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On spectra of a certain class of quadratic operator pencils with one-dimensional linear part

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Abstract

We consider a class of quadratic operator pencils that occur in many problems of physics. The part of such a pencil linear with respect to the spectral parameter describes viscous friction in problems of small vibrations of strings and beams. Patterns in the location of eigenvalues of such pencils are established. If viscous friction (damping) is pointwise, then the operator in the linear part of the pencil is one-dimensional. For this case, rules in the location of purely imaginary eigenvalues are found.

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References

  1. M. G. Krein and A. A. Nudelman, “On direct and inverse problems for frequencies of boundary dissipation of inhomogeneous string,” Dokl. Akad. Nauk SSSR, 247,No. 5, 1046–1049 (1979).

    MathSciNet  Google Scholar 

  2. M. G. Krein and A. A. Nudelman, “On some spectral properties of an inhomogeneous string with dissipative boundary condition,” J. Operator Theory, 22, 369–395 (1989).

    MathSciNet  Google Scholar 

  3. D. Z. Arov, “Realization of a canonical system with dissipative boundary condition at one end of the segment in terms of the coefficient of dynamical compliance,” Sib. Mat. Zh., 16, 440–463 (1975).

    MathSciNet  Google Scholar 

  4. K. Veselic, “On linear vibrational systems with one-dimensional damping,” Appl. Anal., 29, 1–18 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  5. K. Veselic, “On linear vibrational systems with one-dimensional damping. II,” Integral Equat. Oper. Theory, 13, 1–18 (1990).

    Article  MathSciNet  Google Scholar 

  6. S. Cox and E. Zuazua, “The rate at which energy decays in a string damped at one end,” Commun. Part. Different. Equat., 19,No. 1/2. 213–243 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  7. M. A. Shubov, “Asymptotics of resonances and geometry of resonance states in the problem of scattering of acoustic waves by a spherically symmetric inhomogeneity of the density,” Different. Integral Equat., 8,No. 5, 1073–1115 (1995).

    MATH  MathSciNet  Google Scholar 

  8. V. N. Pivovarchik, “Inverse problem for a smooth string with damping at the end,” J. Operator Theory, 38, 243–263 (1997).

    MATH  MathSciNet  Google Scholar 

  9. P. N. Pivovarchik, “Direct and inverse problems for a damped string,” J. Operator Theory, 42, 189–220 (1999).

    MATH  MathSciNet  Google Scholar 

  10. R. Mennicken and V. N. Pivovarchik, “An inverse problem for an inhomogeneous string with an interval of zero density,” Math. Nachr., 259, 1–15 (2003).

    Article  MathSciNet  Google Scholar 

  11. V. N. Pivovarchik, “Direct and inverse three-point Sturm-Liouville problem with parameter-dependent boundary conditions,” Asympt. Anal., 26,No. 3, 4,219–238 (2001).

    MathSciNet  Google Scholar 

  12. M. Möller and V. N. Pivovarchik, “Spectral properties of a fourth-order differential equation,” Z. Anal. Anwendungen., 25,No. 3, 341–366 (2006).

    MATH  Google Scholar 

  13. B. J. Levin, Distribution of Zeros of Entire Functions, American Mathematical Society, Providence, RI (1980).

    Google Scholar 

  14. V. Pivovarchik and H. Woracek, “Shifted Hermite-Biehler functions and their applications,” Integral Equat. Operator Theory, 57, 101–126 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  15. M. G. Krein and H. Langer, “On some mathematical principles in the linear theory of damped oscillations of continua. I, II,” Integral Equat. Operator Theory, 1, 364–399, 539-566 (1978).

    Article  MATH  MathSciNet  Google Scholar 

  16. P. Freitas, M. Grinfeld, and P. Knight, “Stability for finite-dimensional systems with indefinite damping,” Adv. Math. Sci. Appl., 7, 435–460 (1997).

    MathSciNet  Google Scholar 

  17. I. C. Gohberg and M. G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators, American Mathematical Society, Providence, RI (1969).

    MATH  Google Scholar 

  18. I. C. Gohberg, S. Goldberg, and M. A. Kaashoek, Classes of Linear Operators, Vol. 1, Birkhauser, Basel (1990).

    MATH  Google Scholar 

  19. V. N. Pivovarchik, “On positive spectra of one class of polynomial operator pencils,” Integral Equat. Operator Theory, 19, 314–326 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  20. A. S. Markus, Introduction to the Theory of Polynomial Operator Pencils, American Mathematical Society, Providence, RI (1988).

    MATH  Google Scholar 

  21. I. C. Gohberg and E. I. Sigal, “An operator generalization of the logarithmic residue theorem and Rouché theorem,” Mat. Sb., 13,No. 1, 603–625 (1971).

    Article  MATH  Google Scholar 

  22. M. V. Eni, “Stability of the root-number of an analytic operator function and perturbations of its characteristic numbers and eigenvectors,” Sov. Math. Dokl., 8, 542–545 (1967).

    MATH  Google Scholar 

  23. A. G. Kostyuchenko and M. B. Orazov, “Problem of oscillations of an elastic half-cylinder and related self-adjoint quadratic pencils,” Tr. Sem. Petrovskogo, 6, 97–147 (1981).

