Mathematics / Matematik
Permanent URI for this collectionhttps://hdl.handle.net/11147/8
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Article Citation - WoS: 20Citation - Scopus: 23One-Dimensional Semirelativistic Hamiltonian With Multiple Dirac Delta Potentials(American Physical Society, 2017) Erman, Fatih; Gadella, Manuel; Uncu, HaydarIn this paper, we consider the one-dimensional semirelativistic Schrdinger equation for a particle interacting with N Dirac delta potentials. Using the heat kernel techniques, we establish a resolvent formula in terms of an N x N matrix, called the principal matrix. This matrix essentially includes all the information about the spectrum of the problem. We study the bound state spectrum by working out the eigenvalues of the principal matrix. With the help of the Feynman-Hellmann theorem, we analyze how the bound state energies change with respect to the parameters in the model. We also prove that there are at most N bound states and explicitly derive the bound state wave function. The bound state problem for the two-center case is particularly investigated. We show that the ground state energy is bounded below, and there exists a selfadjoint Hamiltonian associated with the resolvent formula. Moreover, we prove that the ground state is nondegenerate. The scattering problem for N centers is analyzed by exactly solving the semirelativistic Lippmann-Schwinger equation. The reflection and the transmission coefficients are numerically and asymptotically computed for the two- center case. We observe the so-called threshold anomaly for two symmetrically located centers. The semirelativistic version of the Kronig-Penney model is shortly discussed, and the band gap structure of the spectrum is illustrated. The bound state and scattering problems in the massless case are also discussed. Furthermore, the reflection and the transmission coefficients for the two delta potentials in this particular case are analytically found. Finally, we solve the renormalization group equations and compute the beta function nonperturbatively.Article On the Number of Bound States of Semirelativistic Hamiltonian With Dirac Delta Potentials in One Dimension(National Research Council of Canada, 2018) Erman, FatihWe 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.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: 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.Article Citation - WoS: 2Citation - Scopus: 3Renormalization of Dirac Delta Potentials Through Minimal Extension of Heisenberg Algebra(IOP Publishing Ltd., 2017) Erman, FatihWe 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.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: 4Citation - Scopus: 5On the Number of Bound States of Point Interactions on Hyperbolic Manifolds(World Scientific Publishing Co. Pte Ltd, 2017) Erman, FatihWe 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, FatihIn 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; 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: 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.