Mathematics / Matematik

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

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Now showing 1 - 4 of 4
  • Article
    Citation - WoS: 4
    Citation - Scopus: 4
    On Schrödinger Operators Modified by Δ Interactions
    (Academic Press, 2023) Akbaş, Kaya Güven; Erman, Fatih; Turgut, O. Teoman
    We 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
    Citation - WoS: 3
    On the Non-Linear Integral Equation Approaches for the Boundary Reconstruction in Double-Connected Planar Domains
    (Ivan Franko National University of Lviv,, 2016) Chapko, R. S.; Yaman, Olha Ivanyshyn; Kanafotskyi, T. S.
    We consider the reconstruction of an interior curve from the given Cauchy data of a harmonic function on the exterior boundary of the planar domain. With the help of Green's function and potential theory the non-linear boundary reconstruction problem is reduced to the system of non-linear boundary integral equations. The three iterative algorithms are developed for its numerical solution. We find the Frechet derivatives for the corresponding operators and show unique solviability of the linearized systems. Full discretization of the systems is realized by a trigonometric quadrature method. Due to the inherited ill-possedness in the obtained system of linear equations we apply the Tikhonov regularization. The numerical results show that the proposed methods give a good accuracy of reconstructions with an economical computational cost.
  • Article
    Citation - WoS: 5
    Citation - Scopus: 5
    The Propagators for Δ and Δ′ Potentials With Time-Dependent Strengths
    (Frontiers Media S.A., 2020) Erman, Fatih; Gadella, Manuel; Uncu, Haydar
    We study the time-dependent Schrodinger equation with finite number of Dirac delta and delta ' potentials with time dependent strengths in one dimension. We obtain the formal solution for generic time dependent strengths and then we study the particular cases for single delta potential and limiting cases for finitely many delta potentials. Finally, we investigate the solution of time dependent Schrodinger equation for delta ' potential with particular forms of the strengths.
  • 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.