minimal extension of Heisenberg algebra. We show that the results are consistent with the other regularization schemes given in the literature.
Citation - WoS: 16
Citation - Scopus: 16
A Singular One-Dimensional Bound State Problem and Its Degeneracies
(Springer Verlag, 2017) Erman, Fatih; Gadella, Manuel; Tunalı, Seçil; Uncu, Haydar
We 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.
Citation - WoS: 1
Citation - Scopus: 1
Recursion Formula for the Green's Function of a Hamiltonian for Several Types of Dirac Delta-Function Potentials in Curved Spaces
(TUBITAK, 2016) Erman, Fatih
In 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.

Mathematics / Matematik

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