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Basis property of eigenfunctions of nonselfadjoint operator pencils generated by the equation of nonhomogeneous damped string. (English) Zbl 0855.47010

Summary: We consider a class of nonselfadjoint quadratic operator pencils generated by the equation, which governs the vibrations of a string with nonconstant bounded density subject to viscous damping with a nonconstant damping coefficient. These pencils depend on a complex parameter \(h\), which enters the boundary conditions. Depending on the values of \(h\), the eigenvalues of the above pencils may describe the resonances in the scattering of elastic waves on an infinite string or the eigenmodes of a finite string. We obtain the asymptotic representations for these eigenvalues. Assuming that the proper multiplicity of each eigenvalue is equal to one, we prove that the eigenfunctions of these pencils form Riesz bases in the weighted \(L^2\)-space, whose weight function is exactly the density of the string. The general case of multiple eigenvalues will be treated in another paper, based on the results of the present work.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
46B15 Summability and bases; functional analytic aspects of frames in Banach and Hilbert spaces
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[1] A.S. Marcus, Introduction to the Spectral Theory of Polynomial Pencils. Transl. of Math. Monographs, Vol. 71, AMS, Providence, RI, (1988).
[2] I.C. Gohberg, M.G. Krein, Introduction to the Theory of Linear Nonselfadjoint Operators. Transl. of Math. Monographs, Vol. 18, AMS, Providence, RI, (1969). · Zbl 0181.13503
[3] Marianna A. Shubov, Asymptotics of Resonances and Eigenvalues for Nonhomogeneous Damped String. Asympt. Anal., 12, (1996), p. 1-48. · Zbl 0864.35108
[4] Marianna A. Shubov, Asymptotics of Resonances and Geometry of Resonance States in the Problem of Scattering of Acoustical Waves by the Spherically Symmetric Inhomogeneity of Density. Dif. Int. Eq. 8 (5), 1995, p. 1073-1115. · Zbl 0827.34075
[5] M.A. Pekker (Marianna A. Shubov), The Nonphysical Sheet for the String Equation. J. Soviet Math., 10, (1978).
[6] M.A. Pekker (Marianna A. Shubov), Resonances in the Scattering of Acoustical Waves by a Spherical Inhomogeneity of the Density. Amer. Math. Soc. Transl., 2, Vol. 115, (1980). · Zbl 0463.35065
[7] S.V. Hruscev, N.K. Nikol?kii, B.S. Pavlov, Unconditional Bases of Exponentials and of Reproducing Kernals. Complex Analysis and Spectral Theory, Lect. Notes Math., Vol. 864, Springer-Verlag, (1981) p. 215-335.
[8] A.M. Sedletskii, On Convergence of Nonharmonic Fourier Series in Systems of Exponentials, Cosines and Sines. Soviet MATH. Dokl, vol. 38, NO. 1, (1989) p. 179-183. · Zbl 0695.42004
[9] S.A. Avdonin and I. Joo, Riesz Bases of Exponentials and Sine-type Functions. Acta Math. Hungarica, 51, No. 1/2, (1988) p. 3-14. · Zbl 0645.42027
[10] R. Young, Introduction to Nonharmonic Fourier Series, Acad. Press, (1980). · Zbl 0493.42001
[11] R. Newton, Scattering Theory of Waves and Particles. 2nd Ed., Springer-Verlag, (1980). · Zbl 0425.62077
[12] B. Ya. Levin, Distribution of Zeroes of Entire Functions. Transl. Math. Monogr., Vol. 5, AMS, Providence, RI, (1964). · Zbl 0152.06703
[13] S. Cox, E. Zuazua, The Rate at Which Energy Decays in a Damped String. Comm. in PDE, Vol. 19, No. 1/2, (1994) p. 213-243. · Zbl 0818.35072
[14] N.K. Bari, A Treatise on a Trigonometric Series. Vols. 1, 2, Macmillian, New York, (1964). · Zbl 0154.06103
[15] V.D. Golovin, Biorthonormal Expansions inL 2 in Linear Combinations of Exponentials. Notes of Kharkov Univ., Math., 30, No. 4, (1964) p. 18-29.
[16] A.S.B. Holland, Introduction to the Theory of Entire Functions. Acad. Press, New York, (1973). · Zbl 0278.30001
[17] B.S. Pavlov, On the completeness of the Set of Resonance States for a System of Differential Equations. Sov. Math. Dokl., 12, (1971). · Zbl 0232.47016
[18] M.A. Shubova, The Serial Structure of Resonances of the Three-Dimensional Schrödinger Operator. Proceed. of the Steklov Institute of Math., 159, (1984) p. 197-217. · Zbl 0576.35028
[19] M.V. Keldysh, On Completeness of the Eigenfunctions of Certain Classes of Nonselfadjoint Linear Operators. Russian Math. Surveys, 27, (1971). · Zbl 0225.47008
[20] Dzh. Alakhverdiev, On Completeness of the System of Eigenelements and Associated Elements for Nonselfadjoint Operators. Soviet Math. Dokl., 6, (1965) p. 102-105 and p. 171-175.
[21] T. Ya. Azizov, On Completeness and the Basis Property for the Eingenvectors and Associate Vectors ofJ-self-adjoint Operators of Class ?(H). Soviet Math. Dokl., 22, (1980).
[22] M.G. Dzhavadov, On Completeness of a Certain Part of the Eigenfunctions of a Nonselfadjoint Differential Operator. Soviet Math. Dokl., 6, (1965). · Zbl 0141.27902
[23] M.G. Dzhavadov, Onm-fold Completeness of half of the Eigenfunctions of Certain Classes of Nonselfadjoint Linear Operators. Russian Math. Serveys, 27, (1971).
[24] E.P. Bogomolova, A.S. Pechentsov, Basis Property of the System of Eigenfunctions of a Boundary Value Problem with a Multiple Root of the Characteristic Polinomial. Vestnik Moscow Univ., Math., Vol. 44, No. 4, (1989) p. 17-22. · Zbl 0687.34021
[25] Marianna A. Shubov, Certain Class of Unconditional Bases in Hilbert Space and its Applications to Functional Model and Scattering Theory. Int. Eq. Oper. Theory, Vol. 13, (1990) p. 750-770. · Zbl 0722.46006
[26] I.C. Gohberg, A.S. Marcus, Some Relations Between Eigenvalues and Matrix Elements of Linear Operators. AMS Transl. (2), 52, (1966) p. 201-216.
[27] R. Boas, Entire Functions. Academic Press, Inc., New York, (1954). · Zbl 0058.30201
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