    MATH  MathSciNet  Google Scholar 

  24. A. I. Markushevich, Theory of Analytic Functions [in Russian], Vol 1, Nauka, Moscow (1968).

    Google Scholar 

  25. V. N. Pivovarchik, “On eigenvalues of a quadratic operator pencil,” Funkts. Anal. Prilozhen., 1989, 25, 80–81.

    MathSciNet  Google Scholar 

  26. A. A. Shkalikov, “Operator pencils arising in elasticity and hydrodynamics: the instability index formula,” Operator Theory: Adv. Appl., 87, 358–385 (1996).

    MathSciNet  Google Scholar 

  27. V. M. Adamyan and V. N. Pivovarchik, “On the spectra of some classes of quadratic operator pencils,” Operator Theory: Adv. Appl., 106, 23–36 (1998).

    MathSciNet  Google Scholar 

  28. V. M. Adamyan, R. Mennicken, and V. N. Pivovarchik, “On the spectral theory of degenerate operator pencils,” Operator Theory: Adv. Appl., 124, 1–19 (2001).

    MathSciNet  Google Scholar 

  29. V. M. Adamyan, H. Langer, and M. Möller, “Compact perturbations of spectra of self-adjoint quadratic operator pencils of definite type,” Integral Equat. Operator Theory, 39, 127–152 (2001).

    Article  Google Scholar 

  30. V. N. Pivovarchik, “On spectra of quadratic operator pencils in the right half-plane,” Mat. Zametki, 45,No. 6, 101–103 (1989).

    MATH  MathSciNet  Google Scholar 

  31. V. N. Pivovarchik, “On the total algebraic multiplicity of the spectrum in the right half-plane for a class of quadratic operator pencils,” St.Peterburg Mat. Zh., 3,No. 2, 447–454 (1992).

    MathSciNet  Google Scholar 

  32. F. Riesz and B. Sz.-Nagy, Lecons d’Analyse Fonctionelle, Acad. Kiadv., Budapest (1952).

    Google Scholar 

  33. C. van der Mee and V. N. Pivovarchik, “Some properties of the eigenvalues of a Schrödinger equation with energy-dependent potential,” Contemp. Math., 307, 305–310 (2002).

    Google Scholar 

  34. A. M. Gomilko and V. N. Pivovarchik, “On bases of eigenfunctions of boundary problem associated with small vibrations of damped nonsmooth inhomogeneous string,” Asympt. Anal., 20,No. 3–4, 301–315 (1999).

    MATH  MathSciNet  Google Scholar 

  35. C. van der Mee and V. N. Pivovarchik, “The Sturm-Liouville inverse spectral problem with boundary conditions depending on the spectral parameter,” Opusc. Math., 25,No. 2, 243–260 (2005).

    MATH  Google Scholar 

  36. S. Hrushev, “The Regge problem for a string, unconditionally convergent eigenfunction expansions, and unconditional bases of exponentials in L 2(-T, T),” J. Operator Theory, 14, 67–85 (1985).

    MathSciNet  Google Scholar 

  37. T. Regge, “Construction of potential from resonances,” Nuovo Cim., 9,No. 3, 491–503 (1958).

    Article  MATH  MathSciNet  Google Scholar 

  38. T. Regge, “Construction of potential from resonances,” Nuovo Cim., 9,No. 5., 671–679 (1958).

    Google Scholar 

  39. A. O. Kravitskii, “On double expansion into series of eigenfunctions of a nonself-adjoint boundary-value problem,” Differents. Uravn., 4,No. 1, 165–177 (1968).

    Google Scholar 

  40. V. M. Kogan, “On double completeness of the set of eigenfunctions and associated functions of the Regge problem,” Funkts. Anal. Prilozhen., 5,No. 3, 70–74 (1971).

    Google Scholar 

  41. A. G. Sergeev, “Asymptotics of Jost functions and eigenvalues of the Regge problem,” Differents. Uravn., 8,No. 5, 925–927 (1972).

    MATH  MathSciNet  Google Scholar 

  42. C. V. van der Mee and V. N. Pivovarchik, “A Sturm-Liouville inverse problem with boundary conditions depending on the spectral parameter,” Funct. Anal. Appl., 36,No. 4, 315–317 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  43. B. Simon, “Resonances in one-dimension and Fredholm determinants,” J. Funct. Anal., 178,No. 2, 396–420 (2000).

    Article  MATH  MathSciNet  Google Scholar 

  44. E. Korotyaev, “Stability for inverse resonance problem,” Res. Notes, 73, 3927–3936 (2004).

    MathSciNet  Google Scholar 

  45. V. N. Pivovarchik and C. van der Mee, “The inverse generalized Regge problem,” Inverse Problems, 17, 1831–1845 (2001).

    Article  MATH  MathSciNet  Google Scholar 

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 59, No. 5, pp. 702–716, May, 2007.

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Pivovarchik, V.N. On spectra of a certain class of quadratic operator pencils with one-dimensional linear part. Ukr Math J 59, 766–781 (2007). https://doi.org/10.1007/s11253-007-0049-5

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  • DOI: https://doi.org/10.1007/s11253-007-0049-5

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