Mathematics / Matematik

Permanent URI for this collectionhttps://hdl.handle.net/11147/8

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Author: search.filters.author.Uncu, HaydarSubject: search.filters.subject.Dirac delta potentials

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Now showing 1 - 3 of 3
  • Article
    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.
  • Article
    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.
  • Article
    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.