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Dubna, August 2009

Dubna, August 2009. International Bogoliubov Conference PROBLEMS IN THEORETICAL AND MATHEMATICAL PHYSICS. Generalized Teukolsky-Starobinsky Identities. Plamen Fiziev Department of Theoretical Physics University of Sofia. Talk at The International Bogoliubov Conference

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Dubna, August 2009

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  1. Dubna, August 2009 International Bogoliubov Conference PROBLEMS IN THEORETICAL AND MATHEMATICAL PHYSICS

  2. Generalized Teukolsky-Starobinsky Identities Plamen Fiziev Department of Theoretical Physics University of Sofia Talk at The International Bogoliubov Conference PROBLEMS IN THEORETICAL AND MATHEMATICAL PHYSICS 25 August 2009

  3. The Nonlinear Mechanics and Wave Mechanics

  4. Heun’s Differential Equation: AKEY for Huge amount of Physical Problems found by Zur Theorie der Riemann'schen Functionen zweiter Ordnung mit Vier Verzweigungs-punkten Math. Ann. 31 (1889) 161-179 Born in Weisbaden April 3, 1859 Died in Karsruhe January 10, 1929

  5. Confluent Heun Equation: Frobeniussolution aound z = 0 : - a recurrence relation

  6. Novel relations for confluent Heun’s functions and their Derivatives, PF:arXiv:0904.0245 [math-ph] Self-adjoint form of confluent Heun’s operator: The comutator: Chain of confluent Heun’s operators: The basic general relation:

  7. The - condition A Novel Identity: => => Note that => =N-polynomial

  8. Teukolsky Master Equation: Small perturbations of spin-weights s =-2,-3/2,-1,-1/2 0,1/2, 1, 3/2, 2 ofKerr and Schwarzschild, background in terms of Weyl invariants x Separatipon of the variables: x TAE: TRE: x Kerr: Schwarzschild: (a=0)

  9. Universal Formof the Exact Solutionsof TAE, TRE and Regge-Wheeler Eq. Since the geodesic equations are solved in elliptic functions PF: arXiv:0902.1277, arXiv:0906.5108 [gr-qc] , For TRE and: For TAE and: x Regge-Wheeler Equation:

  10. Universal formof the Teukolsky-Starobinsky Identities For the above special values of the parameters all solutions turn to be -solutions. As a result the universal identities take place: PF: arXiv:0906.5108 [gr-qc] Generalized Teukolsky- Starobinsky Identities: As a result of amazing new symmetry for N+1=2|s| : + if is a solutions with spin-weight +s, then is a solution of TE with –s!

  11. The Explicit Form of TSI for all -solutions to TRE: Starobinsky Constant

  12. The Explicit Form of TSI for all -solutions to TAE: Starobinsky Constant Disentangled form of TSI for TAE:

  13. The Explicit Form of TSI for all -solutions to RWE: Starobinsky Constant Note that here

  14. New effective method for calculation of Starobinsky constant for all spin-weights s Starobinsky constants for different s coincide up to known factor with the for Taylor series for confluent Heun’s function . Hence,we can calculate Starobinsky constants using recurrence relation : In the case of -solutions:

  15. Thank you for your attention !

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