Grant of Ministery of Research and Innovation, CNCS – UEFISCDI, project number PN-III-P4-ID-PCE-2016-0131, within PNCDI III.

Title: Mathieu and Heun functions in quantum field dynamics

Abstract

The general topic of the project is motivated by the wide international research effort on special functions with a high degree of complexity, which are seen as a modern tool in mathematics and theoretical physics, describing a wide range of phenomena, from quantum optics to black holes.

The project is devoted to the study of quantum particles and fields evolving in various configurations. The obtained solutions to gauge-invariant Schrodinger, Klein-Gordon and Dirac equations are discussed in view of a wide range of applications.

Firstly, we shall focus on Mathieu’s type equations that emerge for periodic potentials. After a careful analysis of the Mathieu functions, especially with respect to their stability, we are going to formulate models for particles inside magnetar’s core and crust, which can be used as a first step in the comparison between theory and observations.

Secondly, we turn to Heun-type equations. As it is known, there are difficulties in the computational techniques and, in order to have an accurate interpretation of the solutions, the properties of the Heun functions must be fully understood. Thus, we intend to clarify some theoretical aspects of the Heun functions and a special attention is to be given to the conditions when these functions are expressed in conjunction with polynomials.

In the second year, we are going to deal with exactly solvable models for fields coupled to gravity in view of their applications in: quantum generators, particle production, astrophysics and cosmology. Finally, a special attention will be given to applications of different types of Heun functions in the study of black hole physics and in gauge theories on thick branes.

Value: 600.000 Lei

Objectives

  • I1. A careful analysis of the Mathieu functions, especially with regard to their stability. For various systems described by Mathieu’s equation, the study of branching points, quantization relations and conserved current density components;
  • I2. The quantum analysis of charged particles in electromagnetic wave-form configurations.
  • II1. Exactly solvable models for fields coupled to gravity in view of their applications in: quantum generators, particle production, astrophysics and cosmology;
  • II2. The formulation of reliable models for particles in magnetars, which can be used as the first step in the comparison between theory and observations;
  • II3. To clarify important aspects of the theory of the Heun functions with respect to: their relation to more simple special functions, their normalization, series expansions and integration techniques.
  • III1. To find new solutions to Kompaneets equation, with astrophysical applications;
  • III2. To find explicit analytic solutions for the scattering of various fields on static and charged black hole backgrounds in four and higher dimensions and see if they can be expressed by means of the Heun functions.

Published papers in ISI journals

  1. Mathieu functions for fermions generated in magnetar’s corona
    M. A. Dariescu and C. Dariescu
    Modern Physics Letters A, 32, No. 32, 1750174 (13 pgs.) (2017).
    DOI: 10.1142/S0217732317501747
    Accession number: WOS: 000412829600005
  2. The SO(3,1) x U(1)-gauge invariant Approach to Charged Bosons in Relativistic Magnetars
    C. Dariescu, M.A. Dariescu and C. Stelea
    General Relativity and Gravitation, 49, No. 12, 153 (2017).
    DOI: https://doi.org/10.1007/s10714-017-2314-8
    WOS:000416789100004
  3. Mathieu and Heun solutions to the Wheeler-De Witt Equation for hyperbolic Universes
    M. A. Dariescu and C.Dariescu
    International J. of Theoretical Physics, 57, No. 3, p. 652–663 (2018).
    DOI: https://doi.org/10.1007/s10773-017-3595-0
    WOS:000424641700003
  4. Coherent Planar Symmetric Spacetimes Generated by Progressive Waves of Nambu-Goldstone Bosons
    C. Dariescu and M.A. Dariescu
    International J. of Theoretical Physics, 57, No.8, p. 2280-2292(2018).
    DOI: https://doi.org/10.1007/s10773-018-3751-1
  5. Elliptic and Heun functions in spatially-flat Friedmann Robertson-Walker Cosmologies
    Denisa–Andreea Mihu
    Romanian Journal of Physics, 63, Nos. 5-6, 109 (2018).
  6. Heun functions describing bosons and fermions on Melvin’s spacetime
    Marina-Aura Dariescu and Ciprian Dariescu
    Advances in High Energy Physics, Volume 2018, Article ID 1953586, 7 pages
    DOI: https://doi.org/10.1155/2018/1953586
  7. Magnetized anisotropic stars
    C. Stelea, M. A. Dariescu, C. Dariescu
    Phys. Rev. D, 97, No. 10, 104059 (2018).
    DOI: https://doi.org/10.1103/PhysRevD.97.104059
    WOS:000433291600020
  8. Double-black hole solutions of theEinstein-Maxwell-Dilaton theory in five dimensions
    C. Stelea
    Phys. Rev. D, 97, 024044 (2018).
    DOI:https://doi.org/10.1103/PhysRevD.97.024044
    WOS:000423434600013
  9. The SO(3,1) x U(1)-gauge covariant Dirac equation in relativistic magnetars
    M. A. Dariescu, C. Dariescu and C. Stelea
    General Relativity and Gravitation, 50, No. 10, 126 (13 pgs,) (2018)
    DOI: https://doi.org/10.1007/s10714-018-2449-2
    WOS: 000444730500002
  10. New bound of the mass-to-radius ratio for electrically charged stars
    C. Stelea, M. A. Dariescu and C. Dariescu
    Phys. Rev. D, 98, No. 12, 124022 (2018)
    DOI:https://doi.org/10.1103/PhysRevD.98.124022
    WOS:000454169700013
  11. Heun-type solutions of the Klein-Gordon and Dirac equations in the Garfinkle-Horowitz-Strominger dilaton black hole background
    M. A. Dariescu, C. Dariescu and C. Stelea
    Advances in High Energy Physics, Volume 2019, Article ID 5769564, 8 pages (2019)
    DOI: 10.1155/2019/5769564
    WOS:000460860300001
  12. Exact stationary solutions to a general form of Kompaneets Equation
    M. A. Dariescu, C. Dariescu, Ghe. Amanoloaei
    Astrophysics (Astrofizika), 62, No. 3, p. 451-462 (2019)
  13. Heun solutions for Dirac fermions in Black Strings Backgrounds
    M. A. Dariescu, C. Dariescu and C. Stelea
    Proceedings of the Romanian Academy, Series A (accepted for publication)
  14. Massless fermions on static general prolate metrics and their Heun solutions
    M. A. Dariescu, C. Dariescu and C. Stelea
    Modern Physics Letters A (accepted for publication)

Oral communications in international conferences. Proceedings

  1. Mathieu Functions Describing Particles Evolving in Electromagnetic Waves
    D. A. Mihu and M. A. Dariescu
    Oral communication in TIM17 Physics Conference, Timisoara, Romania
    AIP Conference Proceedings 1916, 020006-1 (2017)
  2. Parametric induced instabilities of bosons in magnetar’s crust
    M. A. Dariescu, D. A. Mihu and C. Dariescu
    Oral communication in International Scientific Conference Mathematical Modeling. Processes and systems. MATHMODEL 2017, Borovets, Bulgaria
    Conference Proceedings, ISSN 2535-0978, Publisher: SCIENTIFIC TECHNICAL UNION OF MECHANICAL ENGINEERING “INDUSTRY-4.0”, pgs. 16-20.
    International scientific journal “Mathematical Modeling” (Print ISSN 2535-0986, Web ISSN 2603-2929), Issue 3/2017.
  3. Generalized Models of Anisotropic Magnetars
    C. Stelea, M. A. Dariescu and C. Dariescu
    Plenary talk in TIM18 Physics Conference, 24-26 May 2018, Timisoara, Romania
    AIP Conference Proceedings 2071 No. 1, 020002 (8 pag.) (2019) (ISBN: 978-0-7354-1799-1)
  4. Generalized special solutions to modified Kompaneets equation
    D. A. Mihu, M. A. Dariescu and G. Amanoloaei
    Oral communication in TIM18 Physics Conference, 24-26 May 2018, Timisoara, Romania
    AIP Conference Proceedings 2071 No. 1, 020004 (6 pag.) (2019) (ISBN: 978-0-7354-1799-1)

Ph. D. Thesis defended in May 2018.
Title: Mathieu and Heun functions with applications in Astrophysics and Cosmology
Supervisor: Professor Marina-Aura Dariescu
Ph. D. student (member in the Grant’s team): Denisa-Andreea Mihu