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.


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